Greek Music Theory By Will Bruce

Dorian TTS: virile and bellicose

Phrygian TST: agitated and Bacchic

Lydian STT: intimate and lascivious

sunhmmna ==> diezeugmna

(Terpander)         (Pythagoras)~also in use by the Egyptians centuries before.

Major Scale TTS (T) TTS

Dorian (T) Dorian

Nat.Minor Scale TST (T) STT

Phrygian (T) Lydian

Pythagorean Comma = 524288:531441

Difference between Ab and G# = 1/60

whereas solar year (360 days) off by 5 1/4 days creates a gap = 1/69.

Just Intervals are those whose frequencies stand in a simple numerical relationship to one another (Keller).


Some just interval ratios               Prime numbers larger than 5 are                    

according to Pythagoras:                   not used in Western music:

1/1 unison                                            *6/7

1/2 octave                                            *7/8

2/3 perfect 5th                                      *10/11

3/4 perfect 4th                                      *12/13

4/5 major 3rd

8/9 whole tone (2/3:3/4)

15/16 semitone (3/4:4/5)

Other harmonists:

Aristoxenus: Harmonika stoicheia (Harmonic elements)

Ptolemy: Harmonica (base-60 system [divisible by 2,3,4,5,6,10,12,15,20,30...] ~ prevalent in Babylonian mathematics and throughout the ancient world from 500BC-AD150)

Plato: Republic Bk.VIII


Perfect Fifth:

Just Intervals are suitable for monophonic diatonic chant.

Music which modulates into more distant keys encounters some peculiar problems.

 e.g.        C-D > D-E

 (maj3rd) C-E > 4/5 (Pyth. just ratio)      

           12 5ths > 7 octaves


Temperament (adjusting of intervals) therefore is necessary to play in all keys.

Mean-tone: 8 perfect keys; 4 badly mistuned

Werkmeister: 4 5ths narrowed by comma (1691); best 3rds F-A & C-E; good G-B, D-F#,   Bb-D; worse Eb-G, A-C#, E-G#, B-D#. All other 3rds are wolf i.e. cacophonously wide.

Pachebel uses 17 of 24 keys in a set of suites.

Fischer uses 19 in Ariadne Musica (1715). Organist is given the thread of Ariadne as a guide through the labyrinth of remote keys.

J.S. Bachs Wohltemperierte system (1722) of 7 perfect 5ths and 5 tempered 5ths allows comfortable modulation into all 24 keys. cf.

In Bachs system each key has a unique series of tonal relationships. Keys were differentiated in Baroque times by their color and mood:


  D-maj: festive                                        F#min: passionately tense

  E-maj: because of G# & D#, was         G-maj: quiet and thoughtful

              feared and rarely use              G-min: pathos

  E-min: masculine and powerful           A min: masculine, loaded w/ energy

F-maj & C-maj: neutral keys          Bb-min: dark, heavily veiled

  F-min: deep seriousness                      B-min: though a tense maj7th to C-maj, Bach gave it

A-maj: pastoral                                                  consecration in his Mass (BMV232)

Outside of Books I and II of Das Wohltemperierte Clavier Bach does not use:

            C#maj, Db-min, D#min, F#maj, Ab-maj, G#maj, Bb-maj.


Equal Temperament (the dividing of the Pythagorean comma into 12 making all tone relationships homogeneous) had been theorized in Bachs day but wasnt feasible before the advent of electronic tuning devices. In equal temperament, which is the norm today, there is no inherent difference between keys since all semitones are the same size. There are no pure (just) intervals except the octave. Based on the 12th root of 2.


Pythagorass Tuning Experiments:

utilized in the musical example: F-maj. 2-part Invention (BMV794) by J.S.B.

[399a] Plato, Republic

ll kinduneei soi Dvrist lepesyai ka Frugist. ok oda, fhn g, tw rmonaw, ll katleipe kenhn tn rmonan, n te polemik prjei ntow ndreou ka n ps bia rgas& prepntvw n mimsaito fyggouw te ka prosdaw.

[399a] but it would seem that you have left the Dorian and the Phrygian. I don't know the musical modes, I said, but leave us that mode that would fittingly imitate the utterances and the accents of a brave man who is engaged in warfare or in any enforced business.

[398e] ibid.

tnew on yrhndeiw rmonai; lge moi: s gr mousikw. Meijoludist,    fh, ka Suntonoludist ka toiata tinew. okon atai, n d' g,  fairetai; xrhstoi gr ka gunaijn w de pieikew enai, m ti ndrsi.

[398e] are the dirge-like modes of music? Tell me, for you are a musician. The mixed Lydian, he said, and the tense or higher Lydian, and similar modes.These, then, said I, we must do away with. For they are useless even to women who are to make the best of themselves, let alone to men.

Recounted by Plato [36b] Timaeus miolvn d diastsevn ka pitrtvn ka pogdvn genomnvn k totvn  tn desmn n taw prsyen diastsesin,

[36b] And whereas the insertion of these links formed fresh intervals in the former intervals, that is to say, intervals of 3:2 and 4:3 and 9:8.

Modern sources:

Bukofzer, Manfred F. Music in the Baroque Era. New York: W.W. Norton & Co., 1947.

Harnoncourt, Nikolaus. Baroque Music Today: Music as Speech. Portland: Amadeus             Press, 1982.

Keller, Hermann. Das Wohltemperierte Clavier von Johann Sebastian Bach. Brenreiter-            Verlag Kassel, 1965. an excellent website giving a detailed bearing plan for the Baroque tuning of harpsichords created by Dr. Herbert Anton Kellner and assimilated from his book: Wie stimme ich selbst mein Cembalo? Schriftenreihe Das Musikinstrument, Heft 19, Verlag Das Musikinstrument, E. Bochinsky, Frankfurt / Main, 31986, ISBN 3-923639-68-6

McClain, Ernest, G. The Pythagorean Plato: Prelude to the Song Itself. York Beach:                                            Nicolas-Hays, Inc., 1978.