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{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia| 41 equal temperament }}
{{Wikipedia| 41 equal temperament }}
{{EDO intro|41}}
{{ED intro}}


== Theory ==
== Theory ==
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41edo is perhaps the smallest edo with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest one to tune the 9-odd-limit [[consistency|distinctly consistent]], but it is also [[consistency #Consistency to distance d|consistent to distance 2]]. In other words, all intervals in the 9-odd-limit are more in-tune than out of tune. It is also the first edo to either match or improve on 12edo's accuracy of every harmonic up to the 16th, and no interval from the [[11-odd-limit]] except for [[11/10]] and [[20/11]] is represented with more than 10 cents of error in it. Apart from the full 13-limit, it is even more prominent as a 2.3.5.7.11.19.29.31 [[subgroup temperament]] for its size.  
41edo is perhaps the smallest edo with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest one to tune the 9-odd-limit [[consistency|distinctly consistent]], but it is also [[consistency #Consistency to distance d|consistent to distance 2]]. In other words, all intervals in the 9-odd-limit are more in-tune than out of tune. It is also the first edo to either match or improve on 12edo's accuracy of every harmonic up to the 16th, and no interval from the [[11-odd-limit]] except for [[11/10]] and [[20/11]] is represented with more than 10 cents of error in it. Apart from the full 13-limit, it is even more prominent as a 2.3.5.7.11.19.29.31 [[subgroup temperament]] for its size.  


41et is used by the [[Kite Guitar]], see below in [[#Instruments]].
41edo is used by the [[Kite Guitar]], see below in [[#Instruments]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|41|columns=9}}
{{Harmonics in equal|41|columns=11}}
{{Harmonics in equal|41|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 41edo (continued)}}
{{Harmonics in equal|41|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 41edo (continued)}}


=== As a tuning of other temperaments ===
=== As a tuning of other temperaments ===
41edo can be seen as a tuning of the [[garibaldi temperament|garibaldi]] temperament<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref>, as well as [[miracle]], [[magic]], [[superkleismic]], 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 family|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]].
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.
[[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''.
 
[[2460edo]] has potential for a 41edo analog to [[Cent|cents]]. It divides the 41edo step into 60 equal parts, and 60 is a highly composite (a.k.a. antiprime) number, so it contains many other multiples of 41edo, including 205edo, and also contains [[12edo]] among other equal tunings. It also accurately represents [[14afdo|mode 14 of the harmonic series]], as it is consistent all the way up to the 27-odd-limit. This allows for precise detunings in a 41-tone framework to approximate pure just intonation more closely, especially for some higher harmonics. Its step of 1\2460 is called a ''mina''.


== 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
| [[81/80]], [[64/63]], [[49/48]]
| [[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]], [[36/35]], [[33/32]]
| [[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
| [[21/20]], [[22/21]], [[19/18]], [[20/19]]
| [[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
| [[16/15]], [[15/14]], [[14/13]]
| [[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
| [[13/11]], [[19/16]], [[32/27]]
| sF
| {{UDnote|step=10}}
| No
| Ma
|-
| 10
| 292.7
| [[32/27]], [[13/11]], [[19/16]]
| minor 3rd
| 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
| classicminor 3rd
| Km3
| KF
| Nu
| Me
|-
|-
| 12
| 12
| 351.2
| 351.2
| [[11/9]], [[27/22]], [[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
| [[81/64]], [[14/11]], [[24/19]], [[19/15]]
| [[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
| [[27/20]], [[15/11]], [[19/14]]
| [[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
| [[13/9]], [[16/11]], [[19/13]]
| G#, KAb
| {{UDnote|step=22}}
| Pa/Shu
| Se
|-
| 22
| 643.9
| [[16/11]], [[13/9]], [[19/13]]
| mid-5th, upaug 4th
| ~5, ^A4
| ^G#, vvA
| unter/neutral 5th, classic diminished 5th
| u5/N5, KKd5
| uA, KKAb
| Pu/Si
| Su
|-
|-
| 23
| 23
| 673.2
| 673.2
| [[40/27]], [[22/15]], [[28/19]]
| [[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
| [[20/13]], [[32/21]]
| A
| {{UDnote|step=25}}
| Sa
| Sol
|-
| 25
| 731.7
| [[32/21]], [[20/13]]
| up-5th
| ^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
| [[128/81]], [[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
| [[18/11]], [[44/27]], [[13/8]]
| [[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
| [[22/13]], [[27/16]], [[32/19]]
| kB
| {{UDnote|step=31}}
| Lo
| La
|-
| 31
| 907.3
| [[27/16]], [[22/13]], [[32/19]]
| major 6th
| 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
| [[13/7]], [[15/8]], [[28/15]]
| UC/uC#, sDb
| {{UDnote|step=37}}
| Ti
| Tu
|-
| 37
| 1082.9
| [[15/8]], [[28/15]], [[13/7]]
| downmajor 7th
| vM7
| vC#, Db
| classic major 7th, diminished 8ve
| kM7, d8
| kC#, Db
| To
| Ti
|-
|-
| 38
| 38
| 1112.2
| 1112.2
| [[40/21]], [[21/11]], [[36/19]], [[19/10]]
| [[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
| [[48/25]], [[27/14]], [[35/18]], [[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
| [[49/25]], [[63/32]], [[96/49]], [[160/81]]
| SC#, KKDb, u8
| {{UDnote|step=40}}
| Tu
| To
|-
| 40
| 1170.7
| [[160/81]], [[63/32]], [[96/49]]
| dim 8ve
| 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
| 2
| ilo
| 58.5
| (a, b, 0, 0, 1)
| dup-unison, downminor 2nd
| 11/9, 11/6
| ^^1, vm2
| ^^D, vEb
| subminor 2nd, classic aug unison, uber unison
| sm2, kkA1, U1
| sEb, kkD#, UD
| Fro
| Ro
|-
|-
| "
| 3
| lu
| 87.8
| (a, b, 0, 0, -1)
| down-aug 1sn, minor 2nd
| 12/11, 18/11
| vA1, m2
| vD#, Eb
| minor 2nd, comma-narrow augmented unison
| m2, kA1
| Eb, kD#
| Fra
| Rih
|-
|-
| downmajor
| 4
| yo
| 117.1
| (a, b, 1)
| augmented 1sn, upminor 2nd
| 5/4, 5/3
| A1, ^m2
| D#, ^Eb
| classic minor 2nd, augmented unison
| Km2, A1
| KEb, D#
| Fru
| Ra
|-
|-
| major
| 5
| fifthward wa
| 146.3
| (a, b) with b > 1
| mid 2nd
| 9/8, 27/16
| ~2
|-
| ^D#, vvE
| upmajor
| neutral second, super augmented unison
| ru
| N2, SA1
| (a, b, 0, -1)
| UEb/uE, sD#
| 9/7, 12/7
| Ri
|}
| Ru
 
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]]
| 6
! JI chord
| 175.6
! Notes as edosteps
| downmajor 2nd
! Notes of C chord
| vM2
! Written name
| vE
! Spoken name
| classic/comma-wide major 2nd
| kM2
| kE
| Ro
| Reh
|-
|-
| zo (7-over)
| 7
| 6:7:9
| 204.9
| 0-9-24
| major 2nd
| C vEb G
| M2
| Cvm
| E
| C downminor
| major 2nd
| M2
| E
| Ra
| Re
|-
|-
| gu (5-under)
| 8
| 10:12:15
| 234.1
| 0-11-24
| upmajor 2nd
| C ^Eb G
| ^M2
| C^m
| ^E
| C upminor
| supermajor 2nd
|-
| SM2
| ilo (11-over)
| SE
| 18:22:27
| Ru
| 0-12-24
| Ri
| C vvE G
| C~
| C mid
|-
|-
| yo (5-over)
| 9
| 4:5:6
| 263.4
| 0-13-24
| downminor 3rd
| C vE G
| vm3
| Cv
| vF
| C downmajor or C down
| subminor 3rd
| sm3
| sF
| No
| Ma
|-
|-
| ru (7-under)
| 10
| 14:18:21
| 292.7
| 0-15-24
| minor 3rd
| C ^E G
| m3
| C^
| F
| C upmajor or C up
| minor 3rd
|}
| m3
 
| F
Other common triads are
| Na
* 0-10-20 = D F Ab = Dd = D dim
| Meh
* 0-10-21 = D F ^Ab = Dd(^5) = D dim up-five
|-
* 0-10-22 = D F vvA = Dm(~5) = D minor mid-five
| 11
* 0-10-23 = D F vA = Dm(v5) = D minor down-five
| 322.0
* 0-10-24 = D F A = Dm = D minor
| upminor 3rd
* 0-14-24 = D F# A = D = D or D major
| ^m3
* 0-14-25 = D F# ^A = D(^5) = D up-five
| ^F
* 0-14-26 = D F# ^^A = D(^^5) = D half-aug
| classic minor 3rd
* 0-14-27 = D F# vA# = Da(v5) = D aug down-five or perhaps D(v#5) = D downsharp-five
| Km3
* 0-14-28 = D F# A# = Da = D aug
| KF
 
| Nu
For a more complete list, see [[41edo Chord Names]] and [[Ups and downs notation #Chords and chord progressions]].
| Me
 
|-
== Notations ==
| 12
=== Ups and downs notation ===
| 351.2
41edo can 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]]:
| mid 3rd
 
| ~3
{{Sharpness-sharp4}}
| ^^F, vGb
 
| neutral 3rd, sub diminished 4th
The notes within an octave from A are thus:
| N3, sd4
 
| UF/uF#, sGb
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
| Mi
 
| Mu
=== 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:
| 13
 
| 380.5
{{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}}
| downmajor 3rd
 
| vM3
Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.
| vF#, Gb
 
| classic major 3rd, diminished 4th
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.
| kM3, d4
 
| kF#, Gb
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.
| Mo
 
| Mi
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[34edo #Sagittal notation|34edo]].
 
==== Evo flavor ====
<imagemap>
File:41-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
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 ====
<imagemap>
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:
 
[[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 ==
=== Interval mappings ===
{{Q-odd-limit intervals|41}}
 
=== Zeta peak index ===
{| class="wikitable center-all"
|-
|-
! colspan="3" | Tuning
| 14
! colspan="3" | Strength
| 409.8
! colspan="2" | Closest edo
| major 3rd
! colspan="2" | Integer limit
| M3
|-
| F#, ^Gb
! ZPI
| major 3rd, comma-wide diminished 4th
! Steps per octave
| M3, Kd4
! Step size (cents)
| F#, KGb
! Height
| Ma
! Integral
| Maa
! Gap
|-
! Edo
| 15
! Octave (cents)
| 439.0
! Consistent
| upmajor 3rd
! Distinct
| ^M3
|-
| ^F#, vvG
| [[184zpi]]
| supermajor 3rd, classic diminished 4th
| 40.9880783925993
| SM3, KKd4
| 29.2768055263764
| SF#, KKGb
| 7.570230
| Mu
| 1.423937
| Mo
| 17.722623
|-
| 41edo
| 16
| 1200.34902658143
| 468.3
| 16
| down-4th
| 10
| v4
|}
| vG
 
| sub 4th
== Relationship to 12edo ==
| s4
41edo’s [[circle of fifths|circle of 41 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This is possible because 24\41 is on the 7\12 kite in the [[scale tree]]. Stated another way, it is possible because the absolute value of 41edo's [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) is 1.  
| sG
 
| Fo
This "spiral of fifths" can be a useful construct for introducing 41edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.
| Fe
 
|-
There are 12 "-ish" categories, where "-ish" means ±1 edostep. The 6 mid intervals are uncategorized, since they are all so far from 12edo.
| 17
 
| 497.6
The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.
| perfect 4th
 
| P4
[[File:41-edo spiral.png|579x579px]]
| G
 
| perfect 4th
The same spiral, but with notes not intervals:
| P4
 
| G
[[File:41-edo spiral with notes.png|549x549px]]
| Fa
 
| Fa
== Regular temperament properties ==
|-
{| class="wikitable center-4 center-5 center-6"
| 18
|-
| 526.8
! rowspan="2" | [[Subgroup]]
| up-4th
! rowspan="2" | [[Comma list]]
| ^4
! rowspan="2" | [[Mapping]]
| ^G
! rowspan="2" | Optimal<br>8ve stretch (¢)
| comma-wide 4th
! colspan="2" | Tuning error
| K4
|-
| KG
! [[TE error|Absolute]] (¢)
| Fu
! [[TE simple badness|Relative]] (%)
| Fih
|-
|-
| 2.3
| 19
| {{monzo| 65 -41 }}
| 556.1
| {{mapping| 41 65 }}
| mid-4th, downdim 5th
| −0.153
| ~4, vd5
| 0.15
| ^^G, vAb
| 0.52
| uber/neutral 4th, classic augmented 4th
|-
| U4/N4, kkA4
| 2.3.5
| 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
! Spoken name
|-
| zo (7-over)
| 6:7:9
| 0-9-24
| C vEb G
| Cvm
| C downminor
|-
| gu (5-under)
| 10:12:15
| 0-11-24
| C ^Eb G
| C^m
| C upminor
|-
| ilo (11-over)
| 18:22:27
| 0-12-24
| C vvE G
| C~
| C mid
|-
| yo (5-over)
| 4:5:6
| 0-13-24
| C vE G
| Cv
| C downmajor or C down
|-
| ru (7-under)
| 14:18:21
| 0-15-24
| C ^E G
| C^
| C upmajor or C up
|}
 
Other common triads are
* 0-10-20 = D F Ab = Dd = D dim
* 0-10-21 = D F ^Ab = Dd(^5) = D dim up-five
* 0-10-22 = D F vvA = Dm(~5) = D minor mid-five
* 0-10-23 = D F vA = Dm(v5) = D minor down-five
* 0-10-24 = D F A = Dm = D minor
* 0-14-24 = D F# A = D = D or D major
* 0-14-25 = D F# ^A = D(^5) = D up-five
* 0-14-26 = D F# ^^A = D(^^5) = D half-aug
* 0-14-27 = D F# vA# = Da(v5) = D aug down-five or perhaps D(v#5) = D downsharp-five
* 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]].
 
== Notations ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
{{Sharpness-sharp4-szg}}
 
The notes within an octave from A are thus:
 
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
 
=== 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}}
 
Half-sharps and half-flats can be used to avoid double arrows:
{{Ups and downs sharpness|41|true}}
 
=== 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:
 
{{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}}
 
Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.
 
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 Kite's ups and downs notation. The only difference is the use of minor tritone and major tritone.
 
=== Sagittal notation ===
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 ====
{{Sagittal chart|Evo}}
 
==== Evo-SZ flavor ====
{{Sagittal chart|Evo-SZ}}
 
==== Revo flavor ====
{{Sagittal chart}}
 
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]]
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals|41}}
 
== Relationship to 12edo ==
41edo’s [[circle of fifths|circle of 41 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This is possible because 24\41 is on the 7\12 kite in the [[scale tree]]. Stated another way, it is possible because the absolute value of 41edo's [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) is 1.  
 
This "spiral of fifths" can be a useful construct for introducing 41edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.
 
There are 12 "-ish" categories, where "-ish" means ±1 edostep. The 6 mid intervals are uncategorized, since they are all so far from 12edo.
 
The two innermost and two outermost intervals on the spiral are duplicates, reflecting the fact that it is a repeating circle at heart and the spiral shape is only a helpful illusion.
 
[[File:41-edo spiral.png|579x579px]]
 
The same spiral, but with notes not intervals:
 
[[File:41-edo spiral with notes.png|549x549px]]
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 65 -41 }}
| {{Mapping| 41 65 }}
| −0.153
| 0.15
| 0.52
|-
| 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 804: 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 811: 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 818: 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 825: 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 847: 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 855: 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 863: 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 871: 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 879: 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 887: 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 895: 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 903: 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 911: 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 919: 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 927: 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 935: 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 943: 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 951: 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 959: 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 967: 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 975: 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 983: 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 991: 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 999: 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 1,007: 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 1,015: 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 1,023: 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 1,031: 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 1,039: 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 1,047: 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 1,055: 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 1,063: 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 1,071: 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 1,079: 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 1,087: 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 1,095: 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 1,103: 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 1,111: 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 1,119: 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 1,127: 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 1,135: 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 1,143: 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 1,151: 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 1,159: 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 1,167: 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 1,175: 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 1,183: 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 1,191: 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 1,199: 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 1,207: 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 1,215: 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 1,223: 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 1,231: 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 1,239: 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 1,247: 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 1,271: 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 1,279: 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 1,287: 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 1,295: 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 1,303: 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 1,311: 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 1,319: 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 1,327: 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 1,335: 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 1,343: 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 1,366: Line 1,530:
| [[Slendi]]
| [[Slendi]]
| (P8, P4/17)
| (P8, P4/17)
| Pathological 38-tone mos
|  
|-
|-
| 2
| 2
Line 1,460: Line 1,624:
| 17
| 17
| 497.56
| 497.56
| [[Helmholtz]] / [[garibaldi]] / [[cassandra]] / [[andromeda]]<br>[[Kwai]]
| [[Helmholtz (temperament)|Helmholtz]] / [[garibaldi]] / [[cassandra]] / [[andromeda]]<br>[[Kwai]]
| (P8, P5)
| (P8, P5)
| 12-tone mos: 4 3 4 3 3 4 3 4 3 4 3 4 3 3
| 12-tone mos: 4 3 4 3 3 4 3 4 3 4 3 4 3 3
Line 1,480: Line 1,644:
| [[Pluto]]<br>[[Merman]]
| [[Pluto]]<br>[[Merman]]
| (P8, c<sup>3</sup>P4/7)
| (P8, c<sup>3</sup>P4/7)
| Pathological 35-tone mos
|  
|}
|}
== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which [[subgroup]] of [[JI]] we are focusing on.
For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be milder. A tuning that does that is [[ZPI|184zpi]].
For the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as [[65edt]], [[106ed6]], and [[147ed12]].
41edo additionally approximates primes 19, 29, and 31, which all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings.


== Scales and modes ==
== Scales and modes ==
Line 2,050: Line 2,223:
The first 41edo guitar was probably this one, built by [[Erv Wilson]] in the 1960's:
The first 41edo guitar was probably this one, built by [[Erv Wilson]] in the 1960's:


[[File:Erv Wilson's full-41 guitar 2.jpg|none|thumb]]
[[File:Erv Wilson's full-41 guitar 2.jpg|none|thumb|200px]]


Note the new bridge, several inches below the original bridge. The new bridge increases the scale length and spreads the frets out, making the guitar more playable. Erv numbered the frets as seen here, with the 3-limit dorian scale in enlarged numbers.
Note the new bridge, several inches below the original bridge. The new bridge increases the scale length and spreads the frets out, making the guitar more playable. Erv numbered the frets as seen here, with the 3-limit dorian scale in enlarged numbers.


[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|838x838px]]
[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|500px]]


Several more modern guitars:
Several more modern guitars:
<gallery widths=300 heights=200>
File:Melleweijters.com 41edo.jpg|[[Melle Weijters]]' 10-string guitar ([https://melleweijters.com Melleweijters.com])
File:41-EDD_elektrische_gitaar.jpg|41edo electric guitar, by [[Gregory Sanchez]].
File:Ron_Sword_with_a_41ET_Guitar.jpg|41edo classical guitar, by [[Ron Sword]].
</gallery>


[[File:Melleweijters.com 41edo.jpg|frameless|832x832px]]
The [[Kite Guitar]] is a guitar fretting using every other step of 41edo, i.e. 41ed4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers half of 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, although there are certain trade-offs.  


''[[Melle Weijters]]' 10-string guitar ([https://melleweijters.com Melleweijters.com])''
[[File:Caleb's Kite guitar.jpg|none|thumb|200px|Kite guitar]]


[[File:41-EDD_elektrische_gitaar.jpg|alt=41-EDD elektrische gitaar.jpg|560x745px|41-EDD elektrische gitaar.jpg]]
For more photos of Kite guitars, see [[Kite Guitar Photographs]].
 
{{clear}}
''41edo electric guitar, by [[Gregory Sanchez]].''
 
[[File:Ron_Sword_with_a_41ET_Guitar.jpg|alt=Ron_Sword_with_a_41ET_Guitar.jpg|Ron_Sword_with_a_41ET_Guitar.jpg]]
 
''41edo classical guitar, by [[Ron Sword]].''
 
The [[Kite Guitar]] (see also [https://kiteguitar.com KiteGuitar.com] and [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf Kite Tuning]) is a guitar fretting using every other step of 41edo, i.e. 41ed4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers half of 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, although there are certain trade-offs.
 
[[File:Caleb's Kite guitar.jpg|480x640px]]


For more photos of Kite guitars, see [[Kite Guitar Photographs]].
=== Metallophones ===
[[File:41edo Metallophone.png|left|thumb|[https://richiegreene.com/instruments/ 41edo metallophone] spanning three-octaves from vC<sub>5</sub>-^^C<sub>8</sub> by [[Richie Greene]]]]
{{clear}}


=== Keyboards ===
=== Keyboards ===
A possible 41edo keyboard design:
A possible 41edo keyboard design:
[[File:41edo keyboard layout.png|none|thumb|484x484px]]
<gallery widths="300" heights="200">
File:41edo keyboard layout.png
File:TS41 Microtonal MIDI Keyboard (Prototype).jpg|[[User:Tristanbay|Tristan Bay]]'s prototype TS41 MIDI keyboard, laid out in bosanquet with 41 keys per octave
File:Xenachord with 41edo layout.png|[https://richiegreene.com/instruments/ Xenachord] with 41edo layout by [[Richie Greene|Richie]]
</gallery>
See also [[41-edo Keyboards]] for Linnstrument and Harpejji options, as well as DIY options.
{{clear}}


See also [[41-edo Keyboards]] for Lumatone, Linnstrument and Harpejji options, as well as DIY options.
=== Lumatone ===
* [[Lumatone mapping for 41edo]]
See also [[41-edo Keyboards]] for more Lumatone options.
{{clear}}
 
=== Skip fretting ===
* [[Skip fretting system 41 2 11]]
{{clear}}


== 41edo as a Universal Tuning ==
== 41edo as a Universal Tuning ==
Line 2,115: Line 2,299:
Georgian Polyphonic singing can be done in a 41edo context due to its excellent approximations of prime harmonics and neutral third, as well as Pythagorean seconds and sevenths. Asian musical traditions built around pentatonic scales can use both Pythagorean and [[Barbad|Barbad[5]]].
Georgian Polyphonic singing can be done in a 41edo context due to its excellent approximations of prime harmonics and neutral third, as well as Pythagorean seconds and sevenths. Asian musical traditions built around pentatonic scales can use both Pythagorean and [[Barbad|Barbad[5]]].


== Music ==
== Music ==
=== Modern renderings ===
{{Main|{{ROOTPAGENAME}}/Music}}
; {{W|Johann Sebastian Bach}}
 
* [https://www.youtube.com/watch?v=vcsqRDDULq4 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
== See also ==
* [https://www.youtube.com/watch?v=LWd3ZOaAZlY "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [[Magic22 as srutis]] describes a possible use of 41edo for [[indian]] music.  
 
 
; {{W|Nicolaus Bruhns}}
== External links ==
* [https://www.youtube.com/watch?v=8_Rz5kDSDoE ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=DhVrdKowd5Q ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)
 
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=HHn5rrGrVsI ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
 
=== 20th century ===
; [[Joseph Monzo]]
* [https://www.youtube.com/watch?v=N0ca5vdBEpI ''Theme from Invisible Haircut''] (1990)
 
=== 21st century ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=P0vRjzkpOxw FUZZ] (2024)
 
; [[Beheld]]
* [https://www.youtube.com/watch?v=G8hsoaQzRoI ''Subsidence vibe''] (2024)
 
; [[Cameron Bobro]]
* [https://soundcloud.com/cameron-bobro/eveninghorizon-cbobro ''Evening Horizon'']{{dead link}} [https://web.archive.org/web/20201127014810/http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 play]
 
; [[Flora Canou]]
* [https://soundcloud.com/floracanou/sets/notes-of-the-generation ''Notes of the Generation''] (2023) – an 8-piece album in 41et
: "Chaotic Witch #1" · "Party Cubes" · "Big Dreamer Pavilion" · "Lost Cyclops" · "Sky Tree" · "Long Night Ahead" · "Fractocraft" · "After the Generation"
 
; [[Francium]]
* "Tetracotta" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/54Er1Xh83UuePQbuflZzw4 Spotify] | [https://francium223.bandcamp.com/track/tetracotta Bandcamp] | [https://www.youtube.com/watch?v=GN5FTqxhcgc YouTube] – tetracot[13] in 41edo tuning
* "harmon" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/39tYQh4ZQvtyCUIBnLllYB Spotify] | [https://francium223.bandcamp.com/track/harmon Bandcamp] | [https://www.youtube.com/watch?v=IezX2lAjrgw YouTube]
* [https://www.youtube.com/watch?v=4ZLWjUw_O0Q ''We Wish You A Gary Christmas''] (2024) – gary in 41edo tuning
 
; [[Jake Freivald]]
* ''[https://soundcloud.com/jdfreivald/little-magical-object Little Magical Object]'' – magic[19] in 41edo tuning
 
; [[LΛMPLIGHT]]
* [https://youtu.be/cMnuMjXeHrY Caftaphata] (2024) - Also partially in just intonation and 12edo
 
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=UE3FBQBjCPI ''Bohlen–Pierce Fugue for 3 Clarinets in 41EDO BPS9 sLsLsLsLs "Moll II/Pierce"''] (2023)
* [https://www.youtube.com/watch?v=9tMpq2Nvq_Y ''5-Part Bohlen–Pierce Fugue in 41EDO BPS9 sLsLsLssL "Harmonic"''] (2024)
 
; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=tzqbmTmNZsU ''Preludium, for microtonal video game ensemble''] (2023)
* [https://www.youtube.com/watch?v=MUFLiMs8IkQ ''Spring's Arrival'', for synth septet] (2024)
 
; [[Chris Vaisvil]] ([http://www.chrisvaisvil.com/ site])
* [http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 ''The Magic of Belief''] (2013) – magic[19] in 41edo tuning
 
; [[Xeno Ov Eleas]]
* [https://www.youtube.com/watch?v=oQHOltX4Sos ''A Treasure Lost and Must Be Found''] (2022)
 
=== Kite Guitar Recordings ===
 
; [[Kite Giedraitis]]
* [https://soundcloud.com/tallkite/evening-rondo ''Evening Rondo'']
* [https://soundcloud.com/mbirakite/triadic-etude ''Downminor Etude''] (midi demo)
 
; [[Igliashon Jones]]
* [https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-1 ''Modified Kite Guitar Take 1 - Clean'']
* [https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-2 ''Modified Kite Guitar Take 2 - Fuzz'']
 
; [[Pixel Archipelago]]
* [https://pixelarchipelago.bandcamp.com/album/intervallic-prism ''Intervallic Prism''] (2020) – a 7-track album
: "Red" · "Orange" · "Yellow" · "Green" · "Blue" · "Indigo" · "Violet"
 
; [[Aaron Wolf]]
* [https://soundcloud.com/mbirakite/aaron-wolf-12-bar-blues-on-kite-guitar ''12-Bar-Blues on Kite Guitar''] – a simple 12-bar blues
* [https://soundcloud.com/wolftune/fourthward-lang-syne ''Fourthward Lang Syne''] – an arrangement of Auld Lang Syne
 
=== Kite Guitar Videos ===
; [[Timmy Barnett]]
* [https://TallKite.com/KiteGuitar/Downminor&#x20;Etude.m4v ''Downminor Etude''] {{dead link}}
 
; [[Wilckerson Ganda]]
* [https://www.youtube.com/watch?v=gQERKtbkMCE ''Vintage Rock'']
 
; [[Travis Johnson]]
* [https://www.youtube.com/watch?v=eAPzZ9oJYyY ''Evening Rondo'']
 
=== Kite Guitar Scores ===
; [[Kite Guitar originals]]
; [[Kite Guitar translations]]
 
== See also ==
* [https://KiteGuitar.com KiteGuitar.com] for recordings, videos, etc.
* [https://KiteGuitar.com KiteGuitar.com] for recordings, videos, etc.
* [[Lumatone mapping for 41edo]]
* [[Magic22 as srutis]] describes a possible use of 41edo for [[indian]] music.
== External links ==
* [http://www.ronsword.com ''Tetracontamonophonic Scales for Guitar''] by [[Ron Sword]]
* [http://www.ronsword.com ''Tetracontamonophonic Scales for Guitar''] by [[Ron Sword]]
* [https://drive.google.com/open?id=0B3wIGTmjY_VZYllwcHI0d3hEc3M ''Intervals, Scales and Chords in 41EDO''] by [[Cam Taylor]] – a work in progress using just intonation concepts and simplified Sagittal notation.
* [https://drive.google.com/open?id=0B3wIGTmjY_VZYllwcHI0d3hEc3M ''Intervals, Scales and Chords in 41EDO''] by [[Cam Taylor]] – a work in progress using just intonation concepts and simplified Sagittal notation.
Line 2,215: Line 2,313:
<references/>
<references/>


[[Category:3-limit record edos|##]] <!-- 2-digit number -->
[[Category:Magic]]
[[Category:Magic]]
[[Category:Superkleismic]]
[[Category:Superkleismic]]
[[Category:Supermagic]]
[[Category:Keemic]]
[[Category:Tetracot]]
[[Category:Tetracot]]
[[Category:Octacot]]
[[Category:Octacot]]
[[Category:3-limit]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/41edo"