Septimal meantone temperament
In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal temperament, with 31 equal temperament being a more or less optimal tuning for both the 5- and 7-limits. Meantone temperament represents a frequency ratio of approximately 5 by means of four fifths, so that the major third, for instance C-E, is obtained from two tones in succession. Septimal meantone represents the frequency ratio of 56 (7*23) by ten fifths, so that the interval 7:4 is reached by five successive tones. Hence C-A♯, not C-B♭, represents a 7:4 interval in septimal meantone.
- A♯+++ ≈ B
- A♯+++ ≈ B
- C — G — D — A+ — E+ — B+ — F♯++ — C♯++ — G♯++ — D♯++ — A♯+++
- C — ≈G — ≈D — ≈A+ — ≈E+ — ≈B+ — ≈F♯++ — ≈C♯++ — ≈G♯++ — ≈D♯++ — =B
The meantone tuning with pure 5:4 intervals (quarter-comma meantone) has a fifth of size 696.58 cents
Septimal meantone tempers out not only the syntonic comma of 81:80, but also the septimal semicomma of 126:125, and the septimal kleisma of 225:224. Because the septimal semicomma is tempered out, a chord with intervals 6:5-6:5-6:5-7:6, spanning the octave, is a part of the septimal meantone tuning system. This chord might be called the septimal semicomma diminished seventh. Similarly, because the septimal kleisma is tempered out, a chord with intervals of size 5:4-5:4-9:7 spans the octave; this might be called the septimal kleisma augmented triad, and is likewise a characteristic feature of septimal meantone.
Chords of septimal meantone
Septimal meantone of course has major and minor triads, and also diminished triads, which come in both an otonal, 5:6:7 form, as for instance C-E♭-F♯, and an inverted utonal form, as for instance C-D♯-F♯. As previously remarked, it has a septimal diminished seventh chord, which in various inversions can be C-E♭-G♭-B
Septimal meantone can be extended to the 11-limit, but not uniquely. It is possible to take the interval of 11 by means of 18 fifths up and 7 octaves down, so that an 11:4 is made up of nine tones (e.g. C-E
- Composing in Meantone, Xenharmony. (archived version April 2007)
- Siemen Terpstra. "Towards a Theory of Meantone (and 31-et) Harmony". Stichting Huygens-Fokker.