31edo: Difference between revisions

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Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is ~~in fact the sixth~~ [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). ~~It also deals with~~ the [[11-limit]] ~~fairly~~ well~~, and is consistent through it~~, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo and a zeta peak edo]], and represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is consistent through, making it a very tone-efficient melodic approximation of the [[11-limit]], although the 14/11 and 9/7, which are equated (thus tempering [[99/98]]), may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). In the [[13-limit]] it doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

31edo's 12\31 generator (an approximate 21/16) supports [[A-Team]] and yields [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale|8-note "oneirotonic" scales similar to those in 13edo]] but with the 9/8 and 5/4 better in tune; this temperament is also represented by [[13edo]], [[18edo]] and [[44edo]].

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-Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit]] fairly well, and is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
+Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning supporting [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et is relatively quite accurate and is [[The Riemann Zeta Function and Tuning#Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo and a zeta peak edo]], and represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is consistent through, making it a very tone-efficient melodic approximation of the [[11-limit]], although the 14/11 and 9/7, which are equated (thus tempering [[99/98]]), may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). In the [[13-limit]] it doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
 edo's 12\31 generator (an approximate 21/16) supports [[A-Team]] and yields [[13edo#Modes_and_Harmony_in_the_Oneirotonic_Scale|8-note "oneirotonic" scales similar to those in 13edo]] but with the 9/8 and 5/4 better in tune; this temperament is also represented by [[13edo]], [[18edo]] and [[44edo]].