Septimal meantone temperament
Encyclopedia
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 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.
The meantone tuning with pure 5/4 intervals (quarter-comma meantone
) has a fifth of size 696.58 cents
. Similarly, the tuning with pure 7/4 intervals has a fifth of size 696.88 cents . 31 equal temperament
has a fifth of size 696.77 cents , which does excellently for both of them. Obviously, the difference is mainly academic.
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.
under the name German sixth
. It likewise has utonal tetrads, C-E-G-B, which in
the arrangement B-E-G-C becomes Wagner's Tristan chord
. It has also the subminor triad, C-D-G, which is otonal, and the supermajor triad, C-F-G, which is utonal. These can be extended to subminor tetrads, C-D-G-A and supermajor tetrads C-F-G-B.
.
Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
musical intervals by a meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal temperament
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
, with 31 equal temperament
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
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
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
, for instance C-E, is obtained from two tones in succession. Septimal meantone represents the frequency ratio of 56 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
Harmonic seventh
The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of...
in septimal meantone.
The meantone tuning with pure 5/4 intervals (quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
) has a fifth of size 696.58 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
. Similarly, the tuning with pure 7/4 intervals has a fifth of size 696.88 cents . 31 equal temperament
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
has a fifth of size 696.77 cents , which does excellently for both of them. Obviously, the difference is mainly academic.
Theoretical properties
Septimal meantone tempers out not only the syntonic commaSyntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
of 81/80, but also the septimal semicomma
Septimal semicomma
In music, the ratio 126/125 is called the septimal semicomma . It is also called the starling comma after its use in starling temperament, as well as the small septimal comma...
of 126/125, and the septimal kleisma
Septimal kleisma
In music, the ratio 225/224 is called the septimal kleisma .It is a minute comma type interval of approximately 7.7 cents. Factoring it into primes gives 2−5 32 52 7−1, which can be rewritten 2−1 2 . That says that it is the amount that two major thirds of 5/4 and a septimal...
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, 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 harmonyCommon practice period
The common practice period, in the history of Western art music , spanning the Baroque, Classical, and Romantic periods, lasted from c. 1600 to c. 1900.-General characteristics:...
under the name German sixth
Augmented sixth chord
In music theory, an augmented sixth chord contains the interval of an augmented sixth above its "root" or bass tone . This chord has its origins in the Renaissance, further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.-Resolution...
. It likewise has utonal tetrads, C-E-G-B, which in
the arrangement B-E-G-C becomes Wagner's Tristan chord
Tristan chord
The Tristan chord is a chord made up of the notes F, B, D and G. More generally, it can be any chord that consists of these same intervals: augmented fourth, augmented sixth, and augmented ninth above a root...
. It has also the subminor triad, C-D-G, which is otonal, and the supermajor triad, C-F-G, which is utonal. These can be extended to subminor tetrads, C-D-G-A and supermajor tetrads C-F-G-B.
11-limit meantone
Septimal meantone can be extended to the 11-limit, but not in a unique way. 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. The 11 is pure using this method if the fifth is of size 697.30 cents, very close to the fifth of 74 equal temperament. On the other hand, 13 meantone fourths up and two octaves down will also work, and the 11 is pure using this method for a fifth of size 696.05 cents, close to the 696 cents of 50 equal temperament. The two methods are conflated for 31 equal temperament31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...
.