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41edo can be seen as a tuning of the [[magic]] temperament, as well as [[superkleismic]], [[garibaldi]], [[miracle]], and multiple temperaments in the [[tetracot family]].  
41edo can be seen as a tuning of the [[magic]] temperament, as well as [[superkleismic]], [[garibaldi]], [[miracle]], and multiple temperaments in the [[tetracot family]].  


Various 13-limit [[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]].
Various 13-limit [[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; however, telepathy and sorcery merge into one not in 41edo but in [[22edo]].


41edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of Tetracot[7] can also be more accurately interpreted as [[21/19]]—which is equated with [[32/29]] above [[31/28]] below (both very near)—providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.
41edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of Tetracot[7] can also be more accurately interpreted as [[21/19]]—which is equated with [[32/29]] above [[31/28]] below (both very near)—providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.
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=== Subsets and supersets ===
=== Subsets and supersets ===
41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it contains [[41ed4]].  
41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it contains [[41ed4]]. Although not technically subsets, it essentially contains [[88cET]] as every third step and [[13edt]] as every fifth step.


[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo. Its step of 1\205 is called a ''mem''.
[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo. Its step of 1\205 is called a ''mem''.
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== Intervals ==
== Intervals ==
{{See also| 41edo solfege }}
{| class="wikitable center-1 right-2"
 
{| class="wikitable center-1 right-2 center-5 center-6 center-8 center-9"
|-
|-
! #
! #
! Cents
! Cents
! Approximate ratios*
! Approximate ratios*
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^d2)
! [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)
! [[41edo solfege|Kite's<br>solfege]]
! [[41edo solfege|Andrew's<br>solfege]]
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
| perfect unison
| {{UDnote|step=0}}
| P1
| D
| perfect unison
| P1
| D
| Da
| Do
|-
|-
| 1
| 1
| 29.3
| 29.3
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
| up-unison
| {{UDnote|step=1}}
| ^1
| ^D
| comma-wide unison, super unison
| K1/S1
| KD, SD
| Du
| Di
|-
|-
| 2
| 2
| 58.5
| 58.5
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
| dup-unison, downminor 2nd
| {{UDnote|step=2}}
| ^^1, vm2
| ^^D, vEb
| subminor 2nd, classic aug unison, uber unison
| sm2, kkA1, U1
| sEb, kkD#, UD
| Fro
| Ro
|-
|-
| 3
| 3
| 87.8
| 87.8
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
| down-aug 1sn, minor 2nd
| {{UDnote|step=3}}
| vA1, m2
|-
| vD#, Eb
| 4
| minor 2nd, comma-narrow augmented unison
| m2, kA1
| Eb, kD#
| Fra
| Rih
|-
| 4
| 117.1
| 117.1
| [[14/13]], [[15/14]], [[16/15]]
| [[14/13]], [[15/14]], [[16/15]]
| augmented 1sn, upminor 2nd
| {{UDnote|step=4}}
| A1, ^m2
| D#, ^Eb
| classic minor 2nd, augmented unison
| Km2, A1
| KEb, D#
| Fru
| Ra
|-
|-
| 5
| 5
| 146.3
| 146.3
| [[12/11]], [[13/12]]
| [[12/11]], [[13/12]]
| mid 2nd
| {{UDnote|step=5}}
| ~2
| ^D#, vvE
| neutral second, super augmented unison
| N2, SA1
| UEb/uE, sD#
| Ri
| Ru
|-
|-
| 6
| 6
| 175.6
| 175.6
| [[10/9]], [[11/10]], [[21/19]]
| [[10/9]], [[11/10]], [[21/19]]
| downmajor 2nd
| {{UDnote|step=6}}
| vM2
|-
| vE
| 7
| classic/comma-wide major 2nd
| 204.9
| kM2
| kE
| Ro
| Reh
|-
| 7
| 204.9
| [[9/8]]
| [[9/8]]
| major 2nd
| {{UDnote|step=7}}
| M2
| E
| major 2nd
| M2
| E
| Ra
| Re
|-
|-
| 8
| 8
| 234.1
| 234.1
| [[8/7]], [[15/13]]
| [[8/7]], [[15/13]]
| upmajor 2nd
| {{UDnote|step=8}}
| ^M2
| ^E
| supermajor 2nd
| SM2
| SE
| Ru
| Ri
|-
|-
| 9
| 9
| 263.4
| 263.4
| [[7/6]], [[22/19]]
| [[7/6]], [[22/19]]
| downminor 3rd
| {{UDnote|step=9}}
| vm3
|-
| vF
| 10
| subminor 3rd
| 292.7
| sm3
| sF
| No
| Ma
|-
| 10
| 292.7
| [[13/11]], [[19/16]], [[32/27]]
| [[13/11]], [[19/16]], [[32/27]]
| minor 3rd
| {{UDnote|step=10}}
| m3
| F
| minor 3rd
| m3
| F
| Na
| Meh
|-
|-
| 11
| 11
| 322.0
| 322.0
| [[6/5]]
| [[6/5]]
| upminor 3rd
| {{UDnote|step=11}}
| ^m3
| ^F
| classic minor 3rd
| Km3
| KF
| Nu
| Me
|-
|-
| 12
| 12
| 351.2
| 351.2
| [[11/9]], [[16/13]]
| [[11/9]], [[16/13]]
| mid 3rd
| {{UDnote|step=12}}
| ~3
|-
| ^^F, vGb
| 13
| neutral 3rd, sub diminished 4th
| 380.5
| N3, sd4
| UF/uF#, sGb
| Mi
| Mu
|-
| 13
| 380.5
| [[5/4]], [[26/21]]
| [[5/4]], [[26/21]]
| downmajor 3rd
| {{UDnote|step=13}}
| vM3
| vF#, Gb
| classic major 3rd, diminished 4th
| kM3, d4
| kF#, Gb
| Mo
| Mi
|-
|-
| 14
| 14
| 409.8
| 409.8
| [[14/11]], [[19/15]], [[24/19]]
| [[14/11]], [[19/15]], [[24/19]]
| major 3rd
| {{UDnote|step=14}}
| M3
| F#, ^Gb
| major 3rd, comma-wide diminished 4th
| M3, Kd4
| F#, KGb
| Ma
| Maa
|-
|-
| 15
| 15
| 439.0
| 439.0
| [[9/7]], [[32/25]]
| [[9/7]], [[32/25]]
| upmajor 3rd
| {{UDnote|step=15}}
| ^M3
|-
| ^F#, vvG
| 16
| supermajor 3rd, classic diminished 4th
| 468.3
| SM3, KKd4
| SF#, KKGb
| Mu
| Mo
|-
| 16
| 468.3
| [[21/16]], [[13/10]]
| [[21/16]], [[13/10]]
| down-4th
| {{UDnote|step=16}}
| v4
| vG
| sub 4th
| s4
| sG
| Fo
| Fe
|-
|-
| 17
| 17
| 497.6
| 497.6
| [[4/3]]
| [[4/3]]
| perfect 4th
| {{UDnote|step=17}}
| P4
| G
| perfect 4th
| P4
| G
| Fa
| Fa
|-
|-
| 18
| 18
| 526.8
| 526.8
| [[15/11]], [[19/14]], [[27/20]]
| [[15/11]], [[19/14]], [[27/20]]
| up-4th
| {{UDnote|step=18}}
| ^4
|-
| ^G
| 19
| comma-wide 4th
| 556.1
| K4
| KG
| Fu
| Fih
|-
| 19
| 556.1
| [[11/8]], [[18/13]], [[26/19]]
| [[11/8]], [[18/13]], [[26/19]]
| mid-4th, downdim 5th
| {{UDnote|step=19}}
| ~4, vd5
| ^^G, vAb
| uber/neutral 4th, classic augmented 4th
| U4/N4, kkA4
| UG, kkG#
| Fi/Sho
| Fu
|-
|-
| 20
| 20
| 585.4
| 585.4
| [[7/5]], [[45/32]]
| [[7/5]], [[45/32]]
| downaug 4th, dim 5th
| {{UDnote|step=20}}
| vA4, d5
| vG#, Ab
| comma-narrow augmented 4th, diminished 5th
| kA4/d5
| kG#, Ab
| Po/Sha
| Fi
|-
|-
| 21
| 21
| 614.6
| 614.6
| [[10/7]], [[64/45]]
| [[10/7]], [[64/45]]
| aug 4th, updim 5th
| {{UDnote|step=21}}
| A4, ^d5
|-
| G#, ^Ab
| 22
| augmented 4th, comma-wide diminished 5th
| 643.9
| A4/Kd5
| G#, KAb
| Pa/Shu
| Se
|-
| 22
| 643.9
| [[13/9]], [[16/11]], [[19/13]]
| [[13/9]], [[16/11]], [[19/13]]
| mid-5th, upaug 4th
| {{UDnote|step=22}}
| ~5, ^A4
| ^G#, vvA
| unter/neutral 5th, classic diminished 5th
| u5/N5, KKd5
| uA, KKAb
| Pu/Si
| Su
|-
|-
| 23
| 23
| 673.2
| 673.2
| [[22/15]], [[28/19]], [[40/27]]
| [[22/15]], [[28/19]], [[40/27]]
| down-5th
| {{UDnote|step=23}}
| v5
| vA
| comma-narrow 5th
| k5
| kA
| So
| Sih
|-
|-
| 24
| 24
| 702.4
| 702.4
| [[3/2]]
| [[3/2]]
| perfect 5th
| {{UDnote|step=24}}
| P5
|-
| A
| 25
| perfect 5th
| 731.7
| P5
| A
| Sa
| Sol
|-
| 25
| 731.7
| [[20/13]], [[32/21]]
| [[20/13]], [[32/21]]
| up-5th
| {{UDnote|step=25}}
| ^5
| ^A
| super 5th
| S5
| SA
| Su
| Si
|-
|-
| 26
| 26
| 761.0
| 761.0
| [[14/9]], [[25/16]]
| [[14/9]], [[25/16]]
| downminor 6th
| {{UDnote|step=26}}
| vm6
| ^^A, vBb
| subminor 6th, classic augmented 5th
| sm6
| sBb, kkA#
| Flo
| Lo
|-
|-
| 27
| 27
| 790.2
| 790.2
| [[11/7]], [[19/12]], [[30/19]]
| [[11/7]], [[19/12]], [[30/19]]
| minor 6th
| {{UDnote|step=27}}
| m6
|-
| vA#, Bb
| 28
| minor 6th, comma-narrow augmented 5th
| 819.5
| m6
| Bb, kA#
| Fla
| Leh
|-
| 28
| 819.5
| [[8/5]], [[21/13]]
| [[8/5]], [[21/13]]
| upminor 6th
| {{UDnote|step=28}}
| ^m6
| A#, ^Bb
| classic minor 6th, augmented 5th
| Km6, A5
| KBb, A#
| Flu
| Le
|-
|-
| 29
| 29
| 848.8
| 848.8
| [[13/8]], [[18/11]]
| [[13/8]], [[18/11]]
| mid 6th
| {{UDnote|step=29}}
| ~6
| ^A#, vvB
| neutral 6th, super augmented 5th
| N6
| UBb/uB, sA#
| Li
| Lu
|-
|-
| 30
| 30
| 878.0
| 878.0
| [[5/3]]
| [[5/3]]
| downmajor 6th
| {{UDnote|step=30}}
| vM6
|-
| vB
| 31
| classic major 6th
| 907.3
| kM6
| kB
| Lo
| La
|-
| 31
| 907.3
| [[22/13]], [[27/16]], [[32/19]]
| [[22/13]], [[27/16]], [[32/19]]
| major 6th
| {{UDnote|step=31}}
| M6
| B
| major 6th
| M6
| B
| La
| Laa
|-
|-
| 32
| 32
| 936.6
| 936.6
| [[12/7]], [[19/11]]
| [[12/7]], [[19/11]]
| upmajor 6th
| {{UDnote|step=32}}
| ^M6
| ^B
| supermajor 6th
| SM6
| SB
| Lu
| Li
|-
|-
| 33
| 33
| 965.9
| 965.9
| [[7/4]], [[26/15]]
| [[7/4]], [[26/15]]
| downminor 7th
| {{UDnote|step=33}}
| vm7
|-
| vC
| 34
| subminor 7th
| 995.1
| sm7
| sC
| Tho
| Ta
|-
| 34
| 995.1
| [[16/9]]
| [[16/9]]
| minor 7th
| {{UDnote|step=34}}
| m7
| C
| minor 7th
| m7
| C
| Tha
| Teh
|-
|-
| 35
| 35
| 1024.4
| 1024.4
| [[9/5]], [[20/11]], [[38/21]]
| [[9/5]], [[20/11]], [[38/21]]
| upminor 7th
| {{UDnote|step=35}}
| ^m7
| ^C
| classic/comma-wide minor seventh
| Km7
| KC
| Thu
| Te
|-
|-
| 36
| 36
| 1053.7
| 1053.7
| [[11/6]], [[24/13]]
| [[11/6]], [[24/13]]
| mid 7th
| {{UDnote|step=36}}
| ~7
|-
| ^^C, vDb
| 37
| neutral 7th, sub diminished 8ve
| 1082.9
| N7
| UC/uC#, sDb
| Ti
| Tu
|-
| 37
| 1082.9
| [[13/7]], [[15/8]], [[28/15]]
| [[13/7]], [[15/8]], [[28/15]]
| downmajor 7th
| {{UDnote|step=37}}
| vM7
| vC#, Db
| classic major 7th, diminished 8ve
| kM7, d8
| kC#, Db
| To
| Ti
|-
|-
| 38
| 38
| 1112.2
| 1112.2
| [[19/10]], [[21/11]], [[36/19]], [[40/21]]
| [[19/10]], [[21/11]], [[36/19]], [[40/21]]
| major 7th
| {{UDnote|step=38}}
| M7
| C#, ^Db
| major 7th, comma-wide diminished 8ve
| M7, Kd8
| C#, KDb
| Ta
| Taa
|-
|-
| 39
| 39
| 1141.5
| 1141.5
| [[27/14]], [[35/18]], [[48/25]], [[64/33]]
| [[27/14]], [[35/18]], [[48/25]], [[64/33]]
| upmajor 7th
| {{UDnote|step=39}}
| ^M7
|-
| ^C#, vvD
| 40
| supermajor 7th, classic dim 8ve, unter 8ve
| 1170.7
| SM7, KKd8, U8
| SC#, KKDb, u8
| Tu
| To
|-
| 40
| 1170.7
| [[49/25]], [[63/32]], [[96/49]], [[160/81]]
| [[49/25]], [[63/32]], [[96/49]], [[160/81]]
| dim 8ve
| {{UDnote|step=40}}
| v8
| vD
| comma-narrow 8ve, sub 8ve
| k8/s8
| kD, sD
| Do
| Da
|-
|-
| 41
| 41
| 1200.0
| 1200.0
| [[2/1]]
| [[2/1]]
| perfect 8ve
| {{UDnote|step=41}}
| P8
| D
| perfect 8ve
| P8
| D
| Da
| Do
|}
|}
<nowiki>*</nowiki> Based on treating 41edo as a 2.3.5.7.11.13.19 subgroup temperament; other approaches are possible.
<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 ===
=== Proposed interval names and solfèges ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
{{See also| 41edo solfege }}


{| class="wikitable center-all"
{| 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
|-
|-
! Quality
! #
! [[Color notation|Color]]
! Cents
! Monzo format
! 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)
! Examples
! colspan="3" | [[SKULO interval names|SKULO notation]]<br>(K or S = 1, U = 2)
! Kite's<br>solfège
! Andrew's<br>solfège
|-
|-
| downminor
| 0
| zo
| 0.0
| (a, b, 0, 1)
| perfect unison
| 7/6, 7/4
| P1
| D
| perfect unison
| P1
| D
| Da
| Do
|-
|-
| minor
| 1
| fourthward wa
| 29.3
| (a, b) with b < -1
| up-unison
| 32/27, 16/9
| ^1
|-
| ^D
| upminor
| comma-wide unison, super unison
| gu
| K1/S1
| (a, b, -1)
| KD, SD
| 6/5, 9/5
| Du
|-
| Di
| mid
|-
| ilo
| 2
| (a, b, 0, 0, 1)
| 58.5
| 11/9, 11/6
| dup-unison, downminor 2nd
|-
| ^^1, vm2
| "
| ^^D, vEb
| lu
| subminor 2nd, classic aug unison, uber unison
| (a, b, 0, 0, -1)
| sm2, kkA1, U1
| 12/11, 18/11
| sEb, kkD#, UD
|-
| Fro
| downmajor
| Ro
| yo
|-
| (a, b, 1)
| 3
| 5/4, 5/3
| 87.8
|-
| down-aug 1sn, minor 2nd
| major
| vA1, m2
| fifthward wa
| vD#, Eb
| (a, b) with b > 1
| minor 2nd, comma-narrow augmented unison
| 9/8, 27/16
| m2, kA1
|-
| Eb, kD#
| upmajor
| Fra
| ru
| Rih
| (a, b, 0, -1)
|-
| 9/7, 12/7
| 4
|}
| 117.1
 
| augmented 1sn, upminor 2nd
All 41edo chords can be named using ups and downs. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
| A1, ^m2
 
| D#, ^Eb
{| class="wikitable center-all"
| classic minor 2nd, augmented unison
|-
| Km2, A1
! [[Color notation|Color of the 3rd]]
| KEb, D#
! JI chord
| Fru
! Notes as edosteps
| Ra
! Notes of C chord
|-
| 5
| 146.3
| mid 2nd
| ~2
| ^D#, vvE
| neutral second, super augmented unison
| N2, SA1
| UEb/uE, sD#
| Ri
| Ru
|-
| 6
| 175.6
| downmajor 2nd
| vM2
| vE
| classic/comma-wide major 2nd
| kM2
| kE
| Ro
| Reh
|-
| 7
| 204.9
| major 2nd
| M2
| E
| major 2nd
| M2
| E
| Ra
| Re
|-
| 8
| 234.1
| upmajor 2nd
| ^M2
| ^E
| supermajor 2nd
| SM2
| SE
| Ru
| Ri
|-
| 9
| 263.4
| downminor 3rd
| vm3
| vF
| subminor 3rd
| sm3
| sF
| No
| Ma
|-
| 10
| 292.7
| minor 3rd
| m3
| F
| minor 3rd
| m3
| F
| Na
| Meh
|-
| 11
| 322.0
| upminor 3rd
| ^m3
| ^F
| classic minor 3rd
| Km3
| KF
| Nu
| Me
|-
| 12
| 351.2
| mid 3rd
| ~3
| ^^F, vGb
| neutral 3rd, sub diminished 4th
| N3, sd4
| UF/uF#, sGb
| Mi
| Mu
|-
| 13
| 380.5
| downmajor 3rd
| vM3
| vF#, Gb
| classic major 3rd, diminished 4th
| kM3, d4
| kF#, Gb
| Mo
| Mi
|-
| 14
| 409.8
| major 3rd
| M3
| F#, ^Gb
| major 3rd, comma-wide diminished 4th
| M3, Kd4
| F#, KGb
| Ma
| Maa
|-
| 15
| 439.0
| upmajor 3rd
| ^M3
| ^F#, vvG
| supermajor 3rd, classic diminished 4th
| SM3, KKd4
| SF#, KKGb
| Mu
| Mo
|-
| 16
| 468.3
| down-4th
| v4
| vG
| sub 4th
| s4
| sG
| Fo
| Fe
|-
| 17
| 497.6
| perfect 4th
| P4
| G
| perfect 4th
| P4
| G
| Fa
| Fa
|-
| 18
| 526.8
| up-4th
| ^4
| ^G
| comma-wide 4th
| K4
| KG
| Fu
| Fih
|-
| 19
| 556.1
| mid-4th, downdim 5th
| ~4, vd5
| ^^G, vAb
| uber/neutral 4th, classic augmented 4th
| U4/N4, kkA4
| UG, kkG#
| Fi/Sho
| Fu
|-
| 20
| 585.4
| downaug 4th, dim 5th
| vA4, d5
| vG#, Ab
| comma-narrow augmented 4th, diminished 5th
| kA4/d5
| kG#, Ab
| Po/Sha
| Fi
|-
| 21
| 614.6
| aug 4th, updim 5th
| A4, ^d5
| G#, ^Ab
| augmented 4th, comma-wide diminished 5th
| A4/Kd5
| G#, KAb
| Pa/Shu
| Se
|-
| 22
| 643.9
| mid-5th, upaug 4th
| ~5, ^A4
| ^G#, vvA
| unter/neutral 5th, classic diminished 5th
| u5/N5, KKd5
| uA, KKAb
| Pu/Si
| Su
|-
| 23
| 673.2
| down-5th
| v5
| vA
| comma-narrow 5th
| k5
| kA
| So
| Sih
|-
| 24
| 702.4
| perfect 5th
| P5
| A
| perfect 5th
| P5
| A
| Sa
| Sol
|-
| 25
| 731.7
| up-5th
| ^5
| ^A
| super 5th
| S5
| SA
| Su
| Si
|-
| 26
| 761.0
| downminor 6th
| vm6
| ^^A, vBb
| subminor 6th, classic augmented 5th
| sm6
| sBb, kkA#
| Flo
| Lo
|-
| 27
| 790.2
| minor 6th
| m6
| vA#, Bb
| minor 6th, comma-narrow augmented 5th
| m6
| Bb, kA#
| Fla
| Leh
|-
| 28
| 819.5
| upminor 6th
| ^m6
| A#, ^Bb
| classic minor 6th, augmented 5th
| Km6, A5
| KBb, A#
| Flu
| Le
|-
| 29
| 848.8
| mid 6th
| ~6
| ^A#, vvB
| neutral 6th, super augmented 5th
| N6
| UBb/uB, sA#
| Li
| Lu
|-
| 30
| 878.0
| downmajor 6th
| vM6
| vB
| classic major 6th
| kM6
| kB
| Lo
| La
|-
| 31
| 907.3
| major 6th
| M6
| B
| major 6th
| M6
| B
| La
| Laa
|-
| 32
| 936.6
| upmajor 6th
| ^M6
| ^B
| supermajor 6th
| SM6
| SB
| Lu
| Li
|-
| 33
| 965.9
| downminor 7th
| vm7
| vC
| subminor 7th
| sm7
| sC
| Tho
| Ta
|-
| 34
| 995.1
| minor 7th
| m7
| C
| minor 7th
| m7
| C
| Tha
| Teh
|-
| 35
| 1024.4
| upminor 7th
| ^m7
| ^C
| classic/comma-wide minor seventh
| Km7
| KC
| Thu
| Te
|-
| 36
| 1053.7
| mid 7th
| ~7
| ^^C, vDb
| neutral 7th, sub diminished 8ve
| N7
| UC/uC#, sDb
| Ti
| Tu
|-
| 37
| 1082.9
| downmajor 7th
| vM7
| vC#, Db
| classic major 7th, diminished 8ve
| kM7, d8
| kC#, Db
| To
| Ti
|-
| 38
| 1112.2
| major 7th
| M7
| C#, ^Db
| major 7th, comma-wide diminished 8ve
| M7, Kd8
| C#, KDb
| Ta
| Taa
|-
| 39
| 1141.5
| upmajor 7th
| ^M7
| ^C#, vvD
| supermajor 7th, classic dim 8ve, unter 8ve
| SM7, KKd8, U8
| SC#, KKDb, u8
| Tu
| To
|-
| 40
| 1170.7
| dim 8ve
| v8
| vD
| comma-narrow 8ve, sub 8ve
| k8/s8
| kD, sD
| Do
| Da
|-
| 41
| 1200.0
| perfect 8ve
| P8
| D
| perfect 8ve
| P8
| D
| Da
| Do
|}
 
=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
 
{| class="wikitable center-all"
|-
! Quality
! [[Color notation|Color]]
! Monzo format
! Examples
|-
| downminor
| zo
| (a, b, 0, 1)
| 7/6, 7/4
|-
| minor
| fourthward wa
| (a, b) with b < -1
| 32/27, 16/9
|-
| upminor
| gu
| (a, b, -1)
| 6/5, 9/5
|-
| mid
| ilo
| (a, b, 0, 0, 1)
| 11/9, 11/6
|-
| "
| lu
| (a, b, 0, 0, -1)
| 12/11, 18/11
|-
| downmajor
| yo
| (a, b, 1)
| 5/4, 5/3
|-
| major
| fifthward wa
| (a, b) with b > 1
| 9/8, 27/16
|-
| upmajor
| ru
| (a, b, 0, -1)
| 9/7, 12/7
|}
 
All 41edo chords can be named using ups and downs. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
 
{| class="wikitable center-all"
|-
! [[Color notation|Color of the 3rd]]
! JI chord
! Notes as edosteps
! Notes of C chord
! Written name
! Written name
! Spoken name
! Spoken name
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]]. This can be done by combining sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
The notes within an octave from A are thus:


{{Sharpness-sharp4}}
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


The notes within an octave from A are thus:
=== 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}}


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
Half-sharps and half-flats can be used to avoid double arrows:
{{Ups and downs sharpness|41|true}}


=== 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 689: 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 ====
<imagemap>
{{Sagittal chart|Evo}}
File:41-EDO_Evo_Sagittal.svg
 
desc none
==== Evo-SZ flavor ====
rect 80 0 300 50 [[Sagittal_notation]]
{{Sagittal chart|Evo-SZ}}
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:41-EDO_Evo_Sagittal.svg]]
</imagemap>


==== Revo flavor ====
==== Revo flavor ====
<imagemap>
{{Sagittal chart}}
File:41-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 671 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:41-EDO_Revo_Sagittal.svg]]
</imagemap>


We also have a diagram from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], which gives multiple spellings for each pitch, and up to the double-apotome:
We also have a diagram from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], which gives multiple spellings for each pitch, and up to the double-apotome:


[[File:41edo Sagittal.png|800px]]
[[File:41edo Sagittal.png|800px]]
==== Evo-SZ flavor ====
<imagemap>
File:41-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:41-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>


== Approximation to JI ==
== Approximation to JI ==
Line 763: 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 771: 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 778: 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 785: 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 792: 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 799: 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 821: 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 829: 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 837: 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 845: 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 853: 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 861: 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 869: 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 877: 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 885: 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 893: 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 901: 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 909: 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 917: 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 925: 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 933: 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 941: 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 949: 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 957: 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 965: 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 973: 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 981: 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 989: 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 997: 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 1,005: 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 1,013: 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 1,021: 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 1,029: 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 1,037: 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 1,045: 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 1,053: 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 1,061: 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 1,069: 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 1,077: 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 1,085: 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 1,093: 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 1,101: 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 1,109: 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 1,117: 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 1,125: 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 1,133: 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 1,141: 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 1,149: 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 1,157: 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 1,165: 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 1,173: 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 1,181: 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 1,189: 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 1,197: 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 1,205: 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 1,213: 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 1,221: 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 1,245: 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 1,253: 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 1,261: 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 1,269: 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 1,277: 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 1,285: 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 1,293: 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 1,301: 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 1,309: 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 1,317: 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
Line 2,126: Line 2,316:
[[Category:Magic]]
[[Category:Magic]]
[[Category:Superkleismic]]
[[Category:Superkleismic]]
[[Category:Supermagic]]
[[Category:Keemic]]
[[Category:Tetracot]]
[[Category:Tetracot]]
[[Category:Octacot]]
[[Category:Octacot]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/41edo"