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 succesive tones. Hence C-A#, not C-Bb, represents a 7/4 interval in septimal meantone. Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ... Meantone temperament is a system of musical tuning. ... A major third is the larger of two commonly occuring musical intervals that span three diatonic scale degrees. ...

The meantone tuning with pure 5/4 intervals is quarter-comma meantone, with a fifth of size 696.58 cents. Similarly, the tuning with 7/4 intervals pure has a fifth of size 696.78 cents; there is very little difference between them, and in general no discernible differnce between an optimal 5-limit and an optimal 7-limit meantone tuning. The fifth of 31 equal temperament, of size 696.77 cents, does excellently for both of them. Quarter-comma meantone was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. ...

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 meantone 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 augmented triad, and is likewise a characteristic feature of septimal meantone. 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. ...