Download Pythagorean whole tone - Jacobs University Mathematics

Scales
Roederer, Chapter 5, pp. 171 – 181
Cook, Chapter 14, pp. 177 – 185
Cook, Chapter 13, pp. 150 – 152, 157 – 162
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Definition
(For a purely practical purposes)
 A scale is a discrete set of pitches arranged in
such a way as to yield a maximum possible
number of consonant combinations (or
minimizes possible number of dissonance)
when to or more notes of the set are sounded
together.
 (Here a scale is a set of tones with mathematically defined
frequency relationships. This is to distinguish from the various
scale modes, defined by the particular order in which whole
tones and semitones succeed each other.)
A.Diederich – International University Bremen – USC – MMM – Spring 2005
 The solfeggio notation do-re-mi-fa-sol-la-ti-do
is used in the following to indicate relative
positions on the scale (i.e, croma), not actual
pitch.
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Just scale, first steps
The notes do-mi-sol
constitute the major third,
the building stone of
Western music harmony.
Consonant interval
Consonant interval
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Just diatonic scale
Continuing to "fill in" tones, in each step trying to
keep the number of dissonance to a minimum an the
number of consonance to a maximum, yields
A.Diederich – International University Bremen – USC – MMM – Spring 2005
 The intervals 9/8 and 10/9 define whole tones.
 9/8: just diatonic major whole tone
 10/9: just diatonic minor whole tone
 The interval 16/15 defines a semitone.
 8 notes – 28 possible pairs
 16 consonant intervals
 10 dissonant intervals
 2 "out-of- tune" consonances
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Within the just diatonic scale we can form
 three just major triads
 do –mi – sol
 do – fa – la
 re – sol – ti
 two just minor triads
 mi – sol – ti
 do – mi – la
 one out-of-tune triad
 re – fa – la
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Pythagorean scale
 The Pythagorean scale is build up from the so-called
perfect consonances, the just fifth, the just fourth, and
the octave.
 Only one whole tone interval, the Pythagorean whole
tone (9/8)
 Pythagorean diatonic semitone (265/243)
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Problems with these scales
 Only a very limited group of tonalities can be
played with these scales without running into
trouble with out-of-tune consonances. That is
 Both scales impose very serious transposition
and modulation restrictions. (recognized in the
17th century)
 The type of music that can be played is
extremely limited.
A.Diederich – International University Bremen – USC – MMM – Spring 2005
The equally tempered scale
 In the tempered scale the frequency ratio is the
same for all 12 semitones lying between do
and do'.
 Call this ratio s.
 This is the frequency ratio for a tempered
semitone.
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Frequency ratios and values in cents of
musical intervals
A.Diederich – International University Bremen – USC – MMM – Spring 2005
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Example: C major triads in
temperaments
1.
2.
3.
4.
5.
6.
7.
8.
just temperament
mean tone in C
just temperament
equal temperament
just temperament
mean tone in C
mean tone in C#
mean tone in C
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Track 61
Other scales, e.g., the Bohlen Pierce
Scale
Track 62
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Mel scale
 A psychological scale for pitch is the mel scale
proposed by Stevens, Volkman, and Newman
(1937).
 A unit of that scale – a 1000 Hz tone at 40 dB
has a pitch of 1000 mel.
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Relation between Pitch and Frequency
Pitch of 1000 Hz at 40 dB : 1000 mel
(mels = 2410 log[1.6 ¢ 10-3 f +1])
A.Diederich – International University Bremen – USC – MMM – Spring 2005
A piano keyboard normalized to mel scale, that is the
keyborad is warped to match steps which are equal
"distance" on the mel scale
A.Diederich – International University Bremen – USC – MMM – Spring 2005
 At least two different kinds of pitch: the mel scale
measurements and the musical pitch
 The perceived difference between two notes decreases at the
extreme ends of the keyboard.
 Pitches, and differences between them, are not as clear at low
and high frequencies.
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Scales with equal steps on the mel
scale
 Example:
 Down the chromatic mel scale
 Diatonic mel scale in the midrange
 Diatonic mel scale in high range
 Diatonic mel scale on low range
Track 48
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Mel scale music
 Tune in middle mel scale
 The same but in lower mel scale
 Original Bach tune
 Mel version of Bach's tune
Track 49
A.Diederich – International University Bremen – USC – MMM – Spring 2005
 The mel scale is not an appropriate musical
scale.
 Experiments are usually done with pure sine
tones and other sounds that had no standard
musical interval relationships.
 With musical material the psychological scale
resulted in a log frequency scale within the
common musical frequency range.
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Separating height from chroma
 Two components of
pitch:
1 octave
 the pitch height (vertical
position on the pitch
helix)
 the chroma (position
within an octave around
the cylinder
 Either of these
components can be
suppressed
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Example
 Pitches are rising by major seventh, but can be
heard going down chromatically
 Pitches are falling by seventh, but chroma is
rising by a major scale.
Track 50
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Example:
Suppress chroma while leaving height
(To construct: pass noise through a band-pass
filter. The center frequency of the filter
determines the perceived height of the sound)
Track 51
 height only, without chroma change, noise
 height only, without chroma change, string timbers
 height only, no change of chroma, sine tones
 noise example
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Suppress height while leaving chroma
 More difficult: Construct tones with an
ambiguous spectrum, e.g., with harmonics
lying only on octaves, and with a spectrum that
decreases from a maximum near the center to
zero at extremely low and high frequencies.
 After ascending or descending a complete
octave, the final tone is located at the same
place where the scale began.
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Example
Shepard tritone paradox
 tritone can be heard going up or down
 diminished thirds going upward give the context of
"upwardness"
 tritones: now likely heard as going up
 diminished thirds going downward give the
context of "downwardness"
 tritones: now likely heard as going down
A.Diederich – International University Bremen – USC – MMM – Spring 2005
Track 52
Confusing chroma and height
 If chroma underlies pitch, can a melody be
scrambled in terms of height while retaining
chroma?
 If this could be done, would the melody still
recognizable?
 It would seem that if the listener were able to
attend only to chroma and ignore height, the
answer would be "yes".
A.Diederich – International University Bremen – USC – MMM – Spring 2005
 Our perception of melody
actually depends quite
critically on height as well
on chroma.
 Melodies scrambled in
height while retaining
chroma are not readily
recognized.
 A melody constructed with
incorrect chroma, but with
the right shape and height
contour, can be more
recognizable than one that
has been completely
randomly height scrambled
Track 53
A.Diederich – International University Bremen – USC – MMM – Spring 2005