Xenharmonic Wiki

41edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m →Commas: replace inline monzo coding
m Formatting
Line 4: Line 4:
== 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]], 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.
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 [[cents]], an [[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]]) 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]] fairly well, and the [[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]] 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.  


41-ET forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41-ET as the basic [[13-limit|13-limit]] intervals requiring fine tuning +/- 1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41-ET circle in [[205edo|205edo]]. 41-ET is also used by the [[The Kite Guitar|Kite Guitar]], see below in "Instruments".
41-ET forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41-ET as the basic [[13-limit]] intervals requiring fine tuning +/- 1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41-ET circle in [[205edo]]. 41-ET is also used by the [[The Kite Guitar|Kite Guitar]], see below in "Instruments".


41edo is the 13th [[prime_numbers|prime]] edo, following [[37edo|37edo]] and coming before [[43edo|43edo]].
41edo is the 13th [[prime_numbers|prime]] edo, following [[37edo]] and coming before [[43edo]].


== Intervals ==
== Intervals ==
Line 17: Line 17:
! #
! #
! Cents
! Cents
! Approximate Ratios'''*'''
! Approximate Ratios*
! colspan="3" | [[Ups_and_Downs_Notation|Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! Andrew's <br> Solfege Syllables
! Andrew's <br> Solfege Syllables
|-
|-
| 0
| 0
| 0.00
| style="text-align:right" | 0.00
| [[1/1]]
| style="text-align:left" | [[1/1]]
| perfect unison
| perfect unison
| P1
| P1
Line 30: Line 30:
|-
|-
| 1
| 1
| 29.27
| style="text-align:right" | 29.27
| [[81/80]], [[64/63]]
| style="text-align:left" | [[81/80]], [[64/63]]
| up-unison
| up-unison
| ^1
| ^1
Line 38: Line 38:
|-
|-
| 2
| 2
| 58.54
| style="text-align:right" | 58.54
| [[25/24]], [[28/27]], [[33/32]]
| style="text-align:left" | [[25/24]], [[28/27]], [[33/32]]
| double-up unison, downminor 2nd
| double-up unison, downminor 2nd
| ^^1, vm2
| ^^1, vm2
Line 46: Line 46:
|-
|-
| 3
| 3
| 87.805
| style="text-align:right" | 87.80
| [[21/20]], [[22/21]], [[19/18]], [[20/19]]
| style="text-align:left" | [[21/20]], [[22/21]], [[19/18]], [[20/19]]
| down-aug 1sn, minor 2nd
| down-aug 1sn, minor 2nd
| vA1, m2
| vA1, m2
Line 54: Line 54:
|-
|-
| 4
| 4
| 117.07
| style="text-align:right" | 117.07
| [[16/15]], [[15/14]], [[14/13]]
| style="text-align:left" | [[16/15]], [[15/14]], [[14/13]]
| augmented 1sn, upminor 2nd
| augmented 1sn, upminor 2nd
| A1, ^m2
| A1, ^m2
Line 62: Line 62:
|-
|-
| 5
| 5
| 146.34
| style="text-align:right" | 146.34
| [[12/11]], [[13/12]]
| style="text-align:left" | [[12/11]], [[13/12]]
| mid 2nd
| mid 2nd
| ~2
| ~2
Line 70: Line 70:
|-
|-
| 6
| 6
| 175.61
| style="text-align:right" | 175.61
| [[10/9]], [[11/10]], [[21/19]]
| style="text-align:left" | [[10/9]], [[11/10]], [[21/19]]
| downmajor 2nd
| downmajor 2nd
| vM2
| vM2
Line 78: Line 78:
|-
|-
| 7
| 7
| 204.88
| style="text-align:right" | 204.88
| [[9/8]]
| style="text-align:left" | [[9/8]]
| major 2nd
| major 2nd
| M2
| M2
Line 86: Line 86:
|-
|-
| 8
| 8
| 234.15
| style="text-align:right" | 234.15
| [[8/7]], [[15/13]]
| style="text-align:left" | [[8/7]], [[15/13]]
| upmajor 2nd
| upmajor 2nd
| ^M2
| ^M2
Line 94: Line 94:
|-
|-
| 9
| 9
| 263.415
| style="text-align:right" | 263.41
| [[7/6]], [[22/19]]
| style="text-align:left" | [[7/6]], [[22/19]]
| downminor 3rd
| downminor 3rd
| vm3
| vm3
Line 102: Line 102:
|-
|-
| 10
| 10
| 292.68
| style="text-align:right" | 292.68
| [[32/27]], [[13/11]], [[19/16]]
| style="text-align:left" | [[32/27]], [[13/11]], [[19/16]]
| minor 3rd
| minor 3rd
| m3
| m3
Line 110: Line 110:
|-
|-
| 11
| 11
| 321.95
| style="text-align:right" | 321.95
| [[6/5]]
| style="text-align:left" | [[6/5]]
| upminor 3rd
| upminor 3rd
| ^m3
| ^m3
Line 118: Line 118:
|-
|-
| 12
| 12
| 351.22
| style="text-align:right" | 351.22
| [[11/9]], [[27/22]], [[16/13]]
| style="text-align:left" | [[11/9]], [[27/22]], [[16/13]]
| mid 3rd
| mid 3rd
| ~3
| ~3
Line 126: Line 126:
|-
|-
| 13
| 13
| 380.49
| style="text-align:right" | 380.49
| [[5/4]], [[26/21]]
| style="text-align:left" | [[5/4]], [[26/21]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 134: Line 134:
|-
|-
| 14
| 14
| 409.76
| style="text-align:right" | 409.76
| [[81/64]], [[14/11]], [[24/19]], [[19/15]]
| style="text-align:left" | [[81/64]], [[14/11]], [[24/19]], [[19/15]]
| major 3rd
| major 3rd
| M3
| M3
Line 142: Line 142:
|-
|-
| 15
| 15
| 439.02
| style="text-align:right" | 439.02
| [[9/7|9/7,]] [[32/25]]
| style="text-align:left" | [[9/7]], [[32/25]]
| upmajor 3rd
| upmajor 3rd
| ^M3
| ^M3
Line 150: Line 150:
|-
|-
| 16
| 16
| 468.29
| style="text-align:right" | 468.29
| [[21/16]], [[13/10]]
| style="text-align:left" | [[21/16]], [[13/10]]
| down-4th
| down-4th
| v4
| v4
Line 158: Line 158:
|-
|-
| 17
| 17
| 497.56
| style="text-align:right" | 497.56
| [[4/3]]
| style="text-align:left" | [[4/3]]
| perfect 4th
| perfect 4th
| P4
| P4
Line 166: Line 166:
|-
|-
| 18
| 18
| 526.83
| style="text-align:right" | 526.83
| [[15/11]], [[27/20]], [[19/14]]
| style="text-align:left" | [[15/11]], [[27/20]], [[19/14]]
| up-4th
| up-4th
| ^4
| ^4
Line 174: Line 174:
|-
|-
| 19
| 19
| 556.1
| style="text-align:right" | 556.10
| [[11/8]], [[18/13]], [[26/19]]
| style="text-align:left" | [[11/8]], [[18/13]], [[26/19]]
| mid-4th
| mid-4th
| ~4
| ~4
Line 182: Line 182:
|-
|-
| 20
| 20
| 585.37
| style="text-align:right" | 585.37
| [[7/5]]
| style="text-align:left" | [[7/5]]
| downaug 4th, dim 5th
| downaug 4th, dim 5th
| vA4, d5
| vA4, d5
Line 190: Line 190:
|-
|-
| 21
| 21
| 614.63
| style="text-align:right" | 614.63
| [[10/7]]
| style="text-align:left" | [[10/7]]
| aug 4th, updim 5th
| aug 4th, updim 5th
| A4, ^d5
| A4, ^d5
Line 198: Line 198:
|-
|-
| 22
| 22
| 643.90
| style="text-align:right" | 643.90
| [[16/11]], [[13/9]], [[19/13]]
| style="text-align:left" | [[16/11]], [[13/9]], [[19/13]]
| mid-5th
| mid-5th
| ~5
| ~5
Line 206: Line 206:
|-
|-
| 23
| 23
| 673.17
| style="text-align:right" | 673.17
| [[22/15]], [[40/27]], [[28/19]]
| style="text-align:left" | [[22/15]], [[40/27]], [[28/19]]
| down-5th
| down-5th
| v5
| v5
Line 214: Line 214:
|-
|-
| 24
| 24
| 702.44
| style="text-align:right" | 702.44
| [[3/2]]
| style="text-align:left" | [[3/2]]
| perfect 5th
| perfect 5th
| P5
| P5
Line 222: Line 222:
|-
|-
| 25
| 25
| 731.71
| style="text-align:right" | 731.71
| [[32/21]], [[20/13]]
| style="text-align:left" | [[32/21]], [[20/13]]
| up-5th
| up-5th
| ^5
| ^5
Line 230: Line 230:
|-
|-
| 26
| 26
| 760.98
| style="text-align:right" | 760.98
| [[14/9]], [[25/16]]
| style="text-align:left" | [[14/9]], [[25/16]]
| downminor 6th
| downminor 6th
| vm6
| vm6
Line 238: Line 238:
|-
|-
| 27
| 27
| 790.24
| style="text-align:right" | 790.24
| [[128/81]], [[11/7]], [[19/12]], [[30/19]]
| style="text-align:left" | [[128/81]], [[11/7]], [[19/12]], [[30/19]]
| minor 6th
| minor 6th
| m6
| m6
Line 246: Line 246:
|-
|-
| 28
| 28
| 819.51
| style="text-align:right" | 819.51
| [[8/5]], [[21/13]]
| style="text-align:left" | [[8/5]], [[21/13]]
| upminor 6th
| upminor 6th
| ^m6
| ^m6
Line 254: Line 254:
|-
|-
| 29
| 29
| 848.78
| style="text-align:right" | 848.78
| [[18/11]], [[44/27]], [[13/8]]
| style="text-align:left" | [[18/11]], [[44/27]], [[13/8]]
| mid 6th
| mid 6th
| ~6
| ~6
Line 262: Line 262:
|-
|-
| 30
| 30
| 878.05
| style="text-align:right" | 878.05
| [[5/3]]
| style="text-align:left" | [[5/3]]
| downmajor 6th
| downmajor 6th
| vM6
| vM6
Line 270: Line 270:
|-
|-
| 31
| 31
| 907.32
| style="text-align:right" | 907.32
| [[27/16]], [[22/13]], [[32/19]]
| style="text-align:left" | [[27/16]], [[22/13]], [[32/19]]
| major 6th
| major 6th
| M6
| M6
Line 278: Line 278:
|-
|-
| 32
| 32
| 936.59
| style="text-align:right" | 936.59
| [[12/7]], [[19/11]]
| style="text-align:left" | [[12/7]], [[19/11]]
| upmajor 6th
| upmajor 6th
| ^M6
| ^M6
Line 286: Line 286:
|-
|-
| 33
| 33
| 965.85
| style="text-align:right" | 965.85
| [[7/4]], [[26/15]]
| style="text-align:left" | [[7/4]], [[26/15]]
| downminor 7th
| downminor 7th
| vm7
| vm7
Line 294: Line 294:
|-
|-
| 34
| 34
| 995.12
| style="text-align:right" | 995.12
| [[16/9]]
| style="text-align:left" | [[16/9]]
| minor 7th
| minor 7th
| m7
| m7
Line 302: Line 302:
|-
|-
| 35
| 35
| 1024.39
| style="text-align:right" | 1024.39
| [[9/5]], [[20/11]], [[38/21]]
| style="text-align:left" | [[9/5]], [[20/11]], [[38/21]]
| upminor 7th
| upminor 7th
| ^m7
| ^m7
Line 310: Line 310:
|-
|-
| 36
| 36
| 1053.66
| style="text-align:right" | 1053.66
| [[11/6]], [[24/13]]
| style="text-align:left" | [[11/6]], [[24/13]]
| mid 7th
| mid 7th
| ~7
| ~7
Line 318: Line 318:
|-
|-
| 37
| 37
| 1082.93
| style="text-align:right" | 1082.93
| [[15/8]], [[28/15]], [[13/7]]
| style="text-align:left" | [[15/8]], [[28/15]], [[13/7]]
| downmajor 7th
| downmajor 7th
| vM7
| vM7
Line 326: Line 326:
|-
|-
| 38
| 38
| 1112.195
| style="text-align:right" | 1112.20
| [[40/21]], [[21/11]], [[36/19]], [[19/10]]
| style="text-align:left" | [[40/21]], [[21/11]], [[36/19]], [[19/10]]
| major 7th
| major 7th
| M7
| M7
Line 334: Line 334:
|-
|-
| 39
| 39
| 1141.46
| style="text-align:right" | 1141.46
| [[48/25]], [[27/14]], [[64/33]]
| style="text-align:left" | [[48/25]], [[27/14]], [[64/33]]
| upmajor 7th
| upmajor 7th
| ^M7
| ^M7
Line 342: Line 342:
|-
|-
| 40
| 40
| 1170.73
| style="text-align:right" | 1170.73
| [[160/81]], [[63/32]]
| style="text-align:left" | [[160/81]], [[63/32]]
| dim 8ve
| dim 8ve
| v8
| v8
Line 350: Line 350:
|-
|-
| 41
| 41
| 1200
| style="text-align:right" | 1200.00
| 2/1
| style="text-align:left" | 2/1
| perfect 8ve
| perfect 8ve
| P8
| P8
Line 716: Line 716:
|-
|-
| "
| "
|
| style="text-align:center;" |
| style="text-align:center;" | 19.95
| 19.95
| {{Monzo| 10 -11 2 1 }}
| {{Monzo| 10 -11 2 1 }}
| Sazoyoyo
| Sazoyoyo
Retrieved from "https://en.xen.wiki/w/41edo"