Giambattista Doni's
Composizione per il Diatonico EquabileIn 1635 Giambattista Doni (1594-1647) published a trio for viols (not necessarily his own composition) employing Ptolemy's even diatonic ("diatonico equabile") scale, which uses the tetrachord 12/11, 11/10, 10/9. The steps in this scale run 151 cents, 165, 182, with the usual 9/8, 204 cents between the two tetrachords, so that there are virtual (though justly-tuned) quarter-tones. Written for fretted viols, with the mode transposed to different strings, the entire pitch gamut, with E as tonic, is as follows:
Tetrachord Pitch Ratio to E Cents 1 A 4/3 498 Bv 16/11 649 C 8/5 814 D- 16/9 996 2 E 1/1 0 F#v 12/11 151 G 6/5 316 A 4/3 498 3 B 3/2 702 C#v 18/11 853 D 9/5 1018 E 2/1 1200 Notice that only the pitches F#v, Bv, and C#v are quarter-tones. In his score Doni did not distinguish which C's are 8/5 and which are 18/11, nor did he distinguish between the two D's, 16/9 and 9/5, but instructed the players to use whichever form was most consonant (by fifths) with the simultaneous other pitches. The score contains four sizes of neutral (between major and minor), undecimal (eleven-based) thirds:
11/9 being 347.4 cents (C#v/E, Bv/D, F#v/A)
40/33 being 333 cents (D-/F#v, G/Bv)
27/22 being 354.5 cents (D/F#v, A/C#v)
99/80 being 368.9 cents (D/Bv)I provide a virtual harpsichord realization here, and the score (of my version, with my choices of C and D) in Ben Johnston's pitch notation.
Source: Patrizio Barbieri, Enharmonic: Instruments and Music 1470-1900 [Rome: Il Levante Libreria Editrice, 2008], pp. 180-184
If you find any problems with the examples, e-mail me here
