Septimal meantone temperament

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In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the

Choice of temperament

Meantone temperament produces a frequency ratio of approximately 5 by means of four fifths, so that the major third (for instance C–E) is obtained from two successive whole tones.

Septimal meantone produces 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♭

C	G	D	A+	E+	B+	F♯++	C♯++	G♯++	D♯++	A♯+++
C	≈G	≈D	≈A+	≈E+	≈B+	≈F♯++	≈C♯++	≈G♯++	≈D♯++	=B♭

There are several slightly adjusted meantone tunings very nearly equivalent to exact

Temperament goal	Temperament name	Size of fifth	Audio example
pure 5:4 intervals	quarter-comma meantone	696.58 cents	Playⓘ
pure 7:4 intervals	modified quarter tone	696.88 cents	Playⓘ
uniform intervals in every key	31 equal temperament	696.77 cents	Playⓘ

31 equal temperament does excellently for both of the others: It has augmented sixths only 1.1 cent flat of a purely harmonic seventh, and its major third is only 1.2 cents sharper than a pure fifth harmonic; while the fifth is 5.2 cents flat from the third harmonic. The differences are so small that the distinction is mainly academic.^{[citation needed]}

Theoretical properties

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

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, C–E♭–G♭–A, C–E♭–F♯–A or C–D♯–F♯–A. It also has a septimal augmented triad, which in various inversions can be C–E–G♯, C–E–A♭ or C–F♭–A♭. It has both a dominant seventh chord, C–E–G–B♭, and an otonal tetrad, C–E–G–A♯; the latter is familiar in common practice harmony under the name German sixth. It likewise has utonal tetrads, C–E♭–G–B

11-limit meantone

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

Footnotes

References

External links

• "Composing in meantone". Xenharmony. Archived from the original on 1 April 2007.
• Terpstra, Siemen. "Towards a theory of meantone (and 31 ET) harmony". Stichting Huygens-Fokker (huygens-fokker.org).