108edo: Difference between revisions
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The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning. | The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning. | ||
Aside from tuning, 108 is the smallest number with a prime factorization of the form <math>p \ | Aside from tuning, 108 is the smallest number with a prime factorization of the form <math>p \exp p \cdot q \exp q</math>. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer. | ||
[[Category:valentine]] | [[Category:valentine]] | ||
Revision as of 18:59, 5 March 2019
The 108 equal division divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine temperament, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.
Aside from tuning, 108 is the smallest number with a prime factorization of the form [math]\displaystyle{ p \exp p \cdot q \exp q }[/math]. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.