Dorian TTS: virile and bellicose

Phrygian TST: agitated and Bacchic

Lydian STT: intimate and lascivious

sunhmm¡na ==> diezeugm¡na

(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

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 ratios³               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 5^th                                      *10/11

3/4 perfect 4^th                                      *12/13

4/5 major 3^rd

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

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

Other harmonists:

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

Octave:

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

 (maj3^rd) C-E > 4/5 (Pyth. ‘just’ ratio)      

           12 5^ths > 7 octaves

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

Mean-tone: 8 perfect keys; 4 badly mistuned

Werkmeister: 4 5^ths narrowed by ¼ comma (1691); best 3^rds F-A & C-E; good G-B, D-F#,   Bb-D; worse Eb-G, A-C#, E-G#, B-D#. All other 3^rds 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. Bach’s Wohltemperierte system (1722) of 7 perfect 5^ths and 5 tempered 5^ths allows comfortable modulation into all 24 keys. cf. http://ha.kellner.bei.t-online.de/

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

e.g.

  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 maj7^th 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 Bach’s day but wasn’t 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 12^th root of 2.

Pythagoras’s Tuning Experiments:

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

¹ [399a] Plato, Republic

ŽllŒ kinduneæei soi DvristÜ leÛpesyai kaÜ FrugistÛ. oék oäda, ¦fhn ¤gÅ, tŒw �rmonÛaw, ŽllŒ kat‹leipe ¤keÛnhn t¯n �rmonÛan, ¶ ¦n te polemik» pr‹jei öntow ŽndreÛou kaÜ ¤n p‹sú biaÛÄ ¤rgasÛ& prepñntvw ’n mim®saito fyñggouw te kaÜ prosÄdÛaw.

[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.

² [398e] ibid.

tÛnew oïn yrhnÅdeiw �rmonÛai; l¡ge moi: sç gŒr mousikñw. MeijoludistÛ,    ¦fh, kaÜ SuntonoludistÜ kaÜ toiaètaÛ tinew. oékoèn aðtai, ·n d' ¤gÅ,  Žfairet¡ai; �xrhstoi gŒr kaÜ gunaijÜn “w deÝ ¤pieikeÝw eänai, m¯ ÷ti Žndr‹si.

[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.

³Recounted by Plato [36b] Timaeus ²miolÛvn d¢ diast‹sevn kaÜ ¤pitrÛtvn kaÜ ¤pogdñvn genom¡nvn ¤k toætvn  tÇn desmÇn ¤n taÝw prñsyen diast‹sesin,

[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. Bärenreiter-            Verlag Kassel, 1965.

http://ha.kellner.bei.t-online.de/ 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.