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== Intervals ==
== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios*
! [[Kite's ups and downs notation|Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 29.3
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
| {{UDnote|step=1}}
|-
| 2
| 58.5
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
| {{UDnote|step=2}}
|-
| 3
| 87.8
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| {{UDnote|step=3}}
|-
| 4
| 117.1
| [[14/13]], [[15/14]], [[16/15]]
| {{UDnote|step=4}}
|-
| 5
| 146.3
| [[12/11]], [[13/12]]
| {{UDnote|step=5}}
|-
| 6
| 175.6
| [[10/9]], [[11/10]], [[21/19]]
| {{UDnote|step=6}}
|-
| 7
| 204.9
| [[9/8]]
| {{UDnote|step=7}}
|-
| 8
| 234.1
| [[8/7]], [[15/13]]
| {{UDnote|step=8}}
|-
| 9
| 263.4
| [[7/6]], [[22/19]]
| {{UDnote|step=9}}
|-
| 10
| 292.7
| [[13/11]], [[19/16]], [[32/27]]
| {{UDnote|step=10}}
|-
| 11
| 322.0
| [[6/5]]
| {{UDnote|step=11}}
|-
| 12
| 351.2
| [[11/9]], [[16/13]]
| {{UDnote|step=12}}
|-
| 13
| 380.5
| [[5/4]], [[26/21]]
| {{UDnote|step=13}}
|-
| 14
| 409.8
| [[14/11]], [[19/15]], [[24/19]]
| {{UDnote|step=14}}
|-
| 15
| 439.0
| [[9/7]], [[32/25]]
| {{UDnote|step=15}}
|-
| 16
| 468.3
| [[21/16]], [[13/10]]
| {{UDnote|step=16}}
|-
| 17
| 497.6
| [[4/3]]
| {{UDnote|step=17}}
|-
| 18
| 526.8
| [[15/11]], [[19/14]], [[27/20]]
| {{UDnote|step=18}}
|-
| 19
| 556.1
| [[11/8]], [[18/13]], [[26/19]]
| {{UDnote|step=19}}
|-
| 20
| 585.4
| [[7/5]], [[45/32]]
| {{UDnote|step=20}}
|-
| 21
| 614.6
| [[10/7]], [[64/45]]
| {{UDnote|step=21}}
|-
| 22
| 643.9
| [[13/9]], [[16/11]], [[19/13]]
| {{UDnote|step=22}}
|-
| 23
| 673.2
| [[22/15]], [[28/19]], [[40/27]]
| {{UDnote|step=23}}
|-
| 24
| 702.4
| [[3/2]]
| {{UDnote|step=24}}
|-
| 25
| 731.7
| [[20/13]], [[32/21]]
| {{UDnote|step=25}}
|-
| 26
| 761.0
| [[14/9]], [[25/16]]
| {{UDnote|step=26}}
|-
| 27
| 790.2
| [[11/7]], [[19/12]], [[30/19]]
| {{UDnote|step=27}}
|-
| 28
| 819.5
| [[8/5]], [[21/13]]
| {{UDnote|step=28}}
|-
| 29
| 848.8
| [[13/8]], [[18/11]]
| {{UDnote|step=29}}
|-
| 30
| 878.0
| [[5/3]]
| {{UDnote|step=30}}
|-
| 31
| 907.3
| [[22/13]], [[27/16]], [[32/19]]
| {{UDnote|step=31}}
|-
| 32
| 936.6
| [[12/7]], [[19/11]]
| {{UDnote|step=32}}
|-
| 33
| 965.9
| [[7/4]], [[26/15]]
| {{UDnote|step=33}}
|-
| 34
| 995.1
| [[16/9]]
| {{UDnote|step=34}}
|-
| 35
| 1024.4
| [[9/5]], [[20/11]], [[38/21]]
| {{UDnote|step=35}}
|-
| 36
| 1053.7
| [[11/6]], [[24/13]]
| {{UDnote|step=36}}
|-
| 37
| 1082.9
| [[13/7]], [[15/8]], [[28/15]]
| {{UDnote|step=37}}
|-
| 38
| 1112.2
| [[19/10]], [[21/11]], [[36/19]], [[40/21]]
| {{UDnote|step=38}}
|-
| 39
| 1141.5
| [[27/14]], [[35/18]], [[48/25]], [[64/33]]
| {{UDnote|step=39}}
|-
| 40
| 1170.7
| [[49/25]], [[63/32]], [[96/49]], [[160/81]]
| {{UDnote|step=40}}
|-
| 41
| 1200.0
| [[2/1]]
| {{UDnote|step=41}}
|}
<nowiki>*</nowiki> Based on treating 41edo as a 2.3.5.7.11.13.19-subgroup temperament; other approaches are possible.
=== Proposed interval names and solfèges ===
{{See also| 41edo solfege }}
{{See also| 41edo solfege }}


{| class="wikitable center-1 right-2 center-5 center-6 center-8 center-9"
{| class="wikitable center-all right-2 left-3 left-6 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
|-
! #
! #
! Cents
! Cents
! Approximate ratios*
! colspan="3" | [[Kite's ups and downs notation]]<br>([[Kite's thoughts on enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^d2)
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^d2)
! colspan="3" | [[SKULO interval names|SKULO notation]]<br>(K or S = 1, U = 2)
! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)
! Kite's<br>solfège
! [[41edo solfege|Kite's<br>solfege]]
! Andrew's<br>solfège
! [[41edo solfege|Andrew's<br>solfege]]
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| perfect unison
| perfect unison
| P1
| P1
Line 63: Line 282:
| 1
| 1
| 29.3
| 29.3
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
| up-unison
| up-unison
| ^1
| ^1
Line 75: Line 293:
| 2
| 2
| 58.5
| 58.5
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
| dup-unison, downminor 2nd
| dup-unison, downminor 2nd
| ^^1, vm2
| ^^1, vm2
Line 87: Line 304:
| 3
| 3
| 87.8
| 87.8
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| down-aug 1sn, minor 2nd
| down-aug 1sn, minor 2nd
| vA1, m2
| vA1, m2
Line 99: Line 315:
| 4
| 4
| 117.1
| 117.1
| [[14/13]], [[15/14]], [[16/15]]
| augmented 1sn, upminor 2nd
| augmented 1sn, upminor 2nd
| A1, ^m2
| A1, ^m2
Line 111: Line 326:
| 5
| 5
| 146.3
| 146.3
| [[12/11]], [[13/12]]
| mid 2nd
| mid 2nd
| ~2
| ~2
Line 123: Line 337:
| 6
| 6
| 175.6
| 175.6
| [[10/9]], [[11/10]], [[21/19]]
| downmajor 2nd
| downmajor 2nd
| vM2
| vM2
Line 135: Line 348:
| 7
| 7
| 204.9
| 204.9
| [[9/8]]
| major 2nd
| major 2nd
| M2
| M2
Line 147: Line 359:
| 8
| 8
| 234.1
| 234.1
| [[8/7]], [[15/13]]
| upmajor 2nd
| upmajor 2nd
| ^M2
| ^M2
Line 159: Line 370:
| 9
| 9
| 263.4
| 263.4
| [[7/6]], [[22/19]]
| downminor 3rd
| downminor 3rd
| vm3
| vm3
Line 171: Line 381:
| 10
| 10
| 292.7
| 292.7
| [[13/11]], [[19/16]], [[32/27]]
| minor 3rd
| minor 3rd
| m3
| m3
Line 183: Line 392:
| 11
| 11
| 322.0
| 322.0
| [[6/5]]
| upminor 3rd
| upminor 3rd
| ^m3
| ^m3
Line 195: Line 403:
| 12
| 12
| 351.2
| 351.2
| [[11/9]], [[16/13]]
| mid 3rd
| mid 3rd
| ~3
| ~3
Line 207: Line 414:
| 13
| 13
| 380.5
| 380.5
| [[5/4]], [[26/21]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 219: Line 425:
| 14
| 14
| 409.8
| 409.8
| [[14/11]], [[19/15]], [[24/19]]
| major 3rd
| major 3rd
| M3
| M3
Line 231: Line 436:
| 15
| 15
| 439.0
| 439.0
| [[9/7]], [[32/25]]
| upmajor 3rd
| upmajor 3rd
| ^M3
| ^M3
Line 243: Line 447:
| 16
| 16
| 468.3
| 468.3
| [[21/16]], [[13/10]]
| down-4th
| down-4th
| v4
| v4
Line 255: Line 458:
| 17
| 17
| 497.6
| 497.6
| [[4/3]]
| perfect 4th
| perfect 4th
| P4
| P4
Line 267: Line 469:
| 18
| 18
| 526.8
| 526.8
| [[15/11]], [[19/14]], [[27/20]]
| up-4th
| up-4th
| ^4
| ^4
Line 279: Line 480:
| 19
| 19
| 556.1
| 556.1
| [[11/8]], [[18/13]], [[26/19]]
| mid-4th, downdim 5th
| mid-4th, downdim 5th
| ~4, vd5
| ~4, vd5
Line 291: Line 491:
| 20
| 20
| 585.4
| 585.4
| [[7/5]], [[45/32]]
| downaug 4th, dim 5th
| downaug 4th, dim 5th
| vA4, d5
| vA4, d5
Line 303: Line 502:
| 21
| 21
| 614.6
| 614.6
| [[10/7]], [[64/45]]
| aug 4th, updim 5th
| aug 4th, updim 5th
| A4, ^d5
| A4, ^d5
Line 315: Line 513:
| 22
| 22
| 643.9
| 643.9
| [[13/9]], [[16/11]], [[19/13]]
| mid-5th, upaug 4th
| mid-5th, upaug 4th
| ~5, ^A4
| ~5, ^A4
Line 327: Line 524:
| 23
| 23
| 673.2
| 673.2
| [[22/15]], [[28/19]], [[40/27]]
| down-5th
| down-5th
| v5
| v5
Line 339: Line 535:
| 24
| 24
| 702.4
| 702.4
| [[3/2]]
| perfect 5th
| perfect 5th
| P5
| P5
Line 351: Line 546:
| 25
| 25
| 731.7
| 731.7
| [[20/13]], [[32/21]]
| up-5th
| up-5th
| ^5
| ^5
Line 363: Line 557:
| 26
| 26
| 761.0
| 761.0
| [[14/9]], [[25/16]]
| downminor 6th
| downminor 6th
| vm6
| vm6
Line 375: Line 568:
| 27
| 27
| 790.2
| 790.2
| [[11/7]], [[19/12]], [[30/19]]
| minor 6th
| minor 6th
| m6
| m6
Line 387: Line 579:
| 28
| 28
| 819.5
| 819.5
| [[8/5]], [[21/13]]
| upminor 6th
| upminor 6th
| ^m6
| ^m6
Line 399: Line 590:
| 29
| 29
| 848.8
| 848.8
| [[13/8]], [[18/11]]
| mid 6th
| mid 6th
| ~6
| ~6
Line 411: Line 601:
| 30
| 30
| 878.0
| 878.0
| [[5/3]]
| downmajor 6th
| downmajor 6th
| vM6
| vM6
Line 423: Line 612:
| 31
| 31
| 907.3
| 907.3
| [[22/13]], [[27/16]], [[32/19]]
| major 6th
| major 6th
| M6
| M6
Line 435: Line 623:
| 32
| 32
| 936.6
| 936.6
| [[12/7]], [[19/11]]
| upmajor 6th
| upmajor 6th
| ^M6
| ^M6
Line 447: Line 634:
| 33
| 33
| 965.9
| 965.9
| [[7/4]], [[26/15]]
| downminor 7th
| downminor 7th
| vm7
| vm7
Line 459: Line 645:
| 34
| 34
| 995.1
| 995.1
| [[16/9]]
| minor 7th
| minor 7th
| m7
| m7
Line 471: Line 656:
| 35
| 35
| 1024.4
| 1024.4
| [[9/5]], [[20/11]], [[38/21]]
| upminor 7th
| upminor 7th
| ^m7
| ^m7
Line 483: Line 667:
| 36
| 36
| 1053.7
| 1053.7
| [[11/6]], [[24/13]]
| mid 7th
| mid 7th
| ~7
| ~7
Line 495: Line 678:
| 37
| 37
| 1082.9
| 1082.9
| [[13/7]], [[15/8]], [[28/15]]
| downmajor 7th
| downmajor 7th
| vM7
| vM7
Line 507: Line 689:
| 38
| 38
| 1112.2
| 1112.2
| [[19/10]], [[21/11]], [[36/19]], [[40/21]]
| major 7th
| major 7th
| M7
| M7
Line 519: Line 700:
| 39
| 39
| 1141.5
| 1141.5
| [[27/14]], [[35/18]], [[48/25]], [[64/33]]
| upmajor 7th
| upmajor 7th
| ^M7
| ^M7
Line 531: Line 711:
| 40
| 40
| 1170.7
| 1170.7
| [[49/25]], [[63/32]], [[96/49]], [[160/81]]
| dim 8ve
| dim 8ve
| v8
| v8
Line 543: Line 722:
| 41
| 41
| 1200.0
| 1200.0
| [[2/1]]
| perfect 8ve
| perfect 8ve
| P8
| P8
Line 553: Line 731:
| Do
| Do
|}
|}
<nowiki>*</nowiki> Based on treating 41edo as a 2.3.5.7.11.13.19 subgroup temperament; other approaches are possible.


=== Interval quality and chord names in color notation ===
=== Interval quality and chord names in color notation ===
Line 665: Line 842:
* 0-14-28 = D F# A# = Da = D aug
* 0-14-28 = D F# A# = Da = D aug


For a more complete list, see [[41edo Chord Names]] and [[Ups and downs notation #Chords and chord progressions]].
For a more complete list, see [[41edo chord names]] and [[Ups and downs notation #Chords and chord progressions]].


== Notations ==
== Notations ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
41edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Ups and downs sharpness}}
{{Sharpness-sharp4-szg}}


41edo can also be notated with quarter-tone accidentals and [[Alternative symbols for ups and downs notation#Sharp-3|ups and downs]].
The notes within an octave from A are thus:
{{Ups and downs sharpness|41|true}}


Arrows borrowed from extended [[Helmholtz–Ellis notation]] can also be used:
A, B{{sesquiflat2}}, A{{demisharp2}}, B♭, A♯, B{{demiflat2}}, A{{sesquisharp2}}, B, C{{demiflat2}}, B{{demisharp2}}, C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D, E{{sesquiflat2}}, D{{demisharp2}}, E♭, D♯, E{{demiflat2}}, D{{sesquisharp2}}, E, F{{demiflat2}}, E{{demisharp2}}, F, G{{sesquiflat2}}, F{{demisharp2}}, G♭, F♯, G{{demiflat2}}, F{{sesquisharp2}}, G, A{{sesquiflat2}}, G{{demisharp2}}, A♭, G♯, A{{demiflat2}}, G{{sesquisharp2}}, A


{{Sharpness-sharp4}}
=== Kite's ups and downs notation ===
41edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Ups and downs sharpness}}


The notes within an octave from A are thus:
Half-sharps and half-flats can be used to avoid double arrows:  
 
{{Ups and downs sharpness|41|true}}
A, B{{sesquiflat2}}, A{{demisharp2}}, B♭, A♯, B{{demiflat2}}, A{{sesquisharp2}}, B, C{{demiflat2}}, B{{demisharp2}}, C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D, E{{sesquiflat2}}, D{{demisharp2}}, E♭, D♯, E{{demiflat2}}, D{{sesquisharp2}}, E, F{{demiflat2}}, E{{demisharp2}}, F, G{{sesquiflat2}}, F{{demisharp2}}, G♭, F♯, G{{demiflat2}}, F{{sesquisharp2}}, G, A{{sesquiflat2}}, G{{demisharp2}}, A♭, G♯, A{{demiflat2}}, G{{sesquisharp2}}, A


=== Red-Blue notation ===
=== Red-Blue notation ===
A red-note/blue-note system, similar to the one proposed for [[36edo]], is another option for notating 41edo. This is a special case of Kite's [[color notation]], treating 41edo as a temperament of the 2.3.7 subgroup. We have the "white key" albitonic notes A–G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:
A red-note/blue-note system, similar to the one proposed for [[36edo]], is another option for notating 41edo. This is a special case of [[Kite's color notation]], treating 41edo as a temperament of the 2.3.7 subgroup. We have the "white key" albitonic notes A–G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:


{{colored note|A}}, {{colored note|red|A}}, {{colored note|blue|B♭}}, {{colored note|B♭}}, {{colored note|A♯}}, {{colored note|red|A♯}}, {{colored note|blue|B}}, {{colored note|B}}, {{colored note|red|B}}, {{colored note|blue|C}}, {{colored note|C}}, {{colored note|red|C}}, {{colored note|blue|D♭}}, {{colored note|D♭}}, {{colored note|C♯}}, {{colored note|red|C♯}}, {{colored note|blue|D}}, {{colored note|D}}, {{colored note|red|D}}, {{colored note|blue|E♭}}, {{colored note|E♭}}, {{colored note|D♯}}, {{colored note|red|D♯}}, {{colored note|blue|E}}, {{colored note|E}}, {{colored note|red|E}}, {{colored note|blue|F}}, {{colored note|F}}, {{colored note|red|F}}, {{colored note|blue|G♭}}, {{colored note|G♭}}, {{colored note|F♯}}, {{colored note|red|F♯}}, {{colored note|blue|G}}, {{colored note|G}}, {{colored note|red|G}}, {{colored note|blue|A♭}}, {{colored note|A♭}}, {{colored note|G♯}}, {{colored note|red|G♯}}, {{colored note|blue|A}}, {{colored note|A}}
{{colored note|A}}, {{colored note|red|A}}, {{colored note|blue|B♭}}, {{colored note|B♭}}, {{colored note|A♯}}, {{colored note|red|A♯}}, {{colored note|blue|B}}, {{colored note|B}}, {{colored note|red|B}}, {{colored note|blue|C}}, {{colored note|C}}, {{colored note|red|C}}, {{colored note|blue|D♭}}, {{colored note|D♭}}, {{colored note|C♯}}, {{colored note|red|C♯}}, {{colored note|blue|D}}, {{colored note|D}}, {{colored note|red|D}}, {{colored note|blue|E♭}}, {{colored note|E♭}}, {{colored note|D♯}}, {{colored note|red|D♯}}, {{colored note|blue|E}}, {{colored note|E}}, {{colored note|red|E}}, {{colored note|blue|F}}, {{colored note|F}}, {{colored note|red|F}}, {{colored note|blue|G♭}}, {{colored note|G♭}}, {{colored note|F♯}}, {{colored note|red|F♯}}, {{colored note|blue|G}}, {{colored note|G}}, {{colored note|red|G}}, {{colored note|blue|A♭}}, {{colored note|A♭}}, {{colored note|G♯}}, {{colored note|red|G♯}}, {{colored note|blue|A}}, {{colored note|A}}
Line 692: Line 869:
The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.
The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.


If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups and downs notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone.
If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as Kite's ups and downs notation. The only difference is the use of minor tritone and major tritone.


=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[34edo #Sagittal notation|34edo]].
41edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation #Spartan single-shaft|Spartan set]], with the apotome equal to 4 edosteps and the limma to 3 edosteps. Since the apotome can be split in two and the [[243/242|rastma]] is tempered out, a Stein–Zimmermann half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:
 
{| class="wikitable" style="text-align: center;"
! colspan="2" |Steps
! '''0'''
! 1
! 2
! 3
! '''4'''
|-
! rowspan="3" |Symbol
! Evo-SZ
| rowspan="3" | <big>{{sagittal| |//| }}</big>
| rowspan="3" | <big>{{sagittal| /| }}</big>
| <big>{{Sagittal| t }}</big>
| rowspan="2" | <big>{{sagittal| \! }}{{sagittal| # }}</big>
| rowspan="2" | <big>{{sagittal| # }}</big>
|-
! Evo
| rowspan="2" | <big>{{sagittal| /|\ }}</big>
|-
! Revo
| <big>{{sagittal| ||\ }}</big>
| <big>{{sagittal| /||\ }}</big>
|}
The following enharmonics from the Spartan set are present (comma tempered out):
* {{Sagittal| //| }} = {{sagittal| /|) }} = {{sagittal| /|\ }} ([[325/324]], [[352/351]])
* {{Sagittal| /| }} = {{sagittal| |) }} ([[225/224]])
* {{Sagittal| |( }} = {{sagittal| |//| }} ([[5120/5103]])
 
See [[Sagittal notation #Revo|apotome complements]] for equivalent accidental pairs.
 
Featured below is the 41edo gamut notated using the best accidental approximants; in this case, pai/pao and pakai/pakao; the same sagittal sequence as [[34edo #Sagittal notation|34edo]].


==== Evo flavor ====
==== Evo flavor ====
Line 742: Line 951:
|-
|-
| 2.3
| 2.3
| {{monzo| 65 -41 }}
| {{Monzo| 65 -41 }}
| {{mapping| 41 65 }}
| {{Mapping| 41 65 }}
| −0.153
| −0.153
| 0.15
| 0.15
Line 750: Line 959:
| 2.3.5
| 2.3.5
| 3125/3072, 20000/19683
| 3125/3072, 20000/19683
| {{mapping| 41 65 95 }}
| {{Mapping| 41 65 95 }}
| +0.734
| +0.734
| 1.26
| 1.26
Line 757: Line 966:
| 2.3.5.7
| 2.3.5.7
| 225/224, 245/243, 1029/1024
| 225/224, 245/243, 1029/1024
| {{mapping| 41 65 95 115 }}
| {{Mapping| 41 65 95 115 }}
| +0.815
| +0.815
| 1.10
| 1.10
Line 764: Line 973:
| 2.3.5.7.11
| 2.3.5.7.11
| 100/99, 225/224, 243/242, 245/242
| 100/99, 225/224, 243/242, 245/242
| {{mapping| 41 65 95 115 142 }}
| {{Mapping| 41 65 95 115 142 }}
| +0.375
| +0.375
| 1.32
| 1.32
Line 771: Line 980:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 100/99, 105/104, 144/143, 196/195, 243/242
| 100/99, 105/104, 144/143, 196/195, 243/242
| {{mapping| 41 65 95 115 142 152 }}
| {{Mapping| 41 65 95 115 142 152 }}
| −0.060
| −0.060
| 1.55
| 1.55
Line 778: Line 987:
| 2.3.5.7.11.13.19
| 2.3.5.7.11.13.19
| 100/99, 105/104, 133/132, 144/143, 171/169, 196/195
| 100/99, 105/104, 133/132, 144/143, 171/169, 196/195
| {{mapping| 41 65 95 115 142 152 174 }}
| {{Mapping| 41 65 95 115 142 152 174 }}
| +0.111
| +0.111
| 1.49
| 1.49
| 5.10
| 5.10
|}
|}
* 41et is lower in relative error than any previous equal temperaments in the 3-, 13- and 19-limit. The next equal temperaments doing better in these subgroups are 53, 53, and 46, respectively. It is even more prominent in the 2.3.5.7.11.19 and 2.3.5.7.11.13.19 subgroup. The next equal temperaments doing better in these subgroups are 72 and 53, respectively.  
* 41et is lower in relative error than any previous equal temperaments in the 3- and 13-limit. The next equal temperament doing better in either subgroup is [[53edo|53]].  
* It is even better in the 2.3.5.7.11.19 and 2.3.5.7.11.13.19 subgroups. The next equal temperaments doing better in these subgroups are [[72edo|72]] and 53, respectively.
* It is also notable in the 7-, 11-, 17-, and 19-limit, with lower absolute errors than any previous equal temperaments.  


=== Commas ===
=== Commas ===
Line 800: Line 1,011:
| <abbr title="36893488147419103232/36472996377170786403">(40 digits)</abbr>
| <abbr title="36893488147419103232/36472996377170786403">(40 digits)</abbr>
| 19.84
| 19.84
| {{monzo| 65 -41 }}
| {{Monzo| 65 -41 }}
| Wa-41
| Wa-41
| 41-edo
| 41-edo
Line 808: Line 1,019:
| <abbr title="1953125/1889568">(14 digits)</abbr>
| <abbr title="1953125/1889568">(14 digits)</abbr>
| 57.27
| 57.27
| {{monzo| -5 -10 9 }}
| {{Monzo| -5 -10 9 }}
| Tritriyo
| Tritriyo
| y<sup>9</sup>
| y<sup>9</sup>
Line 816: Line 1,027:
| [[34171875/33554432|(16 digits)]]
| [[34171875/33554432|(16 digits)]]
| 31.57
| 31.57
| {{monzo| -25 7 6 }}
| {{Monzo| -25 7 6 }}
| Lala-tribiyo
| Lala-tribiyo
| LLy<sup>3</sup>
| LLy<sup>3</sup>
Line 824: Line 1,035:
| [[3125/3072]]
| [[3125/3072]]
| 29.61
| 29.61
| {{monzo| -10 -1 5 }}
| {{Monzo| -10 -1 5 }}
| Laquinyo
| Laquinyo
| Ly<sup>5</sup>
| Ly<sup>5</sup>
Line 832: Line 1,043:
| [[20000/19683|(10 digits)]]
| [[20000/19683|(10 digits)]]
| 27.66
| 27.66
| {{monzo| 5 -9 4 }}
| {{Monzo| 5 -9 4 }}
| Saquadyo
| Saquadyo
| sy<sup>4</sup>
| sy<sup>4</sup>
Line 840: Line 1,051:
| <abbr title="131072000/129140163">(18 digits)</abbr>
| <abbr title="131072000/129140163">(18 digits)</abbr>
| 25.71
| 25.71
| {{monzo| 20 -17 3 }}
| {{Monzo| 20 -17 3 }}
| Sasa-triyo
| Sasa-triyo
| ssy<sup>3</sup>
| ssy<sup>3</sup>
Line 848: Line 1,059:
| [[32805/32768|(10 digits)]]
| [[32805/32768|(10 digits)]]
| 1.95
| 1.95
| {{monzo| -15 8 1 }}
| {{Monzo| -15 8 1 }}
| Layo
| Layo
| Ly
| Ly
Line 856: Line 1,067:
| [[15625/15309|(10 digits)]]
| [[15625/15309|(10 digits)]]
| 35.37
| 35.37
| {{monzo| 0 -7 6 -1 }}
| {{Monzo| 0 -7 6 -1 }}
| Rutribiyo
| Rutribiyo
| ry<sup>6</sup>
| ry<sup>6</sup>
Line 864: Line 1,075:
| <abbr title="854296875/843308032">(18 digits)</abbr>
| <abbr title="854296875/843308032">(18 digits)</abbr>
| 22.41
| 22.41
| {{monzo| -10 7 8 -7 }}
| {{Monzo| -10 7 8 -7 }}
| Lasepru-aquadbiyo
| Lasepru-aquadbiyo
| Lr<sup>7</sup>y<sup>8</sup>
| Lr<sup>7</sup>y<sup>8</sup>
Line 872: Line 1,083:
| [[875/864]]
| [[875/864]]
| 21.90
| 21.90
| {{monzo| -5 -3 3 1 }}
| {{Monzo| -5 -3 3 1 }}
| Zotriyo
| Zotriyo
| zy<sup>3</sup>
| zy<sup>3</sup>
Line 880: Line 1,091:
| [[3125/3087]]
| [[3125/3087]]
| 21.18
| 21.18
| {{monzo| 0 -2 5 -3 }}
| {{Monzo| 0 -2 5 -3 }}
| Triru-aquinyo
| Triru-aquinyo
| r<sup>3</sup>y<sup>5</sup>
| r<sup>3</sup>y<sup>5</sup>
Line 888: Line 1,099:
| <abbr title="179200/177147">(12 digits)</abbr>
| <abbr title="179200/177147">(12 digits)</abbr>
| 19.95
| 19.95
| {{monzo| 10 -11 2 1 }}
| {{Monzo| 10 -11 2 1 }}
| Sazoyoyo
| Sazoyoyo
| szyy
| szyy
Line 896: Line 1,107:
| [[33075/32768|(10 digits)]]
| [[33075/32768|(10 digits)]]
| 16.14
| 16.14
| {{monzo| -15 3 2 2 }}
| {{Monzo| -15 3 2 2 }}
| Labizoyo
| Labizoyo
| Lzzyy
| Lzzyy
Line 904: Line 1,115:
| [[245/243]]
| [[245/243]]
| 14.19
| 14.19
| {{monzo| 0 -5 1 2 }}
| {{Monzo| 0 -5 1 2 }}
| Zozoyo
| Zozoyo
| zzy
| zzy
Line 912: Line 1,123:
| [[4000/3969]]
| [[4000/3969]]
| 13.47
| 13.47
| {{monzo| 5 -4 3 -2 }}
| {{Monzo| 5 -4 3 -2 }}
| Rurutriyo
| Rurutriyo
| rry<sup>3</sup>
| rry<sup>3</sup>
Line 920: Line 1,131:
| <abbr title="823543/819200">(12 digits)</abbr>
| <abbr title="823543/819200">(12 digits)</abbr>
| 9.15
| 9.15
| {{monzo| -15 0 -2 7 }}
| {{Monzo| -15 0 -2 7 }}
| Lasepzo-agugu
| Lasepzo-agugu
| Lz<sup>7</sup>gg
| Lz<sup>7</sup>gg
Line 928: Line 1,139:
| [[1029/1024]]
| [[1029/1024]]
| 8.43
| 8.43
| {{monzo| -10 1 0 3 }}
| {{Monzo| -10 1 0 3 }}
| Latrizo
| Latrizo
| Lz<sup>3</sup>
| Lz<sup>3</sup>
Line 936: Line 1,147:
| [[225/224]]
| [[225/224]]
| 7.71
| 7.71
| {{monzo| -5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| Ruyoyo
| Ruyoyo
| ryy
| ryy
Line 944: Line 1,155:
| [[16875/16807|(10 digits)]]
| [[16875/16807|(10 digits)]]
| 6.99
| 6.99
| {{monzo| 0 3 4 -5 }}
| {{Monzo| 0 3 4 -5 }}
| Quinru-aquadyo
| Quinru-aquadyo
| r<sup>5</sup>y<sup>4</sup>
| r<sup>5</sup>y<sup>4</sup>
Line 952: Line 1,163:
| [[10976/10935|(10 digits)]]
| [[10976/10935|(10 digits)]]
| 6.48
| 6.48
| {{monzo| 5 -7 -1 3 }}
| {{Monzo| 5 -7 -1 3 }}
| Satrizo-agu
| Satrizo-agu
| sz<sup>3</sup>g
| sz<sup>3</sup>g
Line 960: Line 1,171:
| [[5120/5103]]
| [[5120/5103]]
| 5.76
| 5.76
| {{monzo| 10 -6 1 -1 }}
| {{Monzo| 10 -6 1 -1 }}
| Saruyo
| Saruyo
| sry
| sry
Line 968: Line 1,179:
| [[33554432/33480783|(16 digits)]]
| [[33554432/33480783|(16 digits)]]
| 3.80
| 3.80
| {{monzo| 25 -14 0 -1 }}
| {{Monzo| 25 -14 0 -1 }}
| Sasaru
| Sasaru
| ssr
| ssr
Line 976: Line 1,187:
| [[2401/2400]]
| [[2401/2400]]
| 0.72
| 0.72
| {{monzo| -5 -1 -2 4 }}
| {{Monzo| -5 -1 -2 4 }}
| Bizozogu
| Bizozogu
| z<sup>4</sup>gg
| z<sup>4</sup>gg
Line 984: Line 1,195:
| <abbr title="163840/161051">(12 digits)</abbr>
| <abbr title="163840/161051">(12 digits)</abbr>
| 29.72
| 29.72
| {{monzo| 15 0 1 0 -5 }}
| {{Monzo| 15 0 1 0 -5 }}
| Saquinlu-ayo
| Saquinlu-ayo
| s1u<sup>5</sup>y
| s1u<sup>5</sup>y
Line 992: Line 1,203:
| [[245/242]]
| [[245/242]]
| 21.33
| 21.33
| {{monzo| -1 0 1 2 -2 }}
| {{Monzo| -1 0 1 2 -2 }}
| Luluzozoyo
| Luluzozoyo
| 1uuzzy
| 1uuzzy
Line 1,000: Line 1,211:
| [[100/99]]
| [[100/99]]
| 17.40
| 17.40
| {{monzo| 2 -2 2 0 -1 }}
| {{Monzo| 2 -2 2 0 -1 }}
| Luyoyo
| Luyoyo
| 1uyy
| 1uyy
Line 1,008: Line 1,219:
| [[1344/1331]]
| [[1344/1331]]
| 16.83
| 16.83
| {{monzo| 6 1 0 1 -3 }}
| {{Monzo| 6 1 0 1 -3 }}
| Trilu-azo
| Trilu-azo
| 1u<sup>3</sup>z
| 1u<sup>3</sup>z
Line 1,016: Line 1,227:
| [[896/891]]
| [[896/891]]
| 9.69
| 9.69
| {{monzo| 7 -4 0 1 -1 }}
| {{Monzo| 7 -4 0 1 -1 }}
| Saluzo
| Saluzo
| s1uz
| s1uz
Line 1,024: Line 1,235:
| [[65536/65219|(10 digits)]]
| [[65536/65219|(10 digits)]]
| 8.39
| 8.39
| {{monzo| 16 0 0 -2 -3 }}
| {{Monzo| 16 0 0 -2 -3 }}
| Satrilu-aruru
| Satrilu-aruru
| s1u<sup>3</sup>rr
| s1u<sup>3</sup>rr
Line 1,032: Line 1,243:
| [[243/242]]
| [[243/242]]
| 7.14
| 7.14
| {{monzo| -1 5 0 0 -2 }}
| {{Monzo| -1 5 0 0 -2 }}
| Lulu
| Lulu
| 1uu
| 1uu
Line 1,040: Line 1,251:
| [[385/384]]
| [[385/384]]
| 4.50
| 4.50
| {{monzo| -7 -1 1 1 1 }}
| {{Monzo| -7 -1 1 1 1 }}
| Lozoyo
| Lozoyo
| 1ozg
| 1ozg
Line 1,048: Line 1,259:
| [[441/440]]
| [[441/440]]
| 3.93
| 3.93
| {{monzo| -3 2 -1 2 -1 }}
| {{Monzo| -3 2 -1 2 -1 }}
| Luzozogu
| Luzozogu
| 1uzzg
| 1uzzg
Line 1,056: Line 1,267:
| [[1375/1372]]
| [[1375/1372]]
| 3.78
| 3.78
| {{monzo| -2 0 3 -3 1 }}
| {{Monzo| -2 0 3 -3 1 }}
| Lotriruyo
| Lotriruyo
| 1or<sup>3</sup>y
| 1or<sup>3</sup>y
Line 1,064: Line 1,275:
| [[540/539]]
| [[540/539]]
| 3.21
| 3.21
| {{monzo| 2 3 1 -2 -1 }}
| {{Monzo| 2 3 1 -2 -1 }}
| Lururuyo
| Lururuyo
| 1urry
| 1urry
Line 1,072: Line 1,283:
| [[3025/3024]]
| [[3025/3024]]
| 0.57
| 0.57
| {{monzo| -4 -3 2 -1 2 }}
| {{Monzo| -4 -3 2 -1 2 }}
| Loloruyoyo
| Loloruyoyo
| 1ooryy
| 1ooryy
Line 1,080: Line 1,291:
| [[151263/151250|<abbr title="151263/151250">(12 digits)</abbr>]]
| [[151263/151250|<abbr title="151263/151250">(12 digits)</abbr>]]
| 0.15
| 0.15
| {{monzo| -1 2 -4 5 -2 }}
| {{Monzo| -1 2 -4 5 -2 }}
| Luluquinzo-aquadgu
| Luluquinzo-aquadgu
| 1uuz<sup>5</sup>g<sup>4</sup>
| 1uuz<sup>5</sup>g<sup>4</sup>
Line 1,088: Line 1,299:
| [[343/338]]
| [[343/338]]
| 25.42
| 25.42
| {{monzo| -1 0 0 3 0 -2 }}
| {{Monzo| -1 0 0 3 0 -2 }}
| Thuthutrizo
| Thuthutrizo
| 3uuz<sup>3</sup>
| 3uuz<sup>3</sup>
Line 1,096: Line 1,307:
| [[105/104]]
| [[105/104]]
| 16.57
| 16.57
| {{monzo| -3 1 1 1 0 -1 }}
| {{Monzo| -3 1 1 1 0 -1 }}
| Thuzoyo
| Thuzoyo
| 3uzy
| 3uzy
Line 1,104: Line 1,315:
| [[28672/28431|(10 digits)]]
| [[28672/28431|(10 digits)]]
| 14.61
| 14.61
| {{monzo| 12 -7 0 1 0 -1 }}
| {{Monzo| 12 -7 0 1 0 -1 }}
| Sathuzo
| Sathuzo
| s3uz
| s3uz
Line 1,112: Line 1,323:
| [[275/273]]
| [[275/273]]
| 12.64
| 12.64
| {{monzo| 0 -1 2 -1 1 -1 }}
| {{Monzo| 0 -1 2 -1 1 -1 }}
| Thuloruyoyo
| Thuloruyoyo
| 3u1oryy
| 3u1oryy
Line 1,120: Line 1,331:
| [[144/143]]
| [[144/143]]
| 12.06
| 12.06
| {{monzo| 4 2 0 0 -1 -1 }}
| {{Monzo| 4 2 0 0 -1 -1 }}
| Thulu
| Thulu
| 3u1u
| 3u1u
Line 1,128: Line 1,339:
| [[196/195]]
| [[196/195]]
| 8.86
| 8.86
| {{monzo| 2 -1 -1 2 0 -1 }}
| {{Monzo| 2 -1 -1 2 0 -1 }}
| Thuzozogu
| Thuzozogu
| 3uzzg
| 3uzzg
Line 1,136: Line 1,347:
| [[640/637]]
| [[640/637]]
| 8.13
| 8.13
| {{monzo| 7 0 1 -2 0 -1 }}
| {{Monzo| 7 0 1 -2 0 -1 }}
| Thururuyo
| Thururuyo
| 3urry
| 3urry
Line 1,144: Line 1,355:
| [[1188/1183]]
| [[1188/1183]]
| 7.30
| 7.30
| {{monzo| 2 3 0 -1 1 -2 }}
| {{Monzo| 2 3 0 -1 1 -2 }}
| Thuthuloru
| Thuthuloru
| 3uu1or
| 3uu1or
Line 1,152: Line 1,363:
| [[31213/31104]]
| [[31213/31104]]
| 6.06
| 6.06
| {{monzo| -7 -5 0 4 0 1 }}
| {{Monzo| -7 -5 0 4 0 1 }}
| Thoquadzo
| Thoquadzo
| 3oz<sup>4</sup>3
| 3oz<sup>4</sup>3
Line 1,160: Line 1,371:
| [[325/324]]
| [[325/324]]
| 5.34
| 5.34
| {{monzo| -2 -4 2 0 0 1 }}
| {{Monzo| -2 -4 2 0 0 1 }}
| Thoyoyo
| Thoyoyo
| 3oyy
| 3oyy
Line 1,168: Line 1,379:
| [[352/351]]
| [[352/351]]
| 4.93
| 4.93
| {{monzo| 5 -3 0 0 1 -1 }}
| {{Monzo| 5 -3 0 0 1 -1 }}
| Thulo
| Thulo
| 3u1o
| 3u1o
Line 1,176: Line 1,387:
| [[364/363]]
| [[364/363]]
| 4.76
| 4.76
| {{monzo| 2 -1 0 1 -2 1 }}
| {{Monzo| 2 -1 0 1 -2 1 }}
| Tholuluzo
| Tholuluzo
| 3o1uuz
| 3o1uuz
Line 1,184: Line 1,395:
| [[847/845]]
| [[847/845]]
| 4.09
| 4.09
| {{monzo| 0 0 -1 1 2 -2 }}
| {{Monzo| 0 0 -1 1 2 -2 }}
| Thuthulolozogu
| Thuthulolozogu
| 3uu1oozg
| 3uu1oozg
Line 1,192: Line 1,403:
| [[729/728]]
| [[729/728]]
| 2.38
| 2.38
| {{monzo| -3 6 0 -1 0 -1 }}
| {{Monzo| -3 6 0 -1 0 -1 }}
| Lathuru
| Lathuru
| L3ur
| L3ur
Line 1,200: Line 1,411:
| [[2080/2079]]
| [[2080/2079]]
| 0.83
| 0.83
| {{monzo| 5 -3 1 -1 -1 1 }}
| {{Monzo| 5 -3 1 -1 -1 1 }}
| Tholuruyo
| Tholuruyo
| 3o1ury
| 3o1ury
| Ibnsinma
| Ibnsinma, sinaisma
|-
|-
| 13
| 13
| [[4096/4095]]
| [[4096/4095]]
| 0.42
| 0.42
| {{monzo| 12 -2 -1 -1 0 -1 }}
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| Sathurugu
| Sathurugu
| s3urg
| s3urg
| Schismina
| Minisma
|-
|-
| 13
| 13
| [[6656/6655]]
| [[6656/6655]]
| 0.26
| 0.26
| {{monzo| 9 0 -1 0 -3 1 }}
| {{Monzo| 9 0 -1 0 -3 1 }}
| Thotrilo-agu
| Thotrilo-agu
| 3u1o<sup>3</sup>g2
| 3u1o<sup>3</sup>g2
Line 1,224: Line 1,435:
| [[10648/10647|(10 digits)]]
| [[10648/10647|(10 digits)]]
| 0.16
| 0.16
| {{monzo| 3 -2 0 -1 3 -2 }}
| {{Monzo| 3 -2 0 -1 3 -2 }}
| Thuthutrilo-aru
| Thuthutrilo-aru
| 3uu1o<sup>3</sup>r
| 3uu1o<sup>3</sup>r
Line 1,232: Line 1,443:
| [[2187/2176]]
| [[2187/2176]]
| 8.73
| 8.73
| {{monzo| -7 7 0 0 0 0 -1 }}
| {{Monzo| -7 7 0 0 0 0 -1 }}
| Lasu
| Lasu
| L17u
| L17u
Line 1,240: Line 1,451:
| [[256/255]]
| [[256/255]]
| 6.78
| 6.78
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
| Sugu
| Sugu
| 17ug
| 17ug
Line 1,248: Line 1,459:
| [[715/714]]
| [[715/714]]
| 2.42
| 2.42
| {{monzo| -1 -1 1 -1 1 1 -1 }}
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| Sutholoruyo
| Sutholoruyo
| 17u3o1ory
| 17u3o1ory
Line 1,256: Line 1,467:
| [[210/209]]
| [[210/209]]
| 8.26
| 8.26
| {{monzo| 1 1 1 1 -1 0 0 -1 }}
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| Nuluzoyo
| Nuluzoyo
| 19u1uzy
| 19u1uzy
Line 1,264: Line 1,475:
| [[361/360]]
| [[361/360]]
| 4.80
| 4.80
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
| Nonogu
| Nonogu
| 19oog2
| 19oog2
Line 1,272: Line 1,483:
| [[513/512]]
| [[513/512]]
| 3.38
| 3.38
| {{monzo| -9 3 0 0 0 0 0 1 }}
| {{Monzo| -9 3 0 0 0 0 0 1 }}
| Lano
| Lano
| L19o
| L19o
Line 1,280: Line 1,491:
| [[1216/1215]]
| [[1216/1215]]
| 1.42
| 1.42
| {{monzo| 6 -5 -1 0 0 0 0 1 }}
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| Sanogu
| Sanogu
| s19og
| s19og
Line 1,288: Line 1,499:
| [[736/729]]
| [[736/729]]
| 16.54
| 16.54
| {{monzo| 5 -6 0 0 0 0 0 0 1 }}
| {{Monzo| 5 -6 0 0 0 0 0 0 1 }}
| Satwetho
| Satwetho
| s23o
| s23o
Line 1,296: Line 1,507:
| [[145/144]]
| [[145/144]]
| 11.98
| 11.98
| {{monzo| -4 -2 1 0 0 0 0 0 0 1 }}
| {{Monzo| -4 -2 1 0 0 0 0 0 0 1 }}
| Twenoyo
| Twenoyo
| 29oy
| 29oy
Retrieved from "https://en.xen.wiki/w/41edo"