In music,
septimal meantone temperament, also called
standard septimal meantone or simply
septimal meantone, refers to the tempering of
7-limitIn music theory, limit is any of a variety of methods used to characterize the harmonies found in a piece of music, genre of music, or by extension, the harmonies that can be made with a particular scale or class of scales. The term was introduced by Harry Partch, who used it to give an upper...
musical intervals by a
meantone temperamentMeantone 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 a meantone, each fifth is narrowed by the same amount relative to its width in Just Intonation...
tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as
19 equal temperamentIn 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 63.158 cents...
, with
31 equal temperamentIn music, 31 equal temperament , which can be abbreviated 31-TET, 31-EDO, 31-ET, 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 thirdA major third is one of two commonly occurring musical intervals that span three diatonic scale degrees, the other being the minor third. It is denoted 'major' because it is the larger of the two: the major third is a leap of four semitones, the minor third three. The major third is abbreviated...
, for instance C-E, is obtained from two tones in succession.
In music,
septimal meantone temperament, also called
standard septimal meantone or simply
septimal meantone, refers to the tempering of
7-limitIn music theory, limit is any of a variety of methods used to characterize the harmonies found in a piece of music, genre of music, or by extension, the harmonies that can be made with a particular scale or class of scales. The term was introduced by Harry Partch, who used it to give an upper...
musical intervals by a
meantone temperamentMeantone 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 a meantone, each fifth is narrowed by the same amount relative to its width in Just Intonation...
tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as
19 equal temperamentIn 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 63.158 cents...
, with
31 equal temperamentIn music, 31 equal temperament , which can be abbreviated 31-TET, 31-EDO, 31-ET, 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 thirdA major third is one of two commonly occurring musical intervals that span three diatonic scale degrees, the other being the minor third. It is denoted 'major' because it is the larger of the two: the major third is a leap of four semitones, the minor third three. The major third is abbreviated...
, 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 intervalThe 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 either 16:9 or 9:5, or an...
in septimal meantone.
The meantone tuning with pure 5/4 intervals is
quarter-comma meantoneQuarter-comma meantone was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this tuning the perfect fifth is tempered by one quarter of a syntonic comma in order to obtain just major thirds...
, with a fifth of size 696.58
centsThe cent is a logarithmic unit of measure used for musical intervals. Typically cents are used to measure extremely small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is much too small to be heard between successive...
. Similarly, the tuning with 7/4 intervals pure has a fifth of size 696.88 cents ; there is very little difference between them, and in general no discernible difference between an optimal 5-limit and an optimal 7-limit meantone tuning. The fifth of
31 equal temperamentIn music, 31 equal temperament , which can be abbreviated 31-TET, 31-EDO, 31-ET, is the tempered scale derived by dividing the octave into 31 equal-sized steps...
, of size 696.77 cents , does excellently for both of them.
Theoretical properties
Septimal meantone tempers out not only the
syntonic commaIn music theory, the syntonic comma , also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
of 81/80, but also the
septimal semicommaIn 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 kleismaIn music, the ratio 225/224 is called the septimal kleisma .Another name for it is the marvel comma, since the temperament tempering it out is sometimes called the marvel temperament, ....
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 harmonyThe common practice period, in the history of European art music , spanning the Baroque, Classical, and Romantic periods, lasted from about 1600 until about 1900.-General characteristics:...
under the name
German sixthAn augmented sixth chord contains the interval of an augmented sixth above its "root". 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....
. It likewise has utonal tetrads, C-E-G-B, which in
the arrangement B-E-G-C becomes Wagner's
Tristan chordThe 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 second 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 temperamentIn music, 31 equal temperament , which can be abbreviated 31-TET, 31-EDO, 31-ET, is the tempered scale derived by dividing the octave into 31 equal-sized steps...
.
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