Dividing the octave into 55 equal parts creates units of 21.818c, each of which is reasonably close to the syntonic comma at 21.506c. Since the diatonic (major) and chromatic (minor) semitones of 1/6 comma meantone differ by about a syntonic comma, it is possible to see the diatonic semitone as consisting of 5 units (or commas) and the chromatic semitone as consisting of 4 units (or commas).
How this can approximate a “fully extended 1/6 comma” may be seen in the following: | ||||||
|
||||||
The Appendix to the Haynes article gives several relevant quotations, either specifically mentioning the 55 division or speaking of the comma difference between major and minor semitones. |
||||||
The intervals of 1/6 comma may be compared to the 55 division as follows: | ||||||
1/6 comma | 55 division | |||||
Chromatic semitone | 88.55 | 87.27 | ||||
Diatonic semitone | 108.18 | 109.09 | ||||
Whole tone | 196.73 | 196.36 | ||||
Minor 3rd | 304.91 | 305.45 | ||||
Major 3rd | 393.46 | 392.72 | ||||
4th | 501.64 | 501.81 | ||||
5th | 698.36 | 698.18 | ||||
Other regular meantone systems have their octave division approximations as well: | ||||||
1/3 comma meantone equates with the 19 division (this explains Salinas’ championing of so-called “equal temperament”in 1577: equal, but with 19 rather than 12 notes to the octave). | ||||||
1/4 comma meantone equates with the 31 division (this corresponds to tuning directions given by Vicentino (1555) for his “archicembalo,”and to the “enharmonic harpsichord” of Trasuntino (Venice, 1606, now in Bologna), and creates PM3rds above and below every note). | ||||||
1/5 comma meantone equates with the 43 division (this is discussed by some 18th-century theorists). |