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Commas

We saw the first concept of a comma when I mentioned a few pages back that two Pythagorean semitones do not equal up to a Pythagorean whole tone:

Thus two semitones fall short of reaching the same pitch as a 9/8 whole tone.

This notion of commas is an important one as regards intonation. This is a vast topic and can get very complex and convoluted. For our purposes, I wish you only to understand the basic idea of three commas: The Pythagorean comma, the Syntonic (also called the Dydimic) comma, and the "Great Diesis." Colorful names for the following ideas:

The Pythagorean Comma

The Pythagorean Comma is most easily expressed by stating that no power of two can equal any power of 3. If take a very low tone, represented by the lowest C on the piano and assign that the value of 1, and go up to the top C on the piano, we encompass 7 octaves, and since octaves are expressed as the ratio of 2:1, we find ourselves with the number 128, which is 2 to the seventh. If we go back to that low C and play up by perfect fifths on the keyboard, we of course go up the Cycle of Fifths...

...which for the purposes of Pythagorean thought should be notated as all sharp notes, ending on B#. In equal temperament, of course, the B# becomes the enharmonic equivalent of C natural, but in Pythagorean tuning we need to think in all perfect fifths. Thus, we must go up in perfect fifths 12 times. This would be 3/2 raised to the 12 power (1.5 to the 12th). This results in the number 129.74632.

This ratio, 129.75632/128 is the Pythagorean comma. 12 perfect fifths do not equal up to 7 perfect octaves. This is the basic process in tuning equal temperament, where each fifth is reduced by a twelve of a semitone tone, thereby distributing out the comma equally among the twelve pitches of the chromatic scale.

Since symmetry is a big part of our work this semester, it's also useful to look at this cycle in both ascending and descending directions, so that the comma is seen to occur between the low Gb and the high F#:

This shows that we would get the same result, the Pythagorean comma, if we went outwards by perfect fifths in both directions and reached the furthest limit, with F# above and Gb below. The F# frequency and the Gb frequency would not be in a relationship of some power of 2 to each other, thus of course not resulting in a pleasing and perfectly consonant interval, but instead would be off by the distance of the comma.

The Syntonic (Didymic) Comma

The Pythagorean comma is important to understand, because through understanding it, we come to realize both the limits and possibilities of our current musical system, where Gb and F#, for example, are considered by enharmonic equivalents. But from a practical standpoint, it is only important to piano tuners, who have to detune the perfect fifths by precisely 1/12th of this Pythagorean comma in order to distribute the difference between the all the steps of the chromatic scale. This is what we call equal temperament and is the compromise that we are all accustomed to in the sound of the modern piano.

The Syntonic/Didymic comma is the more critical issue, as it directly effects our everyday music making and ability to tune in ensemble playing. It refers to the diffence between a major third that is tuned by perfect fifths with octave reduction, as opposed to the major third based on the ratio of 5:4, which is what also appears in the harmonic series.

The math here is simple. We've already seen that the Pythagorean major third is 81/64. Study the math in the previous page, if this is not clear. Not only was the third from C to E tuned in this way, but the major thirds between F and A and between G and B were also found to have this ratio.

Just intonation,to review, is based on the principle that these thirds sound somehow "out of tune" when performed in harmony with one another (as opposed to their appearance in a strictly melodic situation). This is because it is literally true. When a string vibrates, it generates the pure third above the fundamental, and the pure fifth. Not only does nature seem to provide for this pure interval, it also relates to the inner workings of our ear, where harmonic ratios that reflect low-integer relationships are more easily processed within the ear. The process is to set the major triads that occur within the major scale (we think of them of course as tonic, dominant and subdominant) as possessing the ratio of 4:5:6, in this way:

C

D

E

F

G

A

B

C

D

4

:

5

:

6

 

 

 

 

 

 

 

4

:

5

:

6

 

 

 

 

 

4

:

5

:

6

Since all the major thirds become "pure", the ratios work out to be:

1

2

3

4

5

6

7

8

1/1

9/8

5/4*

4/3

3/2

5/3**

15/8***

2/1

C

D

E

F

G

A

B

C

9/8

10/9

16/15

9/8

10/9

9/8

16/15

 

So what is the Syntonic or Didymic comma?: it is the difference between the Pythagorean third, which is 81/64, and the Just Third, which when expanded out to yield a common denominator, gives us 80/64 (5/4 * 16/16). Thus the ratio 81/80 is the dydmic comma, the slight but perceptable difference in intonation between Pythagorean and Just thirds. The sound of the beats created by playing two sounds that vibrate at a ratio of 81:80 to each other is the sound of the Syntonic comma.

Arcane? Irrelevant? Musically geeky beyond all belief? Maybe, but when you come to really hear the difference between the justly tuned third and the various "other thirds," whether Pythagorean or Equally tempered, you will come to appreciate the difference. It's one of my goals this semester to provide you with that experience.

The Great Diesis

The last tuning concept to grasp is related to the Syntonic comma. The Great Diesis results from the fact that three major thirds, tuned in just intonation, do not add up to a perfect octave:

In musical notation, we can see it as the difference between C to C and the difference between C to E, followed by E to G#, followed by G# to B#, which in equal temperament is equal to C, but in pure Just intonation, is slightly lower.