9edt: Difference between revisions
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== Theory == | |||
It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]]. | It has a decent seventh harmonic ([[7/1]]) which is 12.4 cents sharp, and an excellent [[13/1]] inherited from [[3edt]] which is only 2.6 cents flat. However, the [[5/1]] is 39 cents flat, thus 13 steps of 9edt (approximating the 5/1) can be described as a neutral seventeenth — or if tritave-reduced to 4 steps, a neutral sixth (approximating the 5/3). This neutral sixth has a size of 845 cents, which is between [[8/5]] and [[5/3]]; if this interval is also taken as an approximation to [[13/8]], it would temper out [[40/39]] — making 9edt an exotemperament in the 8.3.5.13 subgroup. Though, 9edt is more well behaved on the 3.7.13 [[subgroup]], of which it tempers out [[351/343]] and [[2197/2187]]. | ||
Following [[4edt]], this is the next edt that supports [[4L_5s_(3/1-equivalent)|lambda]] temperament. For small edts, this property is virtually the same as supporting a 3/1-equivalent 4L 5s [[moment of symmetry]] scale, of which 9edt offers the "equalized" interpretation of L = s, analogous to [[7edo]] in diatonic ([[5L 2s]]) music. | |||
9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]]. | 9edt is the third [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos zeta peak edt]]. | ||
=== Relation to edos === | |||
9edt is related to [[17edo]], by which it may be approximated by playing every third step (the 17edo non-octave whole-tone scale), the discrepancy is only about four cents when it gets to [[3/1]]. | |||
=== Harmonics === | |||
{{Harmonics in equal|9|3|1|}} | |||
{{Harmonics in equal|9|3|1|intervals=prime}} | |||
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| [[3/1]] | | [[3/1]] | ||
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== Music == | == Music == | ||