Download “Tuning and temperament” by Rudolph Rasch Know who and where

Reading guide
Christensen, ch. 7: First Part
“Tuning and temperament” by Rudolph Rasch
Know who and where Rasch is.
Preparation:
?
p. 193
¶0 “quantitative language” = mathematical language
“arithmetic”: When used as an adjective, this word is accented on the third syllable:
ar-ith-MET-ic. Arithmetic methods include addition, subtraction, multiplication, and
division only.
¶1 “root extraction”: not dentistry! This means finding square roots, cube roots, twelfth roots,
etc.
What can be described by root extraction?
Note: We use “tuning” to mean “setting the pitches of an instrument,” no matter what system we
use to tune the instrument. In technical terminology, “tuning” can mean only one system
(or set of systems) for setting pitches. “Temperament” is a different system, a system that
is not properly called “tuning” in this terminology. (See p. 201, ¶1, where Rasch finally
makes this point.) In musical terms, what does temperament mean?
“This was possible with arithmetic methods in exceptional cases only.”: Only when the original
ratio involved two perfect squares (or cubes, etc.), as in 9:4. (9:4 = 3:2 times 3:2, so what
interval is produced by this ratio?)
“Psychologically more realistic picture”: Whether or not you understand logarithms or not, at
least each interval is always represented by the same number, and the larger of two
intervals intervals is represented by what is clearly the larger of two numbers. The system
is much better than ratios in this regard. Quick! Which is larger? 7:4 or 16:9?
¶2 “frequency values”: as in number of vibrations per second. These ratios work the same as
string lengths, only inverted. In a 2:1 octave, the longer string (corresponding to the 2)
makes the lower pitch. If 2:1 represents frequencies, the ratio still produces an octave, but
the higher frequency (corresponding to the 2) produces the higher pitch.
p. 194
¶1 During the days of the beginning of the scientific method, mathematical methods were invented
that could describe and suggest all the various methods of tuning and temperament that
came before and after. Rasch uses what two centuries to represent all the theories of
tuning and temperament?
¶2 What is the standard book on tuning and temperament?
¶3 Just intonation (which will be explained more fully later in the chapter) is based on the
Chapter 7a – p. 2
introduction of what single interval? (Do you remember the key interval in Pythagorean
tuning?
p. 195
¶0 What are the two basic types of temperament?
¶2 “little was lost”: notice that practicality comes into play here as theorists begin to recognize
that “close” is good enough for the human ear. (Notice also the word “rounding” in ¶0.)
¶3 Most authors in the period discussed in this chapter still use what instrument to discuss tuning?
By the Renaissance, even when ratios are used to locate points on the monochord, these points
begin to be identified not by a ratio of small integers, but of large integers. If the whole
string is represented by 10,000, the octave is placed at 5,000, and the fourth at 7,500.
What other common numbers are used for the whole string? What do you think is the
advantage of using a single large number as a reference?
p. 196
¶1 Now you can see why Glarean used 24 × 36 for his full string: it makes it easy to find several
Pythagorean (3:2) fifths.
¶2 “Sharps do not have a place . . . .”: In Pythagorean tuning, enharmonic spelling makes a big
difference. Find the number corresponding to B on Glarean's chart. Use some math (use
a calculator if you must) to find 2/3 of that number; this should be the number for F#.
How does it compare to Glarean's number for Gb?
¶3 “Chromatic” and “enharmonic” here refer to the tetrachord genera from ancient Greek theory.
We haven't stressed these in class because they didn't survive into more familiar musical
styles. But, as you can see, theorists continued to talk about them many hundreds of years
after they quit being used. The beginning of ¶5 should assure you that this part of
Glarean's system is not essential to understanding Pythagorean tuning or Glarean's
depiction of it.
¶5 “The fifths are just”: i.e. “in tune with nature,” in this case, 3:2. Rasch uses the word “true” to
mean the same thing later in the paragraph.
What is the ratio of the syntonic comma? What two intervals differ by a syntonic comma?
“Cents”: Rasch jumps the gun here and uses cents to measure intervals. The cent is a onehundredth of a tempered half step. It is expressed by the number 1; thus the half step is
represented by the number 100. Look at the size in cents of the Pythagorean major third
(about 407.820) and of the just or “true” major third (about 386.314). This means that
the first is about 7.8% larger than four equal-tempered half steps and that the second is
about 14% smaller four equal-tempered half steps. Astonishingly, it takes logarithms to
provide us with these amazingly intuitive numbers.
What is the size of the syntonic comma in cents?
Chapter 7a – p. 3
p. 198
¶0 What are the two shortcomings of Pythagorean tuning (shortcomings in the context of the
harmonic, chromatic music of 17th- and 18th-century Europe)?
What is the ratio of the ditonic comma? What is it the difference between? What is its size in
cents?
¶1 Why was Pythagorean tuning acceptable during the Middle Ages? (In other words, why
weren't the shortcomings so noticeable?
Why did it again receive interest (apart from any possible historical interest) starting in the
nineteenth century?
¶2 What two intervals form the basis for just intonation?
We've already discussed Ramis's introduction of this tuning. What Spanish theorist developed
a just system in 1577?
¶3 Salinas constructs his scale from e to e. Why do you think he did? (Hint: note on the chart
that he classifies notes as diatonic, chromatic, and enharmonic.)
Why does he have two tunings for D? (Look at the diagram on p. 199 as well.)
p. 200
We'll write this scale on the board with both ratios and large numbers. (Look at the small
numbers on the woodcut: they appear to be the same as the ones on p. 199 but divided by
20!) The description tells where all the commas and dieses are, but more important is
where the harmonic fifths and thirds are! So we'll try to find them in class. In the mean
time, study the woodcut a little bit. Can you find the arches that say “COMMA”?
pp. 200-201
The system is impractical for the keyboard with twelve keys per octave. But some keyboards
were made at this time with split keys for the chormatic notes, so you could tune D< to go
well with B and tune E= to go well with G.