MT 007.02 / SL 266.01
Ideas in Mathematics: The Grammar of Numbers

Exercise: Pythagorean chromatic intervals

Into the Pythagorean diatonic scale which we derived earlier, we have now introduced conceptual semitones with which to fill in the whole-tone [=major second] spaces
The exact ratio of these semitones needs to be established and that is, in fact, the purpose of this exercise.

If we view our new scale as:
c - c# - d - d# - e - f - f# - g - g# - a - a# [=b] - c'
this should enable us to redefine intervals exclusively in terms of the number of semitones which go to make them up.

semitone s_p (e.g. c - c#)

1 semitone
 

second [whole tone] 2_p (e.g. c - d, d - e)

2 semitones
 

minor third 3_p (e.g. e - g, a - c', c - d#/e^b)

3 semitones
 

major third III_p (e.g. c - e)

4 semitones
 

fourth IV_p (e.g. c - f)

5 semitones
 

fifth V_p (e.g. c - g, d - a)

7 semitones

So you have the following tasks to check out:

1. The interval f# - b behaves like a fourth (IV_p). Calculate the interval using the b (of known value) as a reference point.
You already know the ratio c - b in a Pythagorean scale (243:128), from which you need only subtract the ratio for a fourth (4:3) in order to get the ratio for the interval c - f#.
How does this fourth f# - b compare with the interval for a natural fourth?

2. Now that you know where f# is on the scale, calculate the ratio for the semitone f# - g. This should come out to the Pythagorean (diatonic) semitone which you have seen before. But you must prove this.

3. Calculate the interval for the semitone f - f#.
Is this resulting interval, known as the Pythagorean chromatic semitone, wider or narrower than a Pythagorean diatonic semitone?

Two different semitones? Uh-huh, but at least it's better than having two different whole tones (seconds), right? [Rhetorical question]

4A. If I go from c to c' in whole steps (9:8), then after six (6) whole steps I will arrive at b# = c':
c - d - e - f# - g# - a# - b#=c'
This scale, known as the whole-tone scale, will figure in some of our later work on geometrical tuning.
Calculate the ratio of such an interval derived from six whole steps (9/8)⁶ and contrast it with that of a normal octave. This difference should come out to equal the Pythagorean comma which you all know and love, i.e. the amount by which twelve fifths exceed(s) seven octaves. cf. Exercise 28, question 5

4b. Calculate the Pythagorean comma also as the difference between a Pythagorean chromatic semitone and a Pythagorean diatonic semitone. In order to make this work you must first identify which is the larger interval.