Septimal kleisma

16:15 diatonic semitone.

16:15 diatonic semitone  Play (help·info).

15:14 major diatonic semitone.

15:14 major diatonic semitone  Play (help·info).

In music, the ratio 225/224 is called the septimal kleisma ( play (help·info)).^[1] 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 (5/4)² (9/7). That says that it is the amount that two major thirds of 5/4 and a septimal major third, or supermajor third, of 9/7 exceeds the octave. If the septimal kleisma is tempered out, as it is for instance in miracle temperament, septimal meantone temperament and in most equal temperaments, for example 12, 19, 22, 31, 41, 53, 72 or 84 equal, then an augmented triad consisting of two major thirds and a supermajor third making up an octave becomes possible.^{[citation needed]} The existence of such a chord, which might be termed the septimal kleisma augmented triad, is a significant feature of a tuning system.^{[citation needed]}

The septimal kleisma can also be viewed as the difference between the diatonic semitone (16:15) and the septimal diatonic semitone (15:14).

See also

• Septimal meantone temperament
• Miracle temperament
• 72 equal temperament

References

1. ^ Haluska, Jan (2003). The Mathematical Theory of Tone Systems. CRC Press. p. xxvii. ISBN 0824747143.