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== Theory ==
== Theory ==


The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cent|cent]]s, an [[interval|interval]] close in size to [[64/63|64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after [[29edo|29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], and is the seventh [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit|11-limit]] fairly well, and the [[13-limit|13-limit]] perhaps close enough for government work, though its [[13/10|13/10]] is 14 cents sharp. Various 13-limit [[Magic_extensions|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.
The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cent|cent]]s, an [[interval|interval]] close in size to [[64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after [[29edo|29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], and is the seventh [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit|11-limit]] fairly well, and the [[13-limit|13-limit]] perhaps close enough for government work, though its [[13/10]] is 14 cents sharp. Various 13-limit [[Magic_extensions|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.


41edo is consistent in the 15 odd limit. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 is debatable. (In comparison, [[31edo|31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.  
41edo is consistent in the 15 odd limit. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 is debatable. (In comparison, [[31edo|31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.  
Line 23: Line 23:
| 0
| 0
| 0.00
| 0.00
| [[1/1|1/1]]
| [[1/1]]
| perfect unison
| perfect unison
| P1
| P1
Line 31: Line 31:
| 1
| 1
| 29.27
| 29.27
| [[81/80|81/80]], [[64/63]]
| [[81/80]], [[64/63]]
| up-unison
| up-unison
| ^1
| ^1
Line 39: Line 39:
| 2
| 2
| 58.54
| 58.54
| [[25/24|25/24]], [[28/27|28/27]], [[33/32|33/32]]
| [[25/24]], [[28/27]], [[33/32]]
| double-up unison, downminor 2nd
| double-up unison, downminor 2nd
| ^^1, vm2
| ^^1, vm2
Line 47: Line 47:
| 3
| 3
| 87.805
| 87.805
| [[21/20|21/20]], [[22/21|22/21]], [[19/18]], [[20/19]]
| [[21/20]], [[22/21]], [[19/18]], [[20/19]]
| down-aug 1sn, minor 2nd
| down-aug 1sn, minor 2nd
| vA1, m2
| vA1, m2
Line 55: Line 55:
| 4
| 4
| 117.07
| 117.07
| [[16/15|16/15]], [[15/14|15/14]], [[14/13]]
| [[16/15]], [[15/14]], [[14/13]]
| augmented 1sn, upminor 2nd
| augmented 1sn, upminor 2nd
| A1, ^m2
| A1, ^m2
Line 63: Line 63:
| 5
| 5
| 146.34
| 146.34
| [[12/11|12/11]], [[13/12]]
| [[12/11]], [[13/12]]
| mid 2nd
| mid 2nd
| ~2
| ~2
Line 71: Line 71:
| 6
| 6
| 175.61
| 175.61
| [[10/9|10/9]], [[11/10|11/10]], [[21/19]]
| [[10/9]], [[11/10]], [[21/19]]
| downmajor 2nd
| downmajor 2nd
| vM2
| vM2
Line 79: Line 79:
| 7
| 7
| 204.88
| 204.88
| [[9/8|9/8]]
| [[9/8]]
| major 2nd
| major 2nd
| M2
| M2
Line 87: Line 87:
| 8
| 8
| 234.15
| 234.15
| [[8/7|8/7]], [[15/13]]
| [[8/7]], [[15/13]]
| upmajor 2nd
| upmajor 2nd
| ^M2
| ^M2
Line 95: Line 95:
| 9
| 9
| 263.415
| 263.415
| [[7/6|7/6]], [[22/19]]
| [[7/6]], [[22/19]]
| downminor 3rd
| downminor 3rd
| vm3
| vm3
Line 103: Line 103:
| 10
| 10
| 292.68
| 292.68
| [[32/27|32/27]], [[13/11]], [[19/16]]
| [[32/27]], [[13/11]], [[19/16]]
| minor 3rd
| minor 3rd
| m3
| m3
Line 111: Line 111:
| 11
| 11
| 321.95
| 321.95
| [[6/5|6/5]]
| [[6/5]]
| upminor 3rd
| upminor 3rd
| ^m3
| ^m3
Line 119: Line 119:
| 12
| 12
| 351.22
| 351.22
| [[11/9|11/9]], [[27/22|27/22]], [[16/13]]
| [[11/9]], [[27/22]], [[16/13]]
| mid 3rd
| mid 3rd
| ~3
| ~3
Line 127: Line 127:
| 13
| 13
| 380.49
| 380.49
| [[5/4|5/4]], [[26/21]]
| [[5/4]], [[26/21]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 135: Line 135:
| 14
| 14
| 409.76
| 409.76
| [[81/64]], [[14/11|14/11]], [[24/19]], [[19/15]]
| [[81/64]], [[14/11]], [[24/19]], [[19/15]]
| major 3rd
| major 3rd
| M3
| M3
Line 143: Line 143:
| 15
| 15
| 439.02
| 439.02
| [[9/7|9/7,]] [[32/25|32/25]]
| [[9/7|9/7,]] [[32/25]]
| upmajor 3rd
| upmajor 3rd
| ^M3
| ^M3
Line 151: Line 151:
| 16
| 16
| 468.29
| 468.29
| [[21/16|21/16]], [[13/10]]
| [[21/16]], [[13/10]]
| down-4th
| down-4th
| v4
| v4
Line 159: Line 159:
| 17
| 17
| 497.56
| 497.56
| [[4/3|4/3]]
| [[4/3]]
| perfect 4th
| perfect 4th
| P4
| P4
Line 167: Line 167:
| 18
| 18
| 526.83
| 526.83
| [[15/11|15/11]], [[27/20|27/20]], [[19/14]]
| [[15/11]], [[27/20]], [[19/14]]
| up-4th
| up-4th
| ^4
| ^4
Line 175: Line 175:
| 19
| 19
| 556.1
| 556.1
| [[11/8|11/8]], [[18/13]], [[26/19]]
| [[11/8]], [[18/13]], [[26/19]]
| mid-4th
| mid-4th
| ~4
| ~4
Line 183: Line 183:
| 20
| 20
| 585.37
| 585.37
| [[7/5|7/5]]
| [[7/5]]
| downaug 4th, dim 5th
| downaug 4th, dim 5th
| vA4, d5
| vA4, d5
Line 191: Line 191:
| 21
| 21
| 614.63
| 614.63
| [[10/7|10/7]]
| [[10/7]]
| aug 4th, updim 5th
| aug 4th, updim 5th
| A4, ^d5
| A4, ^d5
Line 199: Line 199:
| 22
| 22
| 643.90
| 643.90
| [[16/11|16/11]], [[13/9]], [[19/13]]
| [[16/11]], [[13/9]], [[19/13]]
| mid-5th
| mid-5th
| ~5
| ~5
Line 207: Line 207:
| 23
| 23
| 673.17
| 673.17
| [[22/15|22/15]], [[40/27|40/27]], [[28/19]]
| [[22/15]], [[40/27]], [[28/19]]
| down-5th
| down-5th
| v5
| v5
Line 215: Line 215:
| 24
| 24
| 702.44
| 702.44
| [[3/2|3/2]]
| [[3/2]]
| perfect 5th
| perfect 5th
| P5
| P5
Line 223: Line 223:
| 25
| 25
| 731.71
| 731.71
| [[32/21|32/21]], [[20/13]]
| [[32/21]], [[20/13]]
| up-5th
| up-5th
| ^5
| ^5
Line 231: Line 231:
| 26
| 26
| 760.98
| 760.98
| [[14/9|14/9]], [[25/16|25/16]]
| [[14/9]], [[25/16]]
| downminor 6th
| downminor 6th
| vm6
| vm6
Line 239: Line 239:
| 27
| 27
| 790.24
| 790.24
| [[128/81|128/81]], [[11/7|11/7]], [[19/12]], [[30/19]]
| [[128/81]], [[11/7]], [[19/12]], [[30/19]]
| minor 6th
| minor 6th
| m6
| m6
Line 247: Line 247:
| 28
| 28
| 819.51
| 819.51
| [[8/5|8/5]], [[21/13]]
| [[8/5]], [[21/13]]
| upminor 6th
| upminor 6th
| ^m6
| ^m6
Line 255: Line 255:
| 29
| 29
| 848.78
| 848.78
| [[18/11|18/11]], [[44/27|44/27]], [[13/8]]
| [[18/11]], [[44/27]], [[13/8]]
| mid 6th
| mid 6th
| ~6
| ~6
Line 263: Line 263:
| 30
| 30
| 878.05
| 878.05
| [[5/3|5/3]]
| [[5/3]]
| downmajor 6th
| downmajor 6th
| vM6
| vM6
Line 271: Line 271:
| 31
| 31
| 907.32
| 907.32
| [[27/16|27/16]], [[22/13]], [[32/19]]
| [[27/16]], [[22/13]], [[32/19]]
| major 6th
| major 6th
| M6
| M6
Line 279: Line 279:
| 32
| 32
| 936.59
| 936.59
| [[12/7|12/7]], [[19/11]]
| [[12/7]], [[19/11]]
| upmajor 6th
| upmajor 6th
| ^M6
| ^M6
Line 287: Line 287:
| 33
| 33
| 965.85
| 965.85
| [[7/4|7/4]], [[26/15]]
| [[7/4]], [[26/15]]
| downminor 7th
| downminor 7th
| vm7
| vm7
Line 295: Line 295:
| 34
| 34
| 995.12
| 995.12
| [[16/9|16/9]]
| [[16/9]]
| minor 7th
| minor 7th
| m7
| m7
Line 303: Line 303:
| 35
| 35
| 1024.39
| 1024.39
| [[9/5|9/5]], [[20/11|20/11]], [[38/21]]
| [[9/5]], [[20/11]], [[38/21]]
| upminor 7th
| upminor 7th
| ^m7
| ^m7
Line 311: Line 311:
| 36
| 36
| 1053.66
| 1053.66
| [[11/6|11/6]], [[24/13]]
| [[11/6]], [[24/13]]
| mid 7th
| mid 7th
| ~7
| ~7
Line 319: Line 319:
| 37
| 37
| 1082.93
| 1082.93
| [[15/8|15/8]], [[28/15]], [[13/7]]
| [[15/8]], [[28/15]], [[13/7]]
| downmajor 7th
| downmajor 7th
| vM7
| vM7
Line 327: Line 327:
| 38
| 38
| 1112.195
| 1112.195
| [[40/21|40/21]], [[21/11|21/11]], [[36/19]], [[19/10]]
| [[40/21]], [[21/11]], [[36/19]], [[19/10]]
| major 7th
| major 7th
| M7
| M7
Line 335: Line 335:
| 39
| 39
| 1141.46
| 1141.46
| [[48/25|48/25]], [[27/14|27/14]], [[64/33|64/33]]
| [[48/25]], [[27/14]], [[64/33]]
| upmajor 7th
| upmajor 7th
| ^M7
| ^M7
Line 343: Line 343:
| 40
| 40
| 1170.73
| 1170.73
| [[160/81|160/81]], [[63/32]]
| [[160/81]], [[63/32]]
| dim 8ve
| dim 8ve
| v8
| v8
Line 532: Line 532:
! Error (abs., in [[cent|cents]])
! Error (abs., in [[cent|cents]])
|-
|-
| '''[[4/3|4/3]], [[3/2|3/2]]'''
| '''[[4/3]], [[3/2]]'''
| '''0.484'''
| '''0.484'''
|-
|-
| [[9/8|9/8]], [[16/9|16/9]]
| [[9/8]], [[16/9]]
| 0.968
| 0.968
|-
|-
| [[15/14|15/14]], [[28/15|28/15]]
| [[15/14]], [[28/15]]
| 2.370
| 2.370
|-
|-
| [[7/5|7/5]], [[10/7|10/7]]
| [[7/5]], [[10/7]]
| 2.854
| 2.854
|-
|-
| '''[[8/7|8/7]], [[7/4|7/4]]'''
| '''[[8/7]], [[7/4]]'''
| '''2.972'''
| '''2.972'''
|-
|-
| [[7/6|7/6]], [[12/7|12/7]]
| [[7/6]], [[12/7]]
| 3.456
| 3.456
|-
|-
| [[13/11|13/11]], [[22/13|22/13]]
| [[13/11]], [[22/13]]
| 3.473
| 3.473
|-
|-
| [[11/9|11/9]], [[18/11|18/11]]
| [[11/9]], [[18/11]]
| 3.812
| 3.812
|-
|-
| [[9/7|9/7]], [[14/9|14/9]]
| [[9/7]], [[14/9]]
| 3.940
| 3.940
|-
|-
| [[12/11|12/11]], [[11/6|11/6]]
| [[12/11]], [[11/6]]
| 4.296
| 4.296
|-
|-
| '''[[11/8|11/8]], [[16/11|16/11]]'''
| '''[[11/8]], [[16/11]]'''
| '''4.780'''
| '''4.780'''
|-
|-
| [[16/15|16/15]], [[15/8|15/8]]
| [[16/15]], [[15/8]]
| 5.342
| 5.342
|-
|-
| '''[[5/4|5/4]], [[8/5|8/5]]'''
| '''[[5/4]], [[8/5]]'''
| '''5.826'''
| '''5.826'''
|-
|-
| [[6/5|6/5]], [[5/3|5/3]]
| [[6/5]], [[5/3]]
| 6.310
| 6.310
|-
|-
| [[10/9|10/9]], [[9/5|9/5]]
| [[10/9]], [[9/5]]
| 6.794
| 6.794
|-
|-
| [[18/13|18/13]], [[13/9|13/9]]
| [[18/13]], [[13/9]]
| 7.285
| 7.285
|-
|-
| [[14/11|14/11]], [[11/7|11/7]]
| [[14/11]], [[11/7]]
| 7.752
| 7.752
|-
|-
| [[13/12|13/12]], [[24/13|24/13]]
| [[13/12]], [[24/13]]
| 7.769
| 7.769
|-
|-
| '''[[16/13|16/13]], [[13/8|13/8]]'''
| '''[[16/13]], [[13/8]]'''
| '''8.253'''
| '''8.253'''
|-
|-
| [[15/11|15/11]], [[22/15|22/15]]
| [[15/11]], [[22/15]]
| 10.122
| 10.122
|-
|-
| [[11/10|11/10]], [[20/11|20/11]]
| [[11/10]], [[20/11]]
| 10.606
| 10.606
|-
|-
| [[14/13|14/13]], [[13/7|13/7]]
| [[14/13]], [[13/7]]
| 11.225
| 11.225
|-
|-
| [[15/13|15/13]], [[26/15|26/15]]
| [[15/13]], [[26/15]]
| 13.595
| 13.595
|-
|-
| [[13/10|13/10]], [[20/13|20/13]]
| [[13/10]], [[20/13]]
| 14.079
| 14.079
|}
|}
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