I am writing an essay on musical temperaments for which I was doing some research on the history of tuning and how it evolved over time. I understand the Pythagorean tuning system quite well now but I am confused about a few things and would love an explanation for them:

1) What is the difference between a diatonic and chromatic semitone in reference to Pythagorean tuning?

2) Why were these semitones used differently and notated differently?

3)Did they use it in one chromatic scale itself like we do on the keyboard instruments today or were diatonic and chromatic tunings entirely different and never combined into one scale? So for example you could either have a C to C# tuned as a limma or tuned as a apotome but not together on a fixed pitch instrument?

4) During the time of Pythagoras and several centuries after him, did they only use the diatonic and pentatonic scales? Didn't they use smaller intervals back then?

  1. Expressed as ratios, a diatonic semitone (limma) is 256:243, while the chromatic (apotome) is 2187:2048. They correspond to the difference between 5 pythagorean fifths and 3 octaves and 7 fifths and 4 octaves respectively.

  2. Well... They are different! As I said in the previous aswer: in PYTHAGOREAN TUNING, if you climb 5 fifths and USE OUR NOTES' NAMES, you get from Db up to C (C-Db, diatonic semitone, 256:243), but if you climb seven fifths from C, you get to C# (C-C#, chromatic semitone, 2187:2048)

  3. The greek pitch system is very different from ours, and trying to use modern concepts over it can cause a lot of confussions. In ancient greek music you wouldn't have "a C or C#" (see https://en.wikipedia.org/wiki/Musical_system_of_ancient_Greece#The_Pythagoreans). Their pitch space was built out of two pairs of juxtaposed tetrachords (four notes spanning a perfect fourth), where the two inside notes were movable. This produced three genera of tetrachords: diatonic (equivalent to our modern ST, T, T), chromatic (ST, ST, minor 3rd), enharmonic (1/4T, 1/4T, major 3rd), with the outer notes always spanning a pefect 4th (4:3). Beware that what I represent here as a "Tone" "minor third" etc. are the modern equivalents of what ancient greeks used: they are close, but they are not exactly the same. It has nothing to do with tuning or tempering; a tone could be 7:8, 8:9, etc.; a minor third 32:27, 6:5; etc., and the semitones, chromatic or diatonic etc.

And now your question: YES, there was one choice of tuning for the chromatic genus that used both kinds of semitones. It is the following, the traditional Pythagorean tuning of the chromatic genus (https://en.wikipedia.org/wiki/Genus_(music)):

    hypate   parhypate      lichanos                             mese
      4/3       81/64         32/27                               1/1
      | 256/243  |  2187/2048  |              32/27               |
    -498       -408          -294                                 0 cents

Hypate, parhypate, lichanos and mese are the name of the "notes" in the tetrachord, the mese and hypate spanning a fourth. If you "sum" (multiply, really) the three ratios, you get a fourth (4:3)

  1. Of course; they used the enharmonic genus, with intervals smaller than semitones; check the second reference in answer #3

Part of the distinction is just enharmonic spelling. A diatonic semitone is on two different staff positions (two different letters) such as C to Db. A chromatic semitone is the same spelling of the base note, such as C and C#.

Of course, in equal temperament the sound is the same, and the only distinction is in the functional meaning of the note (if even that), but in unequal temperaments they are different distances. In Pythagorean tuning, the diatonic semitone is 256:243 or 90.2¢ while the chromatic semitone is 2187:2048 or 113.7¢. The difference between those two is another instance of the Pythagorean comma.

  • I understand that but I am just having a hard time imagining these two non enharmonic intervals being used at once for a keyboard instrument. During the development of keyed instruments how did they make use of the limma and the apotome together? – user36492 Sep 16 '18 at 19:02
  • 'in unequal temperaments they are different distances": this is not quite correct. Even in an unequal temperament, the distance from C to C# is the same as the distance from C to Db, because C# and Db are the same note (unless you have a split key, of course, but those were fairly rare). – phoog Sep 17 '18 at 1:40

I'm not sure Pythagoras used these terms. The Pythagorean diatonic semitone is the difference between three octaves and five fifths: (2/1)^3=8/1 and (3/2)^5=243/32. The difference illustrates the impossibility of scale construction using only "perfect" intervals. This difference (and others, like the 81/80 problem tuning a guitar) shows the necessity of some sort of temperament.

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