31edo: Difference between revisions

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'''Thirty-one tone equal temperament''', also called '''31-tET''', '''31-EDO''', '''31-et''', or '''tricesimoprimal meantone temperament''', is the scale derived by dividing the octave into 31 [[equal]]ly large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. 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.

'''Thirty-one tone equal temperament''', also called '''31-tET''', '''31-EDO''', '''31-et''', or '''tricesimoprimal meantone temperament''', is the scale derived by dividing the octave into 31 [[equal]]ly large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]].

== Theory ==

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.

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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-'''Thirty-one tone equal temperament''', also called '''31-tET''', '''31-EDO''', '''31-et''', or '''tricesimoprimal meantone temperament''', is the scale derived by dividing the octave into 31 [[equal]]ly large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. 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.
+'''Thirty-one tone equal temperament''', also called '''31-tET''', '''31-EDO''', '''31-et''', or '''tricesimoprimal meantone temperament''', is the scale derived by dividing the octave into 31 [[equal]]ly large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]].
+== Theory ==
+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.
 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]].