Download The History of Tuning

In tuning, it is most easy to hear UNISON, so far as I know everyone
who can hear can do this; next easiest is "OCTAVE", most people have
no problem telling when two pitch are exactly in "octave"; then there
is the "perfect" fourth and fifth, a student can be trained to hear
them and tell when they have tuned exactly to a fourth or fifth. To me this
is the starting point for the tuning of a musical instrument. A Maker of
musical instruments has to know more about Scales than a Player.
Remember: People learned to talk and sing first (poetry and melody), and then
came musical instruments, rhythm, dancing and harmony; finally there was a
large enough body of work for thinkers/philosophers to try to find rules.
In the order of things, it looked like the Chinese Civilization came first. They had
music, music theory, mathematics, medicine, war and politics. China’s people
knew that Japan existed, there was some trade and China sent monks to Japan
to expand their religion. China also traded with the Mediterranean Cultures, but
they were just too far away to have much influence.
Egypt and Sumerian were also a early civilizations. Egypt was established on a
narrow river valley and kept the same name and culture, there was some
language drift. Sumeria was also in a river valley and got its name from a
conquering people that came from the East or North-East. The area that was
Sumeria changed a lot as successive waves of conquering people arrived. Its
name changed from Sumeria, Chadeea, Babylonia, Mesopotamia, Persia, Iran
and Iraq. Then there were the Greeks and the Romans civilizations cam later
than the Egyptian and Sumerian Civilizations.
How about mathematics; there is a fair description of the Egyptian, Greek and
Roman math and math symbols? There should be one for the Sumerian math as
they seemed to have been more advanced and the Egyptians, Greeks and
Romans who borrowed from them. The mathematics and style of writing has an
influence on music philosophy. Although I have not found an authorian source
yet, I suspect that Sumerian and Greek math had a lot in similar to Roman math
and numbers, that is: if an inferior number is placed in front (reading left to right)
the inferior number is to be subtracted from the superior number. Both Greek
and Romans used letters from their alphabet to signify numbers; separate
symbols were a later invention. The Sumerians and the Egyptians had a written
language, the Greeks were just at the verge of creating a written language,
animals smaller than the horse had been domesticated, and the horse had not
yet been domesticated at the time of the first Hyssop invasion of Egypt.
Now to switch to Greek musical Philosophy ... none of the Greek Musicians could
read music or anything, only some of the rich Greek Philosophers had the skill of
reading and writing. The Greek used letters of their alphabet to indicate a
musical pitch. They had a separate letter for every single pitch of every scale
and every mode. There were not enough letters in the Greek alphabet for all
these symbols, so they used letters backwards and up-side-down and even
“broken” letters to get enough symbols.
Figure 2
Staff of Seiklos
Figure 3
Rubbing of the Staff of Seiklos
Example of Greek Music Score
Refer to Figure 3, Greek music was mostly religious poetry that was sung and a
string instrument was used to back the voice. The above example is a religious
poem (in Classical Greek) the smaller symbols above the text poem are the
notes that are intended to be sounded when the poem is sung.
Pythagoras (600BC) was the first, in the West, to write about forming scales for
stringed instruments (well we had to wait for them to invent writing before they
could write anything down). He was part scientist (he conducted test to verify his
theories) and part mystic. The straight story about Pythagoras is difficult to
obtain as all his writings were destroyed and all existing information is secondhand. There is more music history about Pythagoras in the Math and Physic
department than in the Music Department.
Euclid’s comments are from the Math Department: Pythagoras studied in
Babylonia when it was part of Chaldean Empire and returned home to Samos, a
Greek Island and started a school. This first school was not successful; it
attracted only one student, which Pythagoras had to pay to attend. Next,
Pythagoras studied more in Egypt and returned to Croton, a Greek colony in
southern Italy. His second school was very successful. He caused a scandal as
he allowed women to attend his school and he married his best female pupil late
in life. Pythagoras was both a scientist and a mystic, he performed actual
experiments especially in his study of sound, but he had large parts of his school
locked in secrecy. Philolaus was his best pupil. He is credited with the theorem
that the square of the sides of the two shorter sides of a right triangle equal the
square of the longer. Also that the relationship of pitch of a string could be
correlated with length. He was capable of addition, subtraction, multiplication,
division and the calculating of square roots and cube roots of numbers using
Greek notation. But Pythagoras had the religious belief that the whole universe
rested on numbers and their relationship. It was a major blow to his faith and
creditability when it was discovered in his school that the sides of the simplest
right triangle where the smaller sides are equal does not allow the value of the
longer side to be calculated exactly. In simple terms, the square root of two can
not be found as equal to the ratio of any two real numbers. Of course Euclid
also had his problems as well. He favored geometry, but could not trisect an
angle nor double a cube using geometry and there was one of his theorems that
would later be proven to be faulty.
From the Physic Department: Pythagoras used a Monochord for his sound
experiments. He was only interested in string instruments. He (as well as Plato
and Aristotle) spoke out against flute (and whistles and pipes) music as being
bad for the state. All the aristocratic Greek philosophers postulated that all
music should be controlled by the state in the interest of the community;
they further postulated that disorderly music depraves the conduct of man, but
orderly music improves it. Pythagoras was a sage and philosopher that was
trying to find rules for why certain groups of music sounded pleasing.
Pythagoras already knew that sound was a vibration. He investigated the triads
of 6:5:4 string lengths as sounding pleasant. He developed a diatonic music
scale based on this relation.
Example: pick any pitch to start from. (Note that Pythagoras could not know
about the frequency of notes as he had no way to measure time to any
accuracy.) Pythagoras had to work with string length and his monochord. If the
longest string is selected to be 100 units long and the triad is defined to be the
pitch of every other string, the third string is (100/6)X5=83-1/3, the fifth string is
(100/6)X4=66-2/3. Now take the fifth string and make it the first of a new triad, so
the seventh string is (66.6666666/6)X5=55.55555, and the ninth is
(66.666666/6)X4=44.44444. The eight string is the octave of the first string so its
value can be set at one half the length of the first string.
So for the first pass through:
1
100
2
?
3
4
83-1/3 ?
5
6
66-2/3 ?
7
55.5555
8
50.00
9
(string number)
44.444 (relative string length)
The value of the second string is twice the length of the ninth string so
2X44.44444=88.88888.
The value of the second string can be set to the 6 of a new triad. So the forth
string has to equal (88.88888/6)X5=74.074074 and the sixth string equals
(88.88888/6)X4=59.259259.
So finally:
1
2
3
4
5
6
7
8
100.000 88.888 83.333 74.074 66.666 59.259 55.555 50.000
0.111 0.0625 0.111 0.099 0.111 0.06250 0.099
0
203.9 315.64 519.55 701.95 905.87 1017.60 1200
203.9 111.74 203.9 182.4 203.9 111.73 182.4
string number.
relative length of equal strings.
relative step size.
cents from tonic.
step size in cent.
In this scale the Major and Minor Triads are pure, the fifth is pure. The interval to
the forth is slightly narrow (1 cent).
Also Pythagoras is credited with naming the Modes of the Greek Tetrachord after
various Greek Tribes, when asked why, he is credited with stating: “that the way
they sound to me”.
All the above about Pythagoras may not be true.
The Tetrachord (Ancient). is one of the earliest known scales; it is ancient. It
was known in Babylon and Sumeria. This is a scale for the Kithara (Lyre), rather
than a Harp. The Harp was only a minor instrument in Sumeria and shunned by
the mainland Greeks. Tetrachord means four notes in Greek (cord can mean
either a string or note in Greek, in this case it is note because the various
tetrachord scales can be played quite well on a three string Greek Lyre). The
first note of the tetrachord is called the keynote or tonic, and the fourth string is
tuned to a perfect forth above the tonic. Any two equal strings whose length are
in the ratio of 4:3 will produce a perfect fourth (also 498.05 cents). The intervals
of the two notes between the first note and the fourth note are movable and are
moved to allow the stringed instrument to play the various Greek Modes. There
are three tetrachord modes: 1) the Enharmonic, 2) the Chromatic and 3) the
Diatonic (ancient Greek definitions of these words are different from modern
usage). The Enharmonic Mode is formed by two tones and two diesis. The
Chromatic Mode consists of two semitones and a tone and a half. The Diatonic
Mode is formed by two consecutive tones and a semitone. The movable notes
will have to fit in this interval of 498 cents by the following rules:
Quoting Sachs The Rise of Music in the Ancient World, East and West some Greek Modes and Shades:
“according to Aristonuxes:
interval from tonic
from second note to third third to fourth(fixed)
Enharmonic
398
49
49
Chroma malakon
Chroma hemiolion
Chroma toninion
364
348
298
66
74
99
66
74
99
Diatonon malakon
Diatonon syntonon
249
199
149
199
99
99
however Ptolemy disagreed, he chose:
Enharmonic
386
74
38
Chroma malakon
Chroma syntonon
316
267
120
151
62
80
Diatonon malikon
Diatonon toninon
Diatonon syntonon
Diatonon ditoniaion
Diatonon homalon
233
204
182
204
182
182
233
204
204
165
83
62
198
90
151”
Tetrachords can be extended to make larger scales with more range.
Consecutive tetrachords can be arranged two ways 1) the conjunct is where two
Teteachords are connected and share one string (for a total of seven strings) or
2) the disjunct where the two Tetrachords are separate and there is a tone ( 9:8
string length ratio) separation between the two Tetrachords (for a total of eight
strings). Again remember that the Tonic and the Forth are the fixed strings or
notes. The two strings between the tonic and forth are movable pitches;
however, the sum of their step size stays equal to a tone (9:8 step size) plus a
semi-tone (16:15 step size).
The scale The following on the scale is from the Ten Books of
Architecture by Vitruvius. The version available to us today is a summery of a
summary of a translation of a translation of a copy of a copy (after all the average
life of a copy is about a hundred years, so there had to be many copies and
some copyist had to make “improvements”). Vitruvius translated Aristonuxes
Greek into Latin and summarized. Vitruvius was copied and copied, with each
copier changing it a little. Then Dr. Morris Morgan translated the copy from
Valentine Rose’s Latin into English and cut the section on Aristonuxes short
saying what Virtruvis Summery just contained an obvious interpolation.
Valentine Rose’s copy added a grand staff to the section on Aristonuxes to help
explain the  Refer to Figure 4, the scale of Aristonuxe as quoted
from Vitruvius, Book V, Chapter IV, page XX. These scales are also called the
hexatonic and the octatonic scales.
The  of Aristonuxes of Tarentum (330 BC) is an 18-note scale that
spans two octaves. This two-octave span is supposed to be the range of male
voice. Rose, in his version of Vitruvius, stated that the lowest note is the A an
octave plus two notes below middle C and the highest note was the A above
middle C, but he gave no explanation of why he choose it to be this way. The
 is comprised of five tetrachords and a few additional notes. Now refer
to Figure 4, and I will try to walk through how to form this scale (This scale almost
appears to be two scales push together so that the beginning and end are on the
same scale note, A. This results in a slight jumble of notes about in the middle of
the scale). The  scale is comprised of 8 fixed notes and 10 movable
notes. A technique that would be consistence with the technology at this time
would be to use a Monochord. I have built one for this purpose.
Now we will start with the fixed pitch strings and build the Harp scale. Take the
pitch of the tonic from some authoritarian source. Tune the pitch of the
Monochord to unison with this source. Now tune the pitch of the
Proslambanomenus string to unison with the Monochord. Note the Monochord
string’s vibrating length. Reduce this vibrating length by 8/9 and use this new
pitch to tune the Hypaton Hypate Harp String to unison. Note this vibrating
length on the Monochord and reduce it by 3:4 to get the pitch for the Meson
Hypate. Now tune the Meson Hypate Harp String to unison with the Monochord.
If at any time the Monochord runs out of range for the string it is currently using,
a string change will be required. With a string change, when one is going up the
scale, just remove the current string and insert the next lighter string, then go
back to one of the previously tuned strings on the Harp and pick up the correct
pitch and continue. The Synhemmenon Mese string will be 3:4 the vibrating
length of the string tuned to the Meson Hypate, so reduce the vibrating length of
the string in the Monochord and then tune the Synhemmenon Mese string to
unison with the Monochord. Again note the vibrating length on the Monochord
and reduce it by 3:4 to get the pitch step size to get to the Synhemmenon Nete.
Now tune the Synhemmenon Nete Harp string to unison with the Monochord.
Now it appears that a change in technique is required. Retune the Monochord
back to unison with the Synhemmenon Mese. Note the vibrating length on the
Monochord for the Synhemmenon Mese and then reduce this vibrating length to
1:2 to get the octave pitch. Now tune the Hyperbolaeon Nete Harp (highest)
String to unison with the Monochord. Now note the current Monochord vibrating
string length and increase the length by 4:3 to go down a perfect forth to the
Diezbugmenon Nete. Tune the Diezbugmenon Nete Harp String to unison with
the Monochord. Again, note the vibrating length of the Monochord string length
and increase its length further by 4:3 to go down to the Diezbugmenon
Paramese. Tune the Diezbugmenon Paramese Harp string to unison with the
Monochord. This concludes tuning the fix pitch strings, but as noted before this
gives a break in the middle of the scale between the Synmemmenon Tetrachord
and the Diezbugmenon Hexichord.
Figure 4 The  of Aristonuxes
The Synhemmenon Nete (highest note of the lower Tetrachord) is a pitch step
and a half above the Diezbugmenon Paramese (the lowest note of the higher
Tetrachord).
Confusion. To Quote Sachs, from The Rise of Music in the Ancient World, East
and West, pg222: “With all the Greek modes: Aeolian, Boeotian, Dorian, Iastian,
Ionian, Locrain, Lydian, Phryaian, with or without epithets; aneimene, chalara,
hyper, hypo, mixo, syntono, in an ever changing and often contradictory
terminology, impelled the methodical Greek philosophers to organize this chaotic
multiplicity of modes into one consistent system and eliminate those modes that
were not acceptable. As early as 400 BC the kithara had progressed to eleven
strings. Which in some pentatonic sequence covered exactly two octaves and
within had the possibility of representing the new system in its totality.”
The Greater Perfect Scale. The first section I extracted from Peter Fraser’s web
site and added the step sizes. I then follow this with a quotation from Sachs,
who had a better copy of Euclid to work from. Both are interesting.
Figure 5. The Greater Perfect Scale.
The Greater Perfect Scale (300 BC). Quoting Sachs, from The Rise of Music in
the Ancient World, East and West, pg222-223: “The Greek Philosopher and
Mathematician, Euclid first described the Perfect System or Systema teleion.
The Perfect Scale is a 15 note scale that covers just two octaves. However the
perfect system was more than just a double octave; it was perfect as a unique
attempt to organize the musical space from one center, (the note) a. The center
stands in its original octave of Dorian structure, e’-e, which, by adding half an
octave above and half an octave below, is extended to two octaves a’-A. This
new unit could shift both up and down by half an octave either way and thus
cover three octaves.
The notes added above and below the inner octave were named after the three
extreme notes at either end of it: nete, paranete, trite, above; lichanos,
parhypate, hypate below. Only the lowest note was given a name of its own;
the added note, proslambanomenos. The organization of the two octaves was
rather strange. Subdivision were neither octaves nor pentachords, but
tetrachords throughout. This implied two conjuncts at either end of the inner
octave and disjunction or diazeuxis in the middle. Reading downward, the
arrangement results in the tetrachord hyperbolaion, ‘of the exceeding notes’,
conjunct with the tetrachord diezeugmenon, which as the name said, was
disjunct from the tetrachord meson, ‘of the middle’ notes; this in was conjunct
with the tetrachord hypaton, ‘of the low’ (literally ‘high’) notes. The
proslambanomenos, ‘the left over’ note.
Naming of the notes of the Greater Perfect Scale
The somewhat cryptic remark “literally high” refers to a strange inversion in
Greek terminology: nete, the highest note (in our current terminology) means low
in Greek (or neat, short or small). Likewise, hypate, the lowest note means high
(or long or tall) in Greek.
Similarly the Greeks constructed a lesser perfect scale or systema teleion elatton
on the basis of the old heptad of two conjunct tetrachord. It comprised only an
eleventh, from d’ to A. The top tetrachord did not exist, and the disjunct
tetrachord diezugmenon was replaced by a conjunct tetrachord synemmenon:
Naming of notes of the Lesser Perfect Scale
I am not satisfied with Sachs use of letter to indicate the notes in his
representation of the Greater and Lesser Perfect Scale. Also these letters give
the wrong impression as to the intervals between the notes. The Perfect Scale is
a diatonic scale, the intervals between notes are: (Diatonic=unequal step sizes)
Tone Tone Tone Semitone Tone Tone Semitone, then next octave or Tone Tone
Semitone Tone Tone Tone Semitone. However, the Greeks did not use
instrumental music and only need an octave and a few notes to achieve all the
range that they wanted. The purpose of Greek music was to back a male singer
(they did not use female singers).
Quoting Sachs The Rise of Music in the Ancient World, East and West, pg225:
“Ptolemy, who lived in the second century AD, admitted seven keys, the centers
of which ascended diatonically from e to d’. With an attempt to keep “a” as the
common center.
There is a higher group of hyper scales, a lower group of hypo scales, and a
middle group without epithets. At first sight, all of them are similar Dorian keys;
but the modal structures are fundamentally different in the three groups:
The middle scales, based on disjunct tetrachords, have the fifth on the
top and are plagal.
2) The hyper scales based on conjunct tetrachords, with an additional
note above are likewise plagal.
3) The hypo scales based on conjunct tetrachords, with an additional note
below, have the fourth on top, or rather, should have the fourth on top
and be authentic. “
1)
Quote from Sachs The Rise of Music in the Ancient World, East and West,
pg234: “Aristides Quintilianus expressly states that voices did not span more
than two octaves. For this reason he adds that Dorian was the only key sung in
its total range. … Lyre players had similar limitations. … The consequences are
that music space, vague and shapeless in our music, became a palpable reality
in Greece. Each key had its own center, to be sure; but also music space as a
whole had its immovable center which, being the pitch tone was never neglected.
As a result, every melody had two foci; every note or group of notes gravitated
toward two different centers at once, toward the center of the individual key and
toward the center of the immovable perfect system. The first bearing was called
dynamis or mobile force, and the second, thesis or stationary force. A note
changed its dynamis according to its key; its thesis was immovable. The note e’
for example, was in all melodies nete kata thesin or highest stationary note,
whatever the key; but in Mixolydian, it was also mese katra dynamin or mobile
center, and in Lydian trite kata dynamin or mobile third from above. The mobile
and stationary functions coincided only in Dorian.
Look at a Greek melody from the dynamic center - b in Phrygian or d’ in
Mixolydian - finds the modal structure that we call Dorian; descending, the
sequence is to step through two whole tones and a concluding semitone to the
(dynamic) hypate. Things look different from the stationary center. The player,
adjusting his (Lyre to an) a and tuning the outer strings to e’ and e (the usual
range of melodies), realized that each transposition of the (Dorian) scale altered
the structure of his octave, since it shifted the semitone to places where
previously whole tones had been. The central rectangle in the above diagram
encases the resulting structure of all eight keys.
The Phrygian key, by one tone higher than the Dorian, sharped by two notes of
the central octave, c’ and f; thus the two central tetrachords become e’ d’ c#’ and
a g f# e: the semitone moved to the middle of the tetrachord and the original
Dorian octave changed to the Phrygian species. In the same way, the Lydian
key sharped d’ c’ g f and shifted the semitones to the upper ends of the
tetrachord that formed the central octave. Dorian as a key (Dorios tonos) created
a Dorian mode (Doristi harmonia) in the perfect system; a Dorian mode in the
perfect system could only originate if the melody followed the Dorian key. And
the same is true for Phrygian, Lydian, and the rest. Key and mode conditioned
each other and rightly were given the same tribal names.”
Now quoting from Jargenson, page 24: “Tuning to Fifths and Fourths are the
easiest intervals outside of the octave to tune because they exist next in order
after the octave in the Harmonic series. The ease of tuning Fifths and Fourths
along with their greater utility determined that Fifths and Fourths were to be used
for creating most of the practical scales from the beginning of history to the
present.
When the practice of tuning the pentatonic scale by means of a circle of fourths
and fifths became established, the following changes took place:
1) The fourths and fifths were made in tune enough to be used in vertical
sonorities.
2) Two of the wholetones (intervals between) became enlarged to
become a new interval named a minor third.
3) One of the fourths (intervals) became narrow enough to become a new
interval named a major third.
4) The new intervals listed above and their inversions resulted in minor
thirds, major thirds, minor sixths, and major sixths all being added to
the basic harmonic vocabulary of fourths and fifths; however, there
were still no semitones, tritones or major sevenths to make singing
difficult.
5) Strong tonality was developed in melodic compositions because the
scales were made permanently uneven.
6) One major triad along with its lower relative minor triad was
discovered.
Evidentially the phenomenon of Just Intervals was discovered and used for
tuning the fourths and fifths for two reasons: first, it is much easier to tune an
interval without beats than it is to temper an interval and guess how many beats
it should contain; second, the fifths and thirds can not be completely Just at the
same time, but fourths and fifths were chosen to be pure because the vertical
sonorities of the thirds and sixths were not yet important in harmony. The Just
fifth tuning practice must have been centuries old when Pythagoras began his
investigation into music. All the major and minor sixths and also the major and
minor thirds are each altered by one Syntonic comma from Just because all the
fifths are Just. The latter sixths and thirds are known as Pythagorean sixths and
thirds. Since these intervals are altered by the extreme of one Syntonic comma
each, their beat speeds are too fast to be objectionable because they seem to
disappear into a single tone. Pythagorean sixths and thirds have a tone color or
sound uniquely their own. “
Basic Pentatonic Scale (Ancient) (quoting from Jorgenson, now page 25)”In
this tuning there are two triads: namely the F Major and D Minor. There are two
major sixths and one minor third which are wide by one Syntonic comma. There
are two minor thirds which are narrow by a Syntonic comma. These intervals
which are altered by one Syntonic comma are known as Pythagorean intervals.
… This gives Pythagorean triads a certain rhythmic harmoniousness. …
This descending scale has intervals between notes in the ratio: F 27:32 D 8:9
C 27:32 A 8 :9 G 8:9 F. The cent values in round numbers are 294-204-294204-204. The Basic Pentatonic scale contains pure major tones of the 8:9 ratio,
pure fourths of the 3:4 ratio, pure fifths of the 2:3 ratio, minor sevenths of the 9:16
ratio and pure octave of the 1:2 ratio. There are no semitones, tritones or major
sevenths. All the triads and sixths are altered by the ratio 80:81 Syntonic
comma. The minor third ratio is 27:32; the major third ratio is 64:81; the minor
sixth ratio is 81:128 and the major sixth ratio is 16:27.
The line of fifths in the Pentatonic Scale is F C G D A. The bearing octave is F
below middle C to F above middle C. All fifths and fourths are pure. Major sixths
and minor triads are wide, minor thirds are narrow. Tuning steps:
1.
2.
3.
4.
5.
6.
7.
Tune middle C to tuning folk.
Tune F below middle C and F above middle C both pure to middle C.
Tune G below middle C pure to middle C.
Tune D above middle C to G below middle C pure.
Test the intervals for the purpose of finding possible mistake which should be corrected. In the
figure the numbers above the intervals indicate the beat speeds in beats per second in round
numbers.
Tune A below middle C pure to D above middle C.
Test all intervals.
”
Basic Pentatonic
(descending)
(step size)
Celtic Pentatonic
F
G
A
C
D
F
1/1 27/32
4/3 101/64 16/9
2/1
0
294
498 792
996
1200 cent value from tonic.
27/32 8/9
27/32
8/9
8/9
1/1
0
9/8
4/3
3/2
203.9 498.0 701.9
16/9
996.0
2/1
1200 cent value of tones from tonic.
It has been stated that the Celts organized their music in a pentatonic scale.
Since the Celts preferred to develop their memory and refrained from writing
anything down, one has to rely on their enemies for any historical record of their
customs.
It is highly likely that the gap scale Irish Harp music examples that are reported
by Bunting and other collectors are prehistoric Pentatonic Scales transposed to
historic Diatonic Scale or else works where the composer tried to make his music
sound this way.
Enharmonic Scale of Archytas (380 BC) Still quoting Jorgenson, page 31:
“This is a descending scale with the interval ratio of E 4:5 D 35:36 C 27:28 B
8:9 A 4:5 G 35:36 F 27:28 E. The cent value in round numbers for the
interval between pitches is 386 - 49 - 63 - 204 - 386 - 49 - 63.”
Eratosthenes Diatonic (230 BC) is also the Diatonic Ditoniaion published by
Claudius Ptolemy in 140 AD. Quoting Jorgenson, page 31:
“This scale is the basis of Western Music. In this tuning, the three basic primary
triads are developed. These are the G Major, C Major and F Major triads. Also,
the three lower relative minor triads E Minor, A Minor and D Minor are developed.
The formation of these triads creates the basic C Major tonality.
This descending scale has intervals between notes as: E 8:9, D 8:9, C 243:256,
B 8:9, A 8:9, G 8:9, F 243:256, E. The cent value in round numbers for the
step size between pitches are 204-204-90-204-204-204-90. The above scale
contains 243:256 ratio semitones, 512:729 ratio tritones and 128:243 major
sevenths in addition to all the intervals contained in the Basic Pentatonic Scale.
NOTE: in Jorgenson’s method, pure means that an upper partial of one
pitch is in unison with an upper partial of the other note. Even though
Jorgenson did not like electronic tuners, the electronic tuner method of
tuning is faster than and as far as I can see better. In the electronic tuner
method, just tune to the closest 12 Tone Equal Temperament note, then
tune to the correct offset in cent. In the music below, the number above a
pair of notes is the beat frequency of a pair of their upper partials.
Tuning the Eratosthenes Diatonic Scale: In the Eratosthenes diatonic scale there
are six major and minor triads to the octave and all of them contain proportional
beat speeds. All the triads are Pythagorean triads. The line of fifths are F C G D
A E B. The bearing octave is E below middle C. All fifths and fourths are pure.
Major sixths and minor thirds are narrow.
Tuning Steps:
1.
2.
3.
4.
5.
Tune middle C to the Tuning Fork (unison zero beat).
Tune F below middle C pure to middle C (a fifth, a pair of upper partials have a zero beat).
Tune G below middle C pure to middle C (a fourth, a pair of upper partials have a zero beat).
Tune D above middle C pure to G below middle C.
Test
6.
7.
Tune A below middle C pure to D above middle C.
Test
8.
Tune E above middle C and E below middle C both pure to A below middle C. This creates the
interval EF which is the first semitone. The inversion FE is the first major seventh.
9.
Test
1.
Tune B below middle C pure to E above middle C and also check to see that it is pure to the E below
middle C. This creates the interval FB which is the first tritone.
11. Test
The Eratosthenes Diatonic Scale contains the following “Pythagorean” Triads:
The Pythagorean Triads are proportional as follows:
minor root = 0:1:1
minor first inversion = 1:1:0
minor second inversion = 1:0:1
major root position = 0:3:2
major first inversion = 3:4:0
major second inversion = 4:0:3”
Chromatic Scale of Didymium (30 BC) Still quoting Jorgensen, page 31: “This
is a descending scale with the interval ratio E 5:6 D 24:25 C 15:16 B 8:9 A
5:6 G 24:25 F 15:16 E. The cent value in round numbers for the interval
between pitches is 316 - 70 - 112 - 204 - 316 - 70 - 112. Pure minor tones, minor
thirds, major thirds, minor sixths and major sixths. There are pure ratio 24:25
semitones so small that they could be classified as large third tones.
Diatonic Syntonon of Ptolemy (140 AD) Also known as the Diatonic Scale of
Didymium (30BC). Still quoting Jorgenson, page 31: “This is a descending
scale with the interval ratio of E 9:10 D 8:9 C 15:16 B 8:9 A 9:10 G 8:9 F
15:16 E. The cent value in round numbers for the interval between pitches is
182 - 204 - 112 - 204 - 182 - 204 - 112.
In this scale all the ratio 64:81 Pythagorean major thirds and Pythagorean minor
sixths were changed into pure 4:5 major thirds and pure 5:8 minor sixths. Three
of the minor thirds and three of the major sixths were also made pure. The
interval of a fourth between A D was altered by one Syntonic comma to have a
new ratio of 20:27 which is an out of tune wolf interval. The fourth A D was
ruined out of necessary in order to gain all pure thirds and sixths. There are both
major and minor whole tones in this scale. The three primary triads: G, C, and
F are in Just intonation. Ptolemy’s Diatonic Syntonon is the ancestor of the
Common Model Just Tuning as outlines in Marpug’s Monochord No.1. The
Ptolemy diatonic scale is more harmonious than the Eratosthenes diatonic scale,
but lacks the utility of the Eratosthenes. This conflict between harmoniousness
and utility lead to the difference between the Meantone temperament and the
Well temperament.”
Diatonic Hemiolon of Claudius Ptolemy (140 AD) Still quoting Jorgenson,
page 31: “This is a descending scale with the interval ratios of E 9:10 D 10:11
C 11:12 B 8:9 A 9:10 G 10:11 F 11:12 E. The cent value in round numbers
for the interval between pitches is 182 - 165 - 151 - 204 - 182 - 165 - 151.”
Music Theory of Anicius Manlius Boethius (520 AD) Boethius read a
reference that stated that Pythagoras had perfected a diatonic scale based on
fifths (not possibly true) and Boethius tried to find this scale. Boethius found
Eratosthenes Diatonic scale instead and incorrectly named it the Pythagorean
Diatonic; so from this time on, to avoid additional confusion Eratosthenes scale is
now called the Medieval Pythagorean Diatonic scale. Boethius also got caught in
the confusion of whether Greek scales ascend or descend and he got it wrong.
Next he got the names of the Greek Modes incorrect. But Borthius was not the
only Medieval Music Theorist that got the Greek Modes names incorrect. The
modes known as the Church Modes are the results of this mistake.
Medieval Pythagorean Scale (of Boethius) This scale is called in error the
Medieval Pythagorean Diatonic Scale. This is wrong for two reasons: one,
Boethius misnamed the Diatonic Scale of Eratosthenes and two this scale is
more than a diatonic scale it has four chromatic notes added: Bb, Eb, F# and
C#. But it is a very useful scale and is a good scale in which to compose.
Quoting Jorgenson, page 45: “This is the Eratosthenes diatonic scale with four
chromatic tones added. There are still no wolf interval. In this tuning there are
eighteen major and minor triads to the octave, and all of them contain
proportional beat speeds. The C# minor, C# major, E major, F minor G# minor
and Ab major triads still do not exist in this scale. The line of Just fifths is: Eb,
Bb, F, C, G, D, A, E, B, F#, C#.
In this scale, there is a new kind of interval created which is known as a schisma
interval. These intervals are each altered from Just intonation by the extremely
small amount of one schisma, which in round numbers is only 2 cent in size.
With Schisma intervals it is almost impossible to tell that they are different from
Just interval. They are altered intervals, but they are never directly tempered by
ear. Schisma intervals exist because they are the end result of lines or groups of
other pure intervals. The diminished fourths: C# to F, F# to Bb and B to Eb all
sound quasi pure because they are actually narrow major thirds. The augmented
seconds: Bb to C#, and Eb to F# both sound quasi pure because they are
actually wide schisma minor thirds.”
Tuning Steps for Medieval Chromatic Pythagorean Scale: (quote from
Jorgenson, page 48)
Use the eleven steps for tuning the Eratosthenes scale (from this paper, page
25) then
12. Tune Bb below middle C pure to F below middle C.
13. Test
14. Tune Eb above middle C pure to Bb below middle C.
15. Test
16. Tune F# below middle C pure to B below middle C.
17. Test
18. Tune C# above middle C pure to F# below middle C.
19. Test
Medieval Modes and Music Theory of Guido of Arezzo (1050 AD)
Developed four line staff and the Ut Re Mi Scale. Established the Church Modes
as Dorian, Phrygian, Lydian, Mixolydian and maybe the Locrian (some authorities
deny that the Locrian is a mode). No matter, the scales of Guido were different
from the correct Greek modes. The correct knowledge was still readily available
in Constantine (Eastern Roman Empire, official language was Greek), so it is a
wonder that no one bothered to go obtain it? Christian Monks and people in
trade were making the trip.
Medieval Twelve Tone Chromatic “Pythagorean” (changes up to 1482)
Quoting Jorgenson, page 53: “The medieval twelve tone chromatic Pythagorean
tuning has been the most universally used tuning method in modern music
history (before the adaptation of the 12 Equal) because of its basic, simple
means of tuning and its great utility musically. If the last note added (G#) had
not happened to create the wolf problem, then the history of tuning might have
ceased at this point. The medieval chromatic Pythagorean tuning contains two
great virtues; first, there are twenty-two useable major and minor triads to the
octave; secondly there are three major triads and three minor triads (schisma
triads) that are very close to Just intonation, the ultimate in harmoniousness.
The later two virtues are the cause of the two separate branches of tuning
evolution which began their division at this point. One branch of the evolution
was initiated by Henricus Grammateusin 1518. The purpose of the Grammatean
branch was to not only retain the twenty-two useable major and minor triads to
the octave, but also to increase the number of useable major and minor triads to
the ultimate of twenty-four to the octave. The Grammarian branch leads through
all the Well Temperaments to the Equal Temperament. The other branch of
evolution was initiated by Bartolomenus Ramis de Pareja in 1482. The purpose
of the Ramis branch was not only to retain the three quasi Just major and three
quasi Just minor triads, but to also transpose them from the rarely used keys to
the more commonly used keys and then to increase the number of these
harmonious type major and minor triads from six to sixteen to the octave. The
Ramis branch lead through all the Meantone Temperaments to Equal
Temperament. At Equal Temperament, both branches met again.”
The line of fifths are: Ab Eb Bb F C G D A E B F# C# .
Tuning (Quoting Jorgenson page 51:)”
1.
2.
3.
4.
5.
Tune C to the tuning fork.
Tune F below middle C pure to middle C.
Tune G below middle C pure to middle C.
Tune D above middle C pure to G below middle C.
Test
6.
7.
Tune A below middle C pure to D above middle C.
Test
8. Tune E above middle C and also E below middle C pure to A below middle C. This creates the
interval E F which is the first semitone. The inversion of E F is F E which is the first major seventh.
9. Test
10. Tune B below middle C pure to E above and E below. This creates the interval F B which is the first
tritone.
11. Test
12. Tune Bb below middle C pure to F below middle C.
13. Test
14. Tune Eb below middle C pure to Bb below middle C.
15. Test
16. Tune F# below middle C pure to B below middle C.
17. Test
18. Tune C# above middle C pure to F# below middle C.
19. Test
20. Tune G# below middle C pure to C# above middle C.
21. Test”
Music Theory of Henricus Glareanus (1547) Quotating from Sachs Our
Musical Heritage pg. 149:
“Music had reached a critical point at which contrapuntal was yielding to
harmonic conception, and the ‘Church Modes’ were changing to the more
modern Major and Minor modes. The Swiss Herricus Glareanus wrote the
Dodekachordon (1547), which presented a ‘twelve mode’ system by adding to
the age-old eight ‘Church Modes’ four additional modes: The Aeolian on A and
the Ionian on C and each with its hypo parallel, that is Hypoaeolian on E and
Hypoionian on G.”
Hericus Grammateus ½ Diatonic Comma Pythagorean Quoting Jorgenson,
page 238: “This is the Eratosthenes diatonic with Meantone, and it is also the
Juan Bermundo’s temperament of 1555 and the Heinrich Schreyber
Temperament of 1518. The diatonic comma is divided in to two parts in order to
dissipate the wolf among the two fifths. The great advantage of the Grammateus
temperament over the other Pythagorean temperaments is there is no harmonic
waste caused by narrow major thirds or wide minor thirds. Thus, the total
number of the beat speeds are reduced. Other special characteristic of the
Grammateus temperament are that it completely preserves the traditional
Eratosthes diatonic scale on the long keys (piano white keys) and the raised keys
(black keys on a piano) are tuned as Meantones between the long keys.
Therefore, the raised keys support tones which are enharmonically free in the
manner of standard Equal temperament. ….
In the theoretically correct method this temperament, twenty of the twenty-four
major and minor triads to the octave contain proportional beat speeds and the
diatonic comma is divided into two equal parts which are shared by two fifths. In
the equal-beating variation of this tuning, twenty-one of the twenty-four major and
minor triads to the octave contain proportional beat speeds, but the diatonic
comma is divided unevenly.
….
Like in the other ½ diatonic comma Pythagorean temperaments, all the major
and minor triads are musically useable. The line of fifths is: Gb Db Ab Eb Bb
tempered F C G D A E B tempered F# C# G# D# A# (in this
diatonic/meantone scale the sharps and flats are enharmonic). Major sixths, the
two tempered fourths and the major thirds are wide. The two tempered fifths and
all the minor thirds are narrow. ….
Unless the fourth interval F Bb sounds like a wolf in the piece of interest, the
equal beating method of the Grammateus temperament is superior to the
theoretically correct.”
Jorgenson gives the advantages and disadvantages of the two methods of
Grammateus to each other and to the other Just tunings on page 238 and 239.
All and all the Grammateus compares favorably. The method of tuning the equal
beating Henricus Grammateus ½ diatonic comma Pythagorean temperament
starts on page 240; the theoretically correct Grammateus method starts on page
242.
Summary of Jorgenson’s technique for equal beating Grammateus: First, use
the eleven steps for tuning the Eratosthenes scale (from this paper, page 25)
then;
12. Tune F# pure to B, and then proceed to lower F# until the interval between F# and B beats exactly the
same as the interval between D F#. Both F# B and D F# should be wide.
13. Test F# - B and F# - D should beat equal (4.6), F - A should beat at 10.9, F# - A should beat 6.9 and
D - F should beat at 10.9.
14. Tune Db pure to F#.
15. Test F# beats 4.6 with D; G beats 6.9 with Db; F beats 10.9 with A; E beats 5.2 with Db; d beats 9.2
with A.
16. Tune Ab pure to Db.
17. Test.
18. Tune Eb pure to Ab.
19. Test.
20. Tune Bb pure to Eb.
21. Test.”
Note: The above, F# to B step as given is difficult as the lowering is very slight
and easy to jump over. When testing if the interval from the natural to the
accidental is wide and the accidental to the natural is narrow, then the F# B
interval got the wrong upper partial. When F# is roughly pure to B it is at 376
hertz and when it is lowered it is at 372.516 hertz.
Music Theory of Gioseffo Zarlino (1558) Quotation from Sachs Our Musical
Heritage pg. 149: “ Gioseffo Zarlino (1517-1590), Cypriano de Rore’s successor
as the chapel master of St. Mark’s in Venice, wrote Le Istitutioni harmoniche. It
established both the major and minor third as the dual fundament of all harmonic
and the arithmetic division of a vibrating string. The harmonic division consisted
of the progression 2:1, 3:2, 4:3, 5:4 … and the arithmetic division, in the
progression 6:1, 6:2, 6:3, 6:4, 6:5 …
From Zarlino, the road of harmonic theory lead to Jean-Phippe Rameau and his
Traite de l’Harmonie of 1722.”
Advice from Jorgenson on Selecting a Just Pythagorean Tuning: (from
page 58)
“In choosing a Just Pythagorean tuning for musical performance, it is important to
analyze the music to see which triads are not used.
If the following Triads
are NOT used
Then tune the instrument
in the following :
B major and minor
E major and minor
A major and minor
D major and minor
Agricola in A major
Erlangen adjusted to D major
Erlangen in G major
Erlangen adjusted to C major or else
Agricola transposed into C major.
Ramis or improved Ramis in F major.
G major and minor
If all triads must be used , then the Hericus Grammateus Pythagorean temperament is recommended for all
early music through Purcell (if music is still diatonic, use the Eratosthenes Diatonic scale). For other music
and eighteenth century music a switch to Well Temperament is recommended.”
The Just Temperaments are octave scales that contain sharps and flats. In
these scales the Sharp is not at the same pitch as the enharmonic flat, the
difference is that the sharp is about 21 cent higher pitch than its enharmonic flat.
If all the sharps and flats were accounted for there would be 17 tones to the
octave. In the derivation, below only one set of enharmonic sharp or flat is
chosen.
In the Solomon de Caus Just, there is a problem; if you try it and it does not
come out right, you are probably correct. I have doubled checked against the
text and what I have is what is given in Jorgenson. The first problem that I have
with the tuning method is inherent to the scale, the E is hard to find the right
partial to zero beat, and I keep going past it and picking the wrong upper partial
and have to start over. The C F G and Bb tones are easy, the C F interval is
a perfect fourth and the second harmonic of C easily beats with the third
harmonic of F. So I had to go back and reset E to near its correct pitch by
playing C D E F. The E F semitone is easy to hear to get E near the right
pitch, it is just that the fifth harmonic of the E is hard to hear against the third
harmonic of the A. Next something is wrong with the step that is to tune G# to
E, I really think this is an error, it gives Ab rather than G# which are not equal (Ab
is 204.4, while G# is 209.3 in this Just tuning). My thinking is this should read
tune G# pure to D. G# being flat makes C# and D# flat as well.
Ramis de Paraja Transposed Pythagorean Just (1482) and Improved Ramis
In order to move the desirable harmonious triads into the more commonly used
keys a series of transpositions were performed. See Jorgenson pages 54 to 56
for how it was done. Ramis final work is named the Just Pythagorean Tuning.
This was carried further by Agricola. At the same time Erlanger was doing
similar work, but his translation was rotating from the other direction.
Quoting Jorgenson, page 63: “This tuning is the medieval Pythagorean tuning
into Cb Major. The line of fifths from the original Pythagorean tuning in Cb
major/Ab minor is Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G (12 tones). …
After the original quasi-Just schisma C# F#, and the B major triads from the
chromatic Pythagorean tuning in C major had been transposed into the C, F,
and Bb schisma major triads of Pythagorean tuning Cb major, either Ramis of his
predecessors improved the quasi-Just schisma triads by changing them into
completely Just triads. Since tonality was now around these Just triads because
of the increased in vertical third and sixth usage, the new Ramis Just
Pythagorean of F major and A minor was created. The great miracle of the
Ramis Just Pythagorean is that the total number of beat speeds of all the
intervals are reduced in number when compared to the number of beat speeds in
the original Pythagorean even though no tempering is done.“ The line of fifths
are: D A E B F# C# schisma Ab Eb Bb F C G.
Tuning (see Jorgenson page 66):
1.
2.
Tune middle C to the tuning folk.
Tune G below middle C pure to middle C.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Tune F below middle C pure to middle C.
Tune Bb below middle C pure to F below middle C.
Test Bb to F for zero beat; G to Bb for a beat of 14.5.
Tune Eb above middle C pure to Bb below middle C.
Test Bb to Eb for zero beat; G to Bb for a 14.5 beat; C to Eb for a 19.4 beat.
Tune Ab below middle C pure to Eb above middle C.
Test Ab to Eb zero beat; that both Ab to C and F to Ab beat at the same speed (12.9); G to Bb beats at
14.5.
Tune A below middle C pure to F below middle C.
Test
Tune D above middle C pure to A below middle C.
Test
Tune E below middle C and E above middle C pure to A below middle C.
Test
Tune B below middle C pure to E above middle C and also E below middle C.
Test
Tune F# below middle C pure to B below middle C.
Test
Tune C# above middle C pure to F# below middle C.”
Improved Ramis Just Pythagorean (12 tone). Quoting Jorgenson page 69: “The line of fifths are D A E
schisma Gb Db Ab Eb Bb F C schisma Abb.
1. Tune middle C to tuning folk.
2. Tune F below middle C pure to middle C.
3. Tune Bb below middle C pure to F below middle C.
4. Tune Eb above middle C pure to Bb below middle C.
5. Test F to C for zero beat; F to Bb for zero beat; Bb to Eb for zero beat; c to Eb for a beat of 19.4.
6. Tune Ab below middle C pure to Eb above middle C.
7. Test Bb to Eb for zero beat; F to Bb for 12.9 beat; C to Eb for 19.4 beat; test that Bb to C and F to Bb
beat at the same speed (12.9).
8. Tune Db above middle C pure to Ab below middle C.
9. Test
10. Tune Gb below middle C pure to Db above middle C.
11. Test
12. Tune Cb below middle C pure to Gb below middle C.
13. Test
14. Tune Abb below middle C pure to Cb below middle C.
15. Test
16. Tune A below middle C pure to F below middle C.
17. Test
18. Tune D above middle C pure to A below middle C.
19. Test
20. Tune E below middle C pure to A below middle C.
21. Test
22. Tune E above middle C pure to E below middle C.
Transposed Agricola Just (1539) To quote Jorgenson: “In the Martin Aricola
Just tuning twenty-two major and minor triads to the octave are useable. Twenty
major and minor triads to the octave contain proportional beat speeds. The wolf
fifth D A is one Syntonic comma narrow. The line of fifths is A E B F# schism
Db Ab Eb Bb F C G D. This tuning is an alteration of the Medieval
Pythagorean tuning transposed into Gb Major. The line of fifths from the original
Pythagorean tuning is: Bbb Fb Cb Gb Db Ab Eb F C G D. The Aricola
tuning restores the just intonation triads to the three primary triads. The
Transposed Agricola is superior to the Ramis Just Pythagorean tuning for the
following reasons:
1) One more primary triad is in the Just intonation. Results in all the primary triads in Just
intonation.
2) The wolf is removed off the dominant triad and placed in the less used super tonic triad.
3) The tonality is in the basic key of C Major.
4) The balance is better because of the schisma fifth.
The minor third F#-A exhibits poor balance, avoid. The bearing octave is E
below Middle C. Except for the pure intervals; major sixths, fourths and major
thirds are wide and fifths and minor thirds are narrow.”
Agricola Just Pythagorean Tuning Transposed into the Acoustic Tonality of A
Major/C# Minor see page 73. (12 tones) The tuning notes are: C, G below to C,
F below to C, D above to G, A below to D, E above and below to A, B below to E,
Bb below to F, G# to E below, C# above to G#, F# below to C#, D# above to G#.
C Major/E Minor see page 77. The tuning notes are: G below to C, D above to
G, F below to C, Bb below to F, Eb above to Bb, Ab below to Eb, Db above to
Ab, A below to F, E above to A, E below to E, B below to E and E, F# below to B.
Erlangen Monochord Just Pythagorean Quoting Jorgenson page 79: “At the
same time as Ramis Just Pythagorean was created by means of five
transpositions through the sharps, the Erlangen Monochord was created by five
transpositions through the flats. In this tuning, twenty-two major and minor triads
to the octave are musically useable. These twenty-two triads contain
proportional beat speeds. The wolf fifth A to E is one Erlanger comma narrow.
The line of fifths is: E B schisma Gb Db Ab Eb Bb F C G schisma Ebb
Bbb … The Erlangen Monochord eliminated all harmonic waste caused by the
narrow schisma major thirds and wide schisma wide minor thirds in the original
Pythagorean by making all these intervals pure except for one minor third. The
harmonic waste on this particular minor third (B to D) was eliminated because the
Erlanger Monochord changed this minor third to be exactly the same amount
narrow as it was wide in the original Pythagorean.
The Erlanger Just Pythagorean tuning is an improvement over both the Ramis
and Agricola tunings because the wolf is reduced by two schisms instead of just
one. The Erlanger tuning reduces the wolf from 8.9 beats to 7.4 beats per
second and without using any tempering. Also the Erlanger tuning is superior to
the Ramis because the wolf is transposed off the dominant triad onto the lessor
used submedinat triad, yet the C major triad is still pure and the G major triad is
almost pure.” See Jorgensen, page 81 for more benefits.
For tuning of the Erlenger Monochord Just Pythagorean Scale in the Acoustic
Tonality of G Major/B Minor see Jorgensen, page 83. Tuning notes are: G below
to C, F above to C, Bb below to F, Eb above to Bb, Ab below to Eb, Db above to
Ab, Gb below to Gb, B below to G, Ebb above to Gb, E above to B, Bbb below to
Ebb.
For C Major/ E Minor see page 85. Tuning notes are: G below to C, F above to
C, Bb below to F, Eb above to Bb, Ab below to Eb, Db above to Ab, Gb below
and above to Db, B below to G, Ebb above to Gb, E above to B, A below to E.
For D Major/F# Minor see page 88. Tuning notes are: G below to C, F above to
C, Bb below to F, Eb above to Bb, Ab below to Eb, Db above to Ab, Gb below to
Db, B below to G, Ebb above to Gb, Bbb below to Ebb, Fb above to Bbb.
Salomon de Caus Just (1615) Quoting Jorgenson, page 92: “The changes,
advantages and disadvantages between the Ramis Just Pythagorean and the
Caus Just Pythagorean are as follows:
1.
2.
3.
4.
5.
The number of Just major thirds was doubled from four to eight to the octave by Caus.
The number of Just minor thirds was doubled from three to six to the octave by Caus.
The number of Just intonation major and minor triads were doubled from six to twelve in the Caus.
Two fifths were ruined and made into wolf fifths in the Caus.
Four of the good Ramis Pythagorean major thirds of each octave were ruined and made into wolf
diminished fourths would not be considered wolf if it were that the human ear can not comprehend the
ratio 7 to 9 large major thirds.
6. Ten of the useable Ramis Just Pythagorean major and minor triads of each octave were ruined and
made into unusable wolf triads in the Caus.”
The line of fifths is: D A E B wolf F# C# D# wolf Bb F C G wolf D A E B
Tuning the Caus Just Intonation in C Major or G Major see Jorgenson, page 93. Tuning notes are: G below
to C, F below to C, Bb below to F, A below to F, D above to A, E below and above to A, B below to E
below, G# below to E below, C# above to G#, F# below to C#, D# above to G#.
Friedrich Wilhelm Marpurg (Just) Monochord No. 1. (1776) Basically adding
the C major sharps and flats to the Diatonic Syntonon of Ptolemy. Jorgenson
considers it inferior to the Caus Tuning. See page 96 for comparison. The line
of fifths is: Eb Bb wolf F C G D wolf A E B F# wolf C# G# wolf Eb Bb. See
Jorgenson page 99 for tuning steps.
Leath Gleas is a quite irregular scale, but very useful for playing Celtic/Gaelic
music on the Harp. There are 30 pitch in this scale, the upper 12 (or melody
pitch) are the same as the Eratosthenes Diatonic, the rest of the strings are
tuned in octave to the upper strings. The scale has its semitones placed like it
was in the key of G Major; with the exception of no low F#. This music only used
concords for Harmony and there is no concord that required an F#, so Irish
Harpers being their way, were not going to have a string they never played, no
string, all lower strings moved up one. In additon there were 2 G just below
middle C. No reason is given, but my oppinion is that they were required for
tuning. The Leath Gleas Scale probably dates to between the 10th century and
the 17th Century.
John Kovac’s Paraguayan Just (1999) Collected from John’s video on Playing
the Harp by Ear, from the tuning section. www.johnkovac.com . This is a very
useful scale for playing solo Harp, it sound good. This is a Paraguayan Harp
Scale and could date to as early as the 17th century.
Selecting a Meantone Temperament to Tune Quoting Jorgenson from page
104: “The trend toward Meantone tuning gained great impetus during the
sixteenth and seventh centuries because music composition in general remained
very limited in the use of scale degrees. Meantone in the tonality of C Major/ A
Minor has the following arrangements: the E , A, D, G, C, F, Bb and Eb major
triads are the good harmonious triads; but the Ab, Db, Gb, and B major triads
and also the F, Bb, Eb and G# minor triads are unusable wolf.”
Quoting from page 108: “The Lodovico Fogliano, the Fogliano-Aron and the
Pietro Aron system each contain the smallest number of total beat speeds of the
musically useable major and minor triads compared to all other Meantone
systems. In other words these are the most harmonious of all the Meantone
systems.
The Fogliano, the Fogliano-Aron, the Pietro Aron and the Francisco Salinas
system are the only Meantone system whereby all the musically useable triads
contain simply proportional beet speeds. In other words these are the most
rhythmically harmonious of all the Meantone systems.
The 1/8 Harrison comma Meantone temperament contains the most
rhymethrically harmonious major triads of all the Meantone temperaments.
Therefore, the best temperament for musicians that play in the ‘Happy Keys’.
The minor triads sound rough.
The Gioseffo Zarlino 2/7 Syntonic comma Meantone temperament contains the
most harmonious minor triads. The Zarlino is the best Meantone for musicians
that play in the ‘Sad Keys’. The major triads sound rough.”
The 1/8 Syntonic Well contains the largest contrast between cord colors. (and
most difficult to tune).
The 1/12 diatonic comma standard equal is the same as the scale that Ling-Lun
calculated in the twenty seventh century BC.
The rest of the 13 Meantone temperaments listed are rated as a matter of taste,
as they are less harmonious, their benefits are listed in Jorgenson on pages 111
and 114.
If there is only one string (set) that can be raised or lowered to make sharps and
flats, then a choice must be made, since sharps and flats in Meantone are not
the same pitch, the choice has to be made as to whether the string is going to be
tuned to a sharp or a flat. A string can not support both a sharp and a flat at the
same time. Enharmonic tuning is acoustically impossible for any Meantone
tuning. Example, quoting from Jorgenson, page 113: “The first step when
preparing to perform a music composition is to read through the composition in
order to discover what selection of sharps and flats is used. The selection of
sharps and flats does not always agree with the key signature. The instrument
must always be tuned to the true tonality of the composition regardless of
whether or not it agrees with the key signature. A composition might have a key
signature of D major, which is marked that all the F and C are to be sharp. But
suppose that the composition uses a Bb instead of A# and the predominate
selection of sharps and flats are Bb, D#, G#, and F#. This latter would indicate
that the acoustic tonality of the composition was in G major (see figure 6, below).
Major Acoustic
Tonality
Selection of Tones
Minor Acoustic
Tonality
Figure 6. Chart of Tonalities for Meantone Temperament.”
… When retuning or altering a tuning to its enharmonic partner, the basic
Meantone rule is that most augmented or diminished intervals are wolf. This
means that if it is desired to change an F# into a Gb, the Gb can not be tuned
from a lower D. This would create a diminished fourth D Gb. Then an F# can
only be tuned from a lower D resulting in a true major third D F#. And if a Gb is
desired it can be only tuned from a higher Bb, any Cb or any Db. Gb can not be
tuned from a B. Only F# can be tuned from a B. In all the Meantone
temperaments, the flats are faster in frequency and higher in pitch then their
equivalent enharmonic sharps. This is opposite from the condition found in the
Pythagorean tuning or the Just Pythagorean tunings whereby flats are lower than
the sharps. Following are the rules for changing enharmonic pairs:
F# must be tuned from a lower D, any B, or any C#.
Gb must be tuned from a lower Bb, any Cb, or any Db.
G# must be tuned from a lower E, any C# or any D#.
Ab must be tuned from a lower C, any Db, or any Eb.
A# must be tuned from a lower F#, any D#, or any E#.
Bb must be tuned from a lower D, any Eb, or any F.
C# must be tuned from a lower A, any F#, or any G#.
Db must be tuned from a lower F, any Gb, or any Ab.
D# must be tuned from a lower B, any G#, or any A#.
Eb must be tuned from a lower G, any Ab, or any Bb.
Remember that in Meantone, the sharp of a lower note is not equal to the flat of the next higher note.
This affects the Meantone tuning method. G# and Ab are two separate pitches. In Meantone
temperament flats are higher than sharps (Ab is higher pitched than G#).
Lodovico Fogliano Just Meantone (1529) “The Ramis Just and Salomon de
Caus Just tuning contained the wolf fifth G D which ruined the dominant major
triad: G B D. This G D fifth was wolf because it was narrow by a Syntonic
comma, this was repaired by dividing this interval and sharing it between fifth G
D and D A, which each became ½ Syntonic comma narrow. This made the G B
D triad musically useable and these two intervals became known as tempered
fifths. This was documented by Lodovico Fogliano in 1529. The description of
the steps to alter the Caus Just or the Ramis Just are contained on Jorgensen
pages 102 and 103. Quoting Jorgenson, page 103: “The great advantage of the
Fogliano Meantone tuning over the Caus Just intonation was the one third
increase in the number of good useable major and minor triads. Also, all the Just
major thirds from Caus were saved and transposed into the acoustic tonality of C
major and A minor. Four of the original Caus Just intonation triads were
rendered somewhat impure (tempered) in the Fogliano Meantone; but
nevertheless, the total number of Just intonation triads in the Fogliano Meantone
is eight which is two more than existed in the Ramis Just Pythagorean tuning.”
The sacrifices paid for correcting the dominant or supertonic wolf triads are not
only the loss of six useable major and minor triads to the octave, but also many
and usually all of the fourths and fifths are altered and no longer Just. In spite of
the latter altering of the fourths and fifths, the increase number of pure or close to
pure major thirds greatly increase the overall harmoniousness of the tuning within
the sixteen good major and minor triads to the octave range. In all the later
development of Meantone temperament, the sixteen good useable major and
minor triads of Fogliano were never increased in number.”
“There are sixteen musically useable major and minor triads to the octave; all of
them contain proportional beat speeds. Eight of the triads are Just. There are
three extra second inversion minor triads that are useable. The line of fifths is: A
E B tem F# tem C# G# wolf Eb tem Bb tem F C G tem D tem A.
The tuning method starts on Jorgenson page 118, 120 and 123”
Tuning notes are: F below to C, G below to C, A below to F, E below to A, B
below to E, Eb below to G, G# below to E, C# above to G#. Now the first
temperament, first tune F# roughly pure to B below middle C, then proceed to
lower F# until the interval Eb F# sounds as beatless as a pure ration 6:7 small
minor third (lots of luck). Continue; D below to F#, D above to D below, Bb below
to D above.
Another method for the Lodovico Fogliano Just Meantone in C Major/A Minor:
F below to C, G below to C, A below to F, E below to A, B below to E, Eb below
to G, G# below to E, C# above to G#. Now the temperament, tune Bb roughly
pure to Eb below, then proceed to lower Bb until the fifth Eb Bb beats exactly the
same speed as the small minor third, Bb C# (2.9). Both Eb Bb and Bb C#
should be narrow. Continue, D above to Bb, D below to D, F# to D.
Fogliano-Aron Just Meantone (1523) Quoting Jorgenson, page 107: “The
Fogliano-Aron Just Meantone tuning without temperament accomplishes more as
far as expanding musical resources are concerned than any other form of Just
tuning. This tuning therefore is the end results of a long history of tuning
beginning with Pythagoras. Even though this is still technically a Just intonation,
all music compositions which can be performed in any of the Meantone systems
can also be performed in this tuning.
Quoting Jorgenson, page 129: “There are sixteen useable major and minor
triads to the octave, and all of them contain proportional beat speeds. Four of
the triads are Just intonation. There are also three extra second inversion minor
triads that are useable. The only Just fifths are C G and E B.”
The tuning method is given on page 129. G below to C, Eb below to G, B below
to G, E below to B, G# below to E. First temperament, tune F below middle C
roughly pure to middle C, then proceed to raise F until the interval F G# sounds
as beatless as a pure 6:7 small minor third. Second temperament, tune F#
below middle C roughly pure to B below middle C, then proceed to lower F# until
the interval Eb F# sounds as beatless as a pure 6:7 small minor third. Continue,
D below to F#, D above to D, A below to F, Bb below to D above, C# to A.
Francisco Salinas 1/3 Syntonic Comma Meantone (1577) Quoting
Jorgenson, page 138: “The Salinas temperament belongs to a negative
Meantone temperament; it represents the furthest backwards development ….”
Page 139: “There are sixteen musically useable major and minor triads; all of
them contain proportional beat speeds. The Syntonic comma is divided into
three equal parts and shared by the three basic fifths. “
The C Major tuning method starts on page 139 or 143. C below to C, Eb below
to C, Eb above to Eb, A below to Eb, F# above to A, F# below to F#, D above to
A, Bb below to D.
First temperament, tune G below middle C roughly pure to D above middle C and
then proceed to raise G until the fifth G D beats exactly the same speed as the
major third Eb G below middle C. Eb G and G D should both be narrow (4.2
speed). Continue, tune G above to G.
Second temperament, raise and retune D above middle C until the major third Bb
D beats at exactly the same speed as the fourth D G above middle C. Both Bb
below middle C to D above middle C; and D above middle C to G above middle
C should be wide and beat at exactly 4.2. Also the interval D above middle C to
G above middle C should beat at exactly 4.2 beats. Both G below middle and C
below middle C; and G below middle C to D above middle C should both beat at
exactly 2.1 beats. The four following pairs on tones should also beat at 4.2
beats: G below to middle C; D above to G above; Eb below to G below; Bb
below to D above.
Continue, tune D below to D above, B below to D below, B (an octave lower than
below middle C) to B below, F below to D above, G# below to B (an octave
below).
Third temperament, raise and retune Bb below middle C pure to G above middle
C.
Continue, tune E above to G below, E below to E above, C# above to E below,
C# below to C# above.
1/8 Harrison comma Meantone (1749) Quoting Jorgenson, page 152: “There
are sixteen musically useable triads to the octave, and eight of them contain
proportional beat speeds. Each fifth is narrowed by roughly 1/8 Harrison comma,
and the ratio of each fifth is 1.494530180. The bearing section is D below middle
C to G above middle C. Major sixths and fourths are wide. Fifths, major thirds,
minor thirds and small minor thirds are narrow. “
The tuning method starts on page 153. Tune F above to C, Bb below to F, D
above to Bb.
First tempering, tune G below middle C roughly pure to middle C, then proceed
to lower G until the fourth G C beats at exactly the same speed as the fifth G D,
4.2 beats.
Continue; tune G above to G below.
Second tempering, raise and retune D above middle C until the major third Bb D
beats at exactly the same speed as the fourth D G above middle C. Both Bb
below D above and D above and G above should beat at exactly 4.2 beats. As
well as the three following pairs: C middle to G above; C middle to G below; D
above to G above.
Continue, tune D below to D above.
Third tempering, tune F below middle C roughly pure to middle C, then proceed
to raise F until the fifth F C beats at exactly the same speed as the minor third, D
F below middle C, 2.1 beats.
Fourth tempering, raise and retune F above middle C to F below.
Fifth tempering, raise and retune Bb below middle C until the major third, Bb D
beats exactly the same speed as the minor third, D F above middle C, 2.3 beats.
Sixth tempering, tune Eb roughly pure to G below middle C and then proceed to
raise Eb until the major third Eb G beats at exactly the same speed as the minor
third, G Bb, 1.8 beats.
Continue; tune Eb above to Eb below.
Seventh tempering, tune A below middle C roughly pure to F below, then
proceed to lower A until the major third F A beats at exactly the same speed as
the minor third A C, 1.2 beats.
Continue, tune F# above to F# below.
Eight tempering, tune B below middle C roughly pure to G below middle C, then
proceed to lower B until the major third G B beats at exactly the same speed as
the minor third B D, 2.1 beats.
Ninth tempering, tune E above middle C roughly pure to middle C, then proceed
to lower E until the major third, C E beats at exactly the same speed as the
minor third E G, 4.2 beats.
Continue, tune E below to E above.
Tenth tempering, tune G# below middle C roughly pure to E below middle C,
then proceed to lower G# until the major third E G# beats exactly the same
speed as the minor third G# B, 2.1 beats.
Eleventh tempering, tune C# above middle C roughly pure to A below middle C,
then proceed to lower C# until the major third A C# beats at exactly the same
speed as the minor third C# E, 2.7 beats.
It helps to look at the musical score at the bottom of Jorgenson page 156 to see
what happened.
Zarlino 2/7 Syntonic Comma Meantone (1558) Quoting Jorgenson, page 168:
“There are sixteen musically useable triads to the octave, and eight of them
contain proportional beat speeds. Each fifth is narrowed by 2/7 Syntonic comma.
The bearing section is A below an octave lower than middle C to E above middle
C. Major sixths and fourths are wide. Fifths, major thirds, minor thirds, and small
minor thirds are narrow.”
Tuning method starts on page 163. Tune C below to middle C, Eb below to
middle C, F below to middle C, Db below to middle C.
First tempering, tune Bb (below an octave below middle C) roughly pure to Eb
below middle C, then proceed to lower Bb until the fourth Bb Eb beats at exactly
the same speed as the fifth Bb F, 2.5 beats.
Continue; tune Bb below to Bb (below an octave below).
Second tempering, retune and lower Eb below middle C until the fourth Bb Eb
beats at exactly the same speed as the major sixth Eb C. The following three
pairs should beat at exactly the same rate: Bb Eb; Eb C; Eb C (an octave
below), 1.6 beats.
Third tempering, retune and raise F below middle C until the major third Db F
beats at exactly the same speed as the fourth F Bb, 2.5 beats.
Fourth tempering, tune G below middle C roughly pure to Eb, then proceed to
lower G until the major third Eb G beats exactly the same as the major sixth Eb
C, 1.6 beats.
Fifth tempering, tune D below middle C roughly pure to Bb (below an octave
below middle C), then proceed to lower D until the major third Bb D beats at
exactly the same speed as the major sixth Bb G, 0.8 beats.
Continue, tune D above to D below.
Sixth tempering, tune A below middle C roughly pure to F below middle C, then
proceed to lower A until the major third beats exactly the same as the major sixth
F D, 1.9 beats.
Continue, tune A (below an octave lower than middle C) to A below.
Seventh tempering, tune E below middle C roughly pure to C below middle C,
then proceed to lower E until the major third C E beats at exactly the same
speed as the major sixth C A, 0.9 beats.
Continue; tune B below to G, B (below an octave lower) to B below.
Eight tempering, tune F# below roughly pure to D below middle C, then proceed
to lower F# until the major third F# D beats exactly same speed as the major
sixth D B, 1.9 beats.
Ninth tempering, retune and lower C# below middle C roughly pure to A (below
an octave below middle C) then proceed to lower C# until the major third A C#
beats at exactly the same speed as the major sixth A F#, 1.0 beats.
Continue, tune C# above to C# below.
Tenth tempering, tune G# below middle C roughly pure to E below middle C,
then proceed to lower G# until the major third E G# beats exactly the same
speed as the major sixth E C#, 0.7 beats.”
It helps to see the music score by Jorgenson page 173 to see what happened.
The 1/8 Syntonic Well (inspite of the name this is a Meantone, 1758)
contains the largest contrast between cord colors. Quoting Jorgenson, page
197: “Tuning the Theoretically Correct Jean Baptiste Romieu 1/8 Syntonic
Comma Well temperament in the Acoustic Tonality of C Major/A Minor. All the
major and minor triads are musically useable. There are no proportional beating
triads. Each fifth is narrowed by 1/8 Syntonic Comma. The bearing octave is E
below middle C to E above middle C. Major sixths, fourths, major thirds and
small minor thirds are wide. Fifths and minor thirds are narrow. This is a
complete temperament with no tunings done except for octave.
Tuning C to tuning fork. (see Jorgenson, starting on page 197 for complete
description).
Temper E above middle C from middle C so that the major third C E is wide and
beating at 8.2 beats per second.
Temper G below middle C from middle C so that the fourth G C is wide and
beats 1.2.
Test, it is essential that the following have correct beat speed: G C 1.2 beats; C
E 8.2 beats; and G E 7.6 beats.
Tune D above middle C roughly pure to A below middle C and then proceed to
raise D until the fourth A D beats 1-1/2 times as fast as the fifth G D. Also the
fourth A D beats approximately 1-1/3 times as fast as the fifth A E. (lots of luck)
Test, G D 0.9 beats; A E 1.0 beats; G C 1.2 beats; F D 1.4 beats; C E 8.2
beats; G E 7.6 beats and A C 10.2 beats.
Tune E below middle C pure to E above middle C.
Test
Temper Bb below middle C from D above middle C so that the major third Bb D
is wide and beating 7.3 per second.
Test
Tune F below middle C roughly pure to Bb below middle C and then proceed to
lower F until the fourth F Bb beats approximately 1-1/3 times as fast as the fifth
F C.
Test
Temper B below middle C from G below middle C so that the major third G B is
wide and beating at 6.1 beats per second.
Test
Temper C# above middle C from A below middle C so that the major third A C#
is wide and beats 6.8.
Test
Tune F# below middle C roughly pure to B below middle C and then proceed to
lower F# until the fourth F# B beats approximately 1-1/2 times as fast as the fifth
E B and also the fourth F# B beats 1-1/3 times as fast as the fifth F# C#. F# B
should be wide and F# C# should be narrow.
Test
Temper G# below middle C from E below middle C so that the major third E G#
is wide and beating at 5.1.
Test
Temper Eb above middle C from Bb below middle C so that the fourth Bb Eb is
wide and beating at 1.5.”
It is helpful to look at the score in Jorgenson, page 200 to see what happened.
The 1/12 diatonic comma standard equal (Ancient) is the same as the scale
that Ling-Lun calculated in the twenty seventh century BC.
Quarter Comma Meantone (1523) (This is the ¼ Syntonic Comma Meantone
by Pietro Aron) Quoting Jorgenson, page 177: “There are sixteen musically
useable triads to the octave, and all of them contain proportional beat speeds.
There are also three extra second inversion minor triads which are useable. The
syntonic comma is divided into four equal parts which are shared by four basic
fifths.”
Selecting a Well Temperament to Tune Quoting Jorgenson, page 245: “Well
Temperament is not synonymous with Equal Temperament. J.S. Bach wrote for
the Well Tempered Clavier, not the Equal Tempered. Well temperament had to
be known in the time of Grammateus, although theorist did not start to write
about it until 1690. ….
There are no wolf intervals in Well Temperament, and all the Triads are musically
useable. Complete freedom of modulation exists in both Well and Equal
temperament. During modulation, Equal temperament lacks the harmonic keycolor changes inherent in the unequal spaced tones of Well temperament.
Modulatory key-coloring was considered essential to all those that rejected equal
temperament. The basic idea of Well temperament is to preserve a Meantone
type of harmonic smoothness in the commonly used keys while allowing one to
modulate with every changing key-color through out the cycle of fourths to the
very brilliant lesser used keys. … In the number of proportional beating triads, all
the Well temperaments are superior to their equivalent comma divisions in the
Meantone temperaments. This gives Well temperament an unrivaled quality of
rhythmic harmoniousness. “
Jorgenson gives the method of creating twenty-eight Well Tempered Scales in
his book.
Quoting Jorgenson, page 248: “When choosing a Well Temperament, refer to
the chart on pages 249 and 250, observe the following facts:
1.
2.
3.
4.
The Bendeler ¼ diatonic and the Werckmister Well Temperaments, which divide the diatonic comma,
are more suitable for music with many sharps and flats than the equivalent Well Temperaments that
divide the Syntonic or Erlanger commas. However these diatonic Well Temperaments contain fewer
pure intervals and less color contrasts than the equivalent Syntonic and Erlanger comma Well
Temperaments
In general, the smaller the number of tempering steps, the more suitable the Well Temperament is for
music with few sharps and flats and it is easier to tune.
In general, the larger the number of tempering steps, the more suitable the Well Temperament is for
music with many sharps and flats and it is harder to tune.
The equal-beating temperament methods are easier to tune than the theoretically correct and are most
often more superior musically.” I have chosen to use the equal-beating.
Johann Phillip Bendeler Temperament #1 Well Temperament (1690) This is
a defective temperament as published. Not only was it printed in the wrong key,
it had the minor third B D smaller than minor third F# A. Andrew Werckmeister
repaired this according to his Correct Temperament #1 published in 1691 and
Friedrich Wilhelm Marpurg solved it a different way in 1776., he tempered the
fifth Eb Bb and then compensated by tuning G pure.” There are several other
methods of correction. The one selected for representation here is the Equalbeating First Correction of the Johann Phillipp Bendeler Temperament #1 Well
Temperament in the Acoustic Tonality of C Major.
Quoting Jorgenson, page 280: “The bearing section is F below middle C to Gb
above middle C. Except for the pure intervals, major sixths, fourths and major
thirds are wide and fifths, minor thirds are narrow.
Tuning Tune middle C to tuning folk; G below to C; F above and below to C; Bb
below to F above; Eb above to Bb; Ab below to Eb; Db above to Ab; Gb below
and Gb above to Db.
Temper D above middle C by roughly tuning pure to F below middle C and then
proceed to raise D until the major sixth F D beats at exactly the same speed as
the major third D F#, 6.2 beats.
Temper A below middle C roughly pure to D above middle C and then proceed to
lower A until the fourth A D beats at exactly the same speed as the major third F
A, 3.1 beats.
Tune E above to A below; B below to E above.”
Salinas Well Temperament Quoting Jorgenson, page 282: “This is the
Bartolemeus Ramis de Pareja Just Temperament improved by the Martin
Agricola philosophy and altered by tempering D and A. It is also the Well
Temperament version of the Franscisco Salinas 1/3 Syntonic comma Meantone
temperament. Plus, in 1806, Lord Charles, Earl of Stanhope, introduced the
Salinas into England as the Stanhope Temperament.”
Tuning, tune middle C to the tuning folk; tune G below to C; F above and F below
to C; Bb below to F above; Eb above to Bb below; Ab below to Eb above; Db
above to Ab below; Gb below and Gb above to Db above; b below to G below; E
above to B below.
Temper D above by tuning roughly pure to F below then proceed to raise D until
the major sixth F D beats at exactly the same speed as the major third D F#
above middle C, 6.2 beats.
Temper A below by tuning roughly pure to D above and then proceed to lower A
until the fourth A D beats at exactly the same speed as the major third F A, 3.1
beats.”
½ Erlanger Comma Well Temperament Quoting Jorgenson, page 266: “ In the
Equal-beating method, twenty-two of the twenty -four major and minor triads per
octave contain proportional beat speeds, and this is the greatest number of
proportional beating triads possible. “
From page 270: “ Tune middle C to the tuning fork.
Tune F above to middle C; Bb below to F; Eb above to Bb.
Test the following pairs: Bb F zero beat; Bb Eb zero beat; C F zero beat; C Eb
19.4 beat.
Tune Ab below to Eb; Db above to Ab; Gb above to Db; Cb below to Gb; Abb
below to Cb below; Rename Abb to G.
Tune G above to G below; E above to middle C; A below to E above.
Temper D above middle C by tuning roughly pure to G above then proceed to
lower D until the fourth D G beats at exactly the same speed as the major third
Bb D.”
Note Erlanger was J. S. Bach’s student. Bach did not publish much about the
temperaments that he used. Erlanger could be improving and publishing what he
learned from Bach.
Andrew Werckmister Well Quoting Jorgenson, page 307: “Tune middle C to
tuning folk.
Tune Bb below to middle C; Eb above to Bb; Ab below to Eb; Db above to Ab;
Gb below to Db; Gb above to Gb below.
Temper D above tune roughly pure to Bb below and then proceed to raise D until
the major third Bb D beats exactly the same speed as the major third D F#
above.
Temper G below roughly pure to middle and then proceed to lower G until the
fourth G C beats at exactly the same speed as the fifth G D.
Temper A below roughly pure to D above then proceed to lower A until the fourth
A D beats at exactly the same speed as the major third F A.
Tune E above middle C pure to A below; B below to E above.
It helps to get this right to be able to do the tests between each step and it helps
to see what happened to look at the score on Jorgenson, page 309. The detail is
important but just tooo long for this article.
In 1681 Andreas Werckmeister published Orget-Probe which contained 4
musical scales that he was promoting as corrections to the Just Musical Scale.
Wreckmeister was a student of J.S. Bach and Werckmeister’s musical thought
just might reflect Bach’s musical thought. Bach is silent on what musical scale
he composed in. But, Bach did write on one of his manuscripts that it was not to
be played in Equal Temperament. The KORG Model OT-120 Electronic Tuner
has the Werckmister scales built in, so using this Tuner to tuning to Werckmister
is easy.
Kirnberger ½ Syntonic Comma Well This is the Martin Agricola Just tuning
transposed into the key of C and altered by one temperament, the resulting Well
temperament is actually in G major. This temperament was published in 1779,
but was know years earlier. This temperament is not quite a Well temperament,
because the fourth from A below middle C to D above middle C is still basically a
wolf.
The Kirnberger selected is just one of the four that Jorgenson describes and is
the Equal-Beating Johann Phillipp Kirnberger Well Temperament Transposed
into the Acoustic Tonality of C Major. Quoting Jorgenson, page 263: “Tune
middle C to the tuning folk.
Tune G below to C; F above to C; Bb below to F; Eb above to Bb below; Ab
below to Eb above; Db above to Ab below; Gb below and Gb above to Db above;
A below to middle C; E above to A below; B below to E above.
Temper D above by tuning roughly pure to Bb below and then proceeding to
raise D until the major third Bb D beats at exactly the same speed as the major
third D F# above, 7.3 beats.”
Again the test that Jorgenson gives and the score on page 265 helps a lot, but
too much for this quote.
Quarter Comma Well Temperament Quoting Jorgenson, page 296: “This is the
eighteenth century temperament of Dr. George Sargent transposed into the
tonality of C major. The theoretically correct method has sixteen of the twentyfour major and minor triads to the octave containing proportional beat speeds
and the diatonic comma is divided into four equal parts which are shared by four
of the fifths. The tempered fifths beat slower than the Bendeler, Salinas or
Harrison Well temperaments. In the equal-beating method there are the same
number of proportional beating triads, but the diatonic comma is not divided
evenly among the four fifths.
Tune middle C to the tuning folk. Tune F above to C; Bb below to F; Eb above to
Bb; Ab below to Eb; Db above to Ab; Gb below to Db; Cb below to Gb.
Temper G below middle C from middle C so that the fourth G C is wide and
beating at 2.7 beats per second. (good luck .. at telling a beat to .7!).
Temper D above middle C from G below so that the fifth G D is narrow and
beating at 2.0 beats per second.
Tune A below to D.”
Irregular Temperaments The background for these miscellaneous scales is the
desire for composers to have new material to explore when their current work is
starting to be the same thing over again. There are primitive scales that existed
for a while and then were overcome by more modern replacements. Only in very
remote regions do these earlier scales have a chance of still existing before
being superseded. The most primitive still in existence in the early twentieth
century is the Javanese with 5 equal tones per octave. Another closely related
but later is the Siamese with seven equal tones per octave. Then there are later
historic scales that did not get enough of a following to be able to gain
dominance over the octave scale and while these lesser historic scales are not
disappearing, they are not gaining any new converts either; two of these scales
are the Arabian at 17 equal tones per harmonic, the Hindu at 22 equal tones per
harmonic. Plus there are numerous modern experimental scales that have not
gone anywhere and have little use in tuning Replica Instruments.
Jorgenson 5/7 Tuning Quoting Jorgenson, page 377: “The five and seven
temperament is a new language based on the melodic series, then the historical
tunings such as the Pythagorean, Just; Meantone and Well Temperament can all
be viewed as dialects of one common language based on the harmonic series...”
Quoting Jorgenson, page 378: “In order to understand what the five and seven
temperament is, it is helpful to review how the standard 12 equal temperament is
arranged on a keyboard. Each of the twelve intervals or tonal distances between
C, C#, D, D#, E, F, F#, G, G#, A, A#, B and C are named semitones or half
steps. In equal temperament they are all the same size. All other intervals are
constructed by adding together various number of these semitone. Notice in
particular that the interval D# F# is 1-1/2 times as large as the interval C# D#.
Also notice that the interval F G is exactly twice as large as the interval E F.
Therefore in the standard equal temperament, neither the tones on the black
keys nor the white keys are equally spaced.
In the five and seven the equal temperament interval D G# has been preserved,
and the frequencies of the tones D and G# are identical to the tones of D and G#
in the equal temperament. However, all the remaining tones have been arranged
as follows:
1.
2.
The intervals between the successive black keys are all made exactly the same size. Thus, D# F# is
made the same size as F# G# and A# C# is made the same size as G# A#. The originals equal
tempered black keys minor thirds become smaller and the original equal tempered black key whole
tones become larger.
The interval between the successive white keys are all made exactly the same. Thus, E F is made the
same size as F G; and B C is the same size as C D. The original white key equal tempered semitones
become larger and the equal tempered white key whole equal tempered whole tones become smaller.
All of this results in the five intervals between the black keys and the seven
intervals between the white keys of the octave being exactly equally spaced.
This latter quality accounts for the name “five and seven”. As in the standard
equal temperament, it is also true in the five and seven that G# is exactly midway
between G and A; Gb and Bb; F and B; E and C; Eb and Db; and finally a
lower D and a upper D. Also, D is exactly midway between Db and Eb; C and E;
B and F; Bb and Gb; A and G and finally a lower G# and a upper G#. Thus G#
and D are the center.”
Quoting Jorgenson, page 380: “The exact number of cents between each tone in
the five and seven temperament is: D 120 Eb 51-3/7 E 171-3/7 F 17-1/7 F# 1542/7 G 85-5/7 G# 85-5/7 A 154-2/7 Bb 17-1/7 B 171-3/7 C 51-3/7 C# 120 D. …
then the exact amount in cents that each tone deviates from the same tone in 12
equal temperament is
D Eb
E
F
0 +20 -28 4/7 +42 6/7
F#
G
G#
-40 +14 2/7 0
A
Bb
B
C
C# D
-14 2/7 +40 -42 6/7 +28 4/7 -20 0”
If the Jorgenson five and seven is tuned on one piano and 12 equal temperament
on a second, the two playing together sound interesting. Also a two keyboard
stack of electronic keyboards is interesting.
Partch’s 43 The 43 pitches first in ratio of small integers from the tonic: 1:1
81:80 33:32 21:20 16:15 12:11 11:10 10:9 9:8 8:7 7:6 32:27 6:5 11:9 5:4
14:11 9:7 21:16 4:3 27:20 11:8 7:5 10:7 16:11 40:27 3:2 32:21 14:9 11:7
8:5 18:11 5:3 27:16 12:7 7:4 16:9 9:5 20:11 11:6 15:8 40:21 64:33
160:81 2:1.
Next these same 43 pitches in cents from the tonic: 0 21.5 53.2 84.5 111.7
150.6 165.0 182.4 203.9 231.2 266.9 294.1 315.6 347.4 ??? (do 5:4) 417.5
435.1 470.8 496.0 519.5 551.3 582.5 617.5 648.7 680.5 702.0 729.2
764.9 782.5 813.7 852.6 884.4 905.9 933.1 968.8 996.1 1017.6 1035.0
1049.4 1088.3 1115.5 1146.8 1178.5 1200.
19-Tone Equal The cents from the tonic are: 0 63.2 126.3 189.5 193.1 252.6
315.8 378.9 442.1 505.3 568.4 631.6 694.7 757.9 821.1 884.2 947.4
1010.5 1073.7 1136.8 1200.
5-Limit Just The 12 tones first in ratio of small integers: 1:1 16:15 9:8 6:5 5:4
4:3 45:32 3:2 8:5 5:3 16:9 15:8 2:1.
Now the same 12 tones in cent from tonic: 0 111.7 203.9 315.6 386.3 498.0
590.2 702.0 813.7 884.4 996.1 1088.3 1200.
Now the common names of these 12 tones: tonic, minor second, major second,
minor third, major third, perfect fourth, diminished fifth, perfect fifth, minor sixth,
major sixth, minor seventh, major seventh, octave.
True Blues This is really a pentatonic scale trying to be an octave scale, so
some of the tone positions move around. The main ratios in small integers are:
1:1 6:5 4:3 3:2 7:4 2:1, with two additional sometimes tones are: 7:6 and 5:4.
The cents from the tonic are: 0 (266.9) 315.6 (386.3) 498.0 702.0 968.8
1200.
Music Scale by Vallotti Francesco Antonio Vallotti (11 June 1697 – 10
January 1780) was an Italian composer, music theorist, and organist. He studied
with G. A. Bissone at the church of St. Eusebius, and joined the Franciscan order
in 1716. He was ordained as a priest in 1720. In 1722 he became an organist at
St. Antonio in Padua, and would eventually become maestro there in 1730,
succeeding maestro Calegari, and would hold that position for the next fifty
years. Here he would meet and work with another theorist and composer
named Giuseppe Tartini. Vallotti spent a great deal of thought on the theory
of harmony and counterpoint. His theoretical endeavors would culminate in 1779
with the publishing of his 167-page, four volume work, Della scienza teorica e
pratica della moderna musica (On the scientific theory and practice of modern
music). The Valotti temperament has six perfect fifth intervals and six intervals
with –1/6 comma. The perfect fifths are B-F#, F#-C#, C#-G#, G#-Eb, Eb-Bb and
Bb-F. The –1/6 comma fifths are C-G, G-D, D-A, A-E, E-B and F-C. The
frequency ratio for a perfect fifth is 1.5 and for a –1/6 comma is 1.49662.
Music Scale by Young is a well temperament devised by Thomas Young,
which he included in a letter to the Royal Society of London written July 9, 1799.
It was read January 16, 1800 and included in the Society's Philosophical
Transactions published that year. Young outlined a practical method to "make
the harmony most perfect in those keys which are the most frequently used," by
tuning upwards from C a sequence of six pure fourths, as well as "six equally
imperfect fifths," in other words six progressively purer flat fifths. His goal was to
give better major thirds in more commonly used keys, but to not have any
unplayable keys. The KORG Model OT-120 Electronic Tuner has Young’s Scale
built in, so using this Tuner to tuning to the Young Scale is easy.
TEMPO For almost all of history, the speed that music is played has been left to
the performer. The ancient Greek Chorus took a step forward and then back
while singing, this might have been an effort of beating time. The ancient Greek
had singing, music and religion joined together; music was mostly used to back
the human voice, there were even Greek Law against purely instrumental music.
There was ancient Greek written music, using Greek Letters in various forms, it
was not used much among musicians as none of them could read. Nothing
much happened for a thousand years, then in 1050 AD, Guido de Arezzo invents
the Gran Staff and the ut re mi... scale. Arezzo’s Grand staff has only 4 staves,
no bars or measures; but it was the start. The importance was that musical
performance is projected to the position of symbols on a drawing. Later, the first
tempo symbols on Arezzo’s Gran Staff were Red and Black ellipses that
indicated long and short notes to match long and short vowels in singing. Due to
the pressure to symbolize more that came from the rapid expansion of church
organs, more tempo marks were soon added. Then things got real complicated:
like a demisemiquaver is half of half of half a crotchet, there are 4 crotchets in a
semibreve. So how many demisemiquavers are there in a semibreve?
Before the 19th Century, it was not possible to divide time down to a minute. In
1452, Galileo Galilei worked out the math for the period of a pendulum. In
1696, Etienne Loulié used an adjustable pendulum in the construction of the first
metronome, too expensive and impractical. In 1812. Dietrich Nikolaus Winkel
invented a more practical metronome. In 1816, Johann Maelzel incorporating
Winkel's ideas, and started manufacturing the metronome under his own name.
By this date, music time could be measured down to beats per minute. But no
more.
Frequency of Pitch For almost all of history, music philosophers had to use the
Monochord to measure pitch. Before the mid 19th Century measurement could
not be made to a fraction of a second. Anything ancient that talks about
Frequency is not correct and should be looked at more closely.
When someone states that: in 1619 Pretorius gave the ‘concert pitch’ of
Northern Germany as A = 567. Schnitger’s organ in St Jacobi in Hamberg
(1688) was tuned A = 489. Silbermann’s organ in Strassburg Cathedral,
Germany (1713) had a pitch A = 393. At Trinity College, Cambridge, Father
Smith’s organ was tuned (1752) to A = 522.5. By 1859 the French government’s
commission of musicians and physicists advised a standard pitch, A = 435. The
Covent Gardens Orchestra in 1879 was playing a pitch of A = 450. At the same
time America’s ‘concert pitch’ was A = 461.6. All this may not be true, because
how could they measure these frequencies?
Now in America Standard A= 440 and Concert Pitch A = 432; while in England
the standard of C = 522 causes A = 438.9.
Standard Music Scales Since the mid-20th Century the standard Musical Scale
for all Western Musical Instruments has become the 12-tone Equal
Temperament. But what was the standard Musical Scale for each place and
each period in history before this? The Church Organ at St. Ann Church in
Belfast Ireland during the time of Edward Bunting’s apprenticeship is of major
interest to Harpers trying to understand Bunting’s musical score.
Quoting from an article by Betty Truitt in The Folk Harp Journal #28, page 4: “Did
you know? In 1619 Pretorius gave the ‘concert pitch’ of Northern Germany as A =
567. Schnitger’s organ in S. Jacobi in Hamberg (1688) was tuned A = 489.
Silbermann’s organ in Strassburg Cathedral, Germany (1713) had a pitch A =
393. At Trinity College, Cambridge, Father Smith’s organ was tuned (1752) to A
= 522.5. By1859 the French government’s commission of musicians and
physicists advised a standard pitch, A = 435. The Covent Gardens Orchestra in
1879 was playing a pitch of A = 450. At the same time America’s ‘concert pitch’
went as high as A = 461.6. Now in America A= 440, while in England the
standard of C = 522 causes A = 438.9.” Plus this may not be true as there is no
way that they could have measured frequency at this point in history. But
anyway, this points out one of the problems with using letters to indicate a
note ... one has no information as to what frequency this symbol represents. Is
an A equal to 440 or 432 vibrations per second.
Size of Commas
A musical comma is the difference between two sets of Just intervals or
combinations of Just intervals. A comma always creates wolf tones. A comma
always prevents an instrument from having all its tones being Justly in tune at the
same time. The common ratio Just intervals are never multiples of each other. If
one Just interval is not a multiple of another Just interval, then the sum of the
series of one kind of interval will never equal the sum of any other kind of Just
interval series no matter how far into infinity both series might be extended. One
proof: 2:1 is the ratio for an octave (Just or otherwise); 3:2 is the ratio for a Just
fifth; given A and B are whole numbers (arbitrary steps in a series) so (2/1)**A =
(3/2) **B , then 2**(A+B) = 3**(B); but this is a contradiction, because 2 raised to
any integer power has to be an even number and 3 raised to any integer power
must be odd. There are an infinite number of possible musical comma.
Name
Aron Comma
Didymic (Syntonic, Ptolemic) Comma
Diatonic (Pythagorean) Comma
Harrison Comma
Javanese Comma
Jorgenson Comma
Kirnberger Comma
Kirinberger plus Schisma Comma
Salinas Comma
Schisma (Comma)
Verheijen Comma
Size in Cent
74
22
24
51
90
63.15
26
28
50
2
98