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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
| major 3rd, comma-wide diminished 4th
| M3, Kd4
| F#, KGb
| Ma
| Maa
|-
|-
! ZPI
| 15
! Steps per octave
| 439.0
! Step size (cents)
| upmajor 3rd
! Height
| ^M3
! Integral
| ^F#, vvG
! Gap
| supermajor 3rd, classic diminished 4th
! Edo
| SM3, KKd4
! Octave (cents)
| SF#, KKGb
! Consistent
| Mu
! Distinct
| Mo
|-
|-
| [[184zpi]]
| 40.9880783925993
| 29.2768055263764
| 7.570230
| 1.423937
| 17.722623
| 41edo
| 1200.34902658143
| 16
| 16
| 10
| 468.3
|}
| down-4th
 
| v4
== Relationship to 12edo ==
| vG
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.
| sub 4th
 
| s4
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.
| sG
 
| Fo
There are 12 "-ish" categories, where "-ish" means ±1 edostep. The 6 mid intervals are uncategorized, since they are all so far from 12edo.
| Fe
 
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]]
| 17
! rowspan="2" | [[Comma list]]
| 497.6
! rowspan="2" | [[Mapping]]
| perfect 4th
! rowspan="2" | Optimal<br>8ve stretch (¢)
| P4
! colspan="2" | Tuning error
| G
| perfect 4th
| P4
| G
| Fa
| Fa
|-
|-
! [[TE error|Absolute]] (¢)
| 18
! [[TE simple badness|Relative]] (%)
| 526.8
| up-4th
| ^4
| ^G
| comma-wide 4th
| K4
| KG
| Fu
| 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
| UG, kkG#
| Fi/Sho
| Fu
|-
|-
| 2.3.5
| 20
| 3125/3072, 20000/19683
| 585.4
| {{mapping| 41 65 95 }}
| downaug 4th, dim 5th
| +0.734
| vA4, d5
| 1.26
| vG#, Ab
| 4.31
| comma-narrow augmented 4th, diminished 5th
| kA4/d5
| kG#, Ab
| Po/Sha
| Fi
|-
|-
| 2.3.5.7
| 21
| 225/224, 245/243, 1029/1024
| 614.6
| {{mapping| 41 65 95 115 }}
| aug 4th, updim 5th
| +0.815
| A4, ^d5
| 1.10
| G#, ^Ab
| 3.76
| augmented 4th, comma-wide diminished 5th
| A4/Kd5
| G#, KAb
| Pa/Shu
| Se
|-
|-
| 2.3.5.7.11
| 22
| 100/99, 225/224, 243/242, 245/242
| 643.9
| {{mapping| 41 65 95 115 142 }}
| mid-5th, upaug 4th
| +0.375
| ~5, ^A4
| 1.32
| ^G#, vvA
| 4.51
| unter/neutral 5th, classic diminished 5th
| u5/N5, KKd5
| uA, KKAb
| Pu/Si
| Su
|-
|-
| 2.3.5.7.11.13
| 23
| 100/99, 105/104, 144/143, 196/195, 243/242
| 673.2
| {{mapping| 41 65 95 115 142 152 }}
| down-5th
| −0.060
| v5
| 1.55
| vA
| 5.29
| comma-narrow 5th
| k5
| kA
| So
| Sih
|-
|-
| 2.3.5.7.11.13.19
| 24
| 100/99, 105/104, 133/132, 144/143, 171/169, 196/195
| 702.4
| {{mapping| 41 65 95 115 142 152 174 }}
| perfect 5th
| +0.111
| P5
| 1.49
| A
| 5.10
| perfect 5th
|}
| P5
* 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.
| A
 
| Sa
=== Commas ===
| Sol
41et [[tempering out|tempers out]] the following [[comma]]s using its patent [[val]], {{val| 41 65 95 115 142 152 168 174 185 199 203 }}.
 
{| class="commatable wikitable center-1 center-2 right-3 center-6"
|-
|-
! [[Harmonic limit|Prime<br>limit]]
| 25
! [[Ratio]]<ref>Ratios with more than 8 digits are presented by placeholders with informative hints</ref>
| 731.7
! [[Cents]]
| up-5th
! [[Monzo]]
| ^5
! colspan="2" | [[Color name]]
| ^A
! Name(s)
| super 5th
| S5
| SA
| Su
| Si
|-
|-
| 3
| 26
| <abbr title="36893488147419103232/36472996377170786403">(40 digits)</abbr>
| 761.0
| 19.84
| downminor 6th
| {{monzo| 65 -41 }}
| vm6
| Wa-41
| ^^A, vBb
| 41-edo
| subminor 6th, classic augmented 5th
| [[41-comma]]
| sm6
| sBb, kkA#
| Flo
| Lo
|-
|-
| 5
| 27
| <abbr title="1953125/1889568">(14 digits)</abbr>
| 790.2
| 57.27
| minor 6th
| {{monzo| -5 -10 9 }}
| m6
| Tritriyo
| vA#, Bb
| y<sup>9</sup>
| minor 6th, comma-narrow augmented 5th
| [[Shibboleth comma]]
| m6
| Bb, kA#
| Fla
| Leh
|-
|-
| 5
| 28
| [[34171875/33554432|(16 digits)]]
| 819.5
| 31.57
| upminor 6th
| {{monzo| -25 7 6 }}
| ^m6
| Lala-tribiyo
| A#, ^Bb
| LLy<sup>3</sup>
| classic minor 6th, augmented 5th
| [[Ampersand comma]]
| Km6, A5
| KBb, A#
| Flu
| Le
|-
|-
| 5
| 29
| [[3125/3072]]
| 848.8
| 29.61
| mid 6th
| {{monzo| -10 -1 5 }}
| ~6
| Laquinyo
| ^A#, vvB
| Ly<sup>5</sup>
| neutral 6th, super augmented 5th
| Magic comma
| N6
| UBb/uB, sA#
| Li
| Lu
|-
|-
| 5
| 30
| [[20000/19683|(10 digits)]]
| 878.0
| 27.66
| downmajor 6th
| {{monzo| 5 -9 4 }}
| vM6
| Saquadyo
| vB
| sy<sup>4</sup>
| classic major 6th
| [[Tetracot comma]]
| kM6
| kB
| Lo
| La
|-
|-
| 5
| 31
| <abbr title="131072000/129140163">(18 digits)</abbr>
| 907.3
| 25.71
| major 6th
| {{monzo| 20 -17 3 }}
| M6
| Sasa-triyo
| B
| ssy<sup>3</sup>
| major 6th
| [[Roda]]
| M6
| B
| La
| Laa
|-
|-
| 5
| 32
| [[32805/32768|(10 digits)]]
| 936.6
| 1.95
| upmajor 6th
| {{monzo| -15 8 1 }}
| ^M6
| Layo
| ^B
| Ly
| supermajor 6th
| [[Schisma]]
| SM6
| SB
| Lu
| Li
|-
|-
| 7
| 33
| [[15625/15309|(10 digits)]]
| 965.9
| 35.37
| downminor 7th
| {{monzo| 0 -7 6 -1 }}
| vm7
| Rutribiyo
| vC
| ry<sup>6</sup>
| subminor 7th
| Arcturus comma, great BP diesis
| sm7
| sC
| Tho
| Ta
|-
|-
| 7
| 34
| <abbr title="854296875/843308032">(18 digits)</abbr>
| 995.1
| 22.41
| minor 7th
| {{monzo| -10 7 8 -7 }}
| m7
| Lasepru-aquadbiyo
| C
| Lr<sup>7</sup>y<sup>8</sup>
| minor 7th
| [[Blackjackisma]]
| m7
|-
| C
| 7
| Tha
| [[875/864]]
| Teh
| 21.90
| {{monzo| -5 -3 3 1 }}
| Zotriyo
| zy<sup>3</sup>
| Keema
|-
|-
| 7
| 35
| [[3125/3087]]
| 1024.4
| 21.18
| upminor 7th
| {{monzo| 0 -2 5 -3 }}
| ^m7
| Triru-aquinyo
| ^C
| r<sup>3</sup>y<sup>5</sup>
| classic/comma-wide minor seventh
| Gariboh comma
| Km7
| KC
| Thu
| Te
|-
|-
| 7
| 36
| <abbr title="179200/177147">(12 digits)</abbr>
| 1053.7
| 19.95
| mid 7th
| {{monzo| 10 -11 2 1 }}
| ~7
| Sazoyoyo
| ^^C, vDb
| szyy
| neutral 7th, sub diminished 8ve
| [[Tolerma]]
| N7
| UC/uC#, sDb
| Ti
| Tu
|-
|-
| 7
| 37
| [[33075/32768|(10 digits)]]
| 1082.9
| 16.14
| downmajor 7th
| {{monzo| -15 3 2 2 }}
| vM7
| Labizoyo
| vC#, Db
| Lzzyy
| classic major 7th, diminished 8ve
| [[Mirwomo comma]]
| kM7, d8
| kC#, Db
| To
| Ti
|-
|-
| 7
| 38
| [[245/243]]
| 1112.2
| 14.19
| major 7th
| {{monzo| 0 -5 1 2 }}
| M7
| Zozoyo
| C#, ^Db
| zzy
| major 7th, comma-wide diminished 8ve
| Sensamagic comma
| M7, Kd8
| C#, KDb
| Ta
| Taa
|-
|-
| 7
| 39
| [[4000/3969]]
| 1141.5
| 13.47
| upmajor 7th
| {{monzo| 5 -4 3 -2 }}
| ^M7
| Rurutriyo
| ^C#, vvD
| rry<sup>3</sup>
| supermajor 7th, classic dim 8ve, unter 8ve
| Octagar comma
| SM7, KKd8, U8
| SC#, KKDb, u8
| Tu
| To
|-
|-
| 7
| 40
| <abbr title="823543/819200">(12 digits)</abbr>
| 1170.7
| 9.15
| dim 8ve
| {{monzo| -15 0 -2 7 }}
| v8
| Lasepzo-agugu
| vD
| Lz<sup>7</sup>gg
| comma-narrow 8ve, sub 8ve
| [[Quince comma]]
| k8/s8
| kD, sD
| Do
| Da
|-
|-
| 7
| 41
| [[1029/1024]]
| 1200.0
| 8.43
| perfect 8ve
| {{monzo| -10 1 0 3 }}
| P8
| Latrizo
| D
| Lz<sup>3</sup>
| perfect 8ve
| Gamelisma
| P8
|-
| D
| 7
| Da
| [[225/224]]
| Do
| 7.71
|}
| {{monzo| -5 2 2 -1 }}
 
| Ruyoyo
=== Interval quality and chord names in color notation ===
| ryy
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
| Marvel comma
 
{| class="wikitable center-all"
|-
|-
| 7
! Quality
| [[16875/16807|(10 digits)]]
! [[Color notation|Color]]
| 6.99
! Monzo format
| {{monzo| 0 3 4 -5 }}
! Examples
| Quinru-aquadyo
| r<sup>5</sup>y<sup>4</sup>
| [[Mirkwai comma]]
|-
|-
| 7
| downminor
| [[10976/10935|(10 digits)]]
| zo
| 6.48
| (a, b, 0, 1)
| {{monzo| 5 -7 -1 3 }}
| 7/6, 7/4
| Satrizo-agu
| sz<sup>3</sup>g
| [[Hemimage comma]]
|-
|-
| 7
| minor
| [[5120/5103]]
| fourthward wa
| 5.76
| (a, b) with b < -1
| {{monzo| 10 -6 1 -1 }}
| 32/27, 16/9
| Saruyo
| sry
| Hemifamity comma
|-
|-
| 7
| upminor
| [[33554432/33480783|(16 digits)]]
| gu
| 3.80
| (a, b, -1)
| {{monzo| 25 -14 0 -1 }}
| 6/5, 9/5
| Sasaru
| ssr
| [[Garischisma]]
|-
|-
| 7
| mid
| [[2401/2400]]
| ilo
| 0.72
| (a, b, 0, 0, 1)
| {{monzo| -5 -1 -2 4 }}
| 11/9, 11/6
| Bizozogu
| z<sup>4</sup>gg
| Breedsma
|-
|-
| 11
| "
| <abbr title="163840/161051">(12 digits)</abbr>
| lu
| 29.72
| (a, b, 0, 0, -1)
| {{monzo| 15 0 1 0 -5 }}
| 12/11, 18/11
| Saquinlu-ayo
| s1u<sup>5</sup>y
| [[Thuja comma]]
|-
|-
| 11
| downmajor
| [[245/242]]
| yo
| 21.33
| (a, b, 1)
| {{monzo| -1 0 1 2 -2 }}
| 5/4, 5/3
| Luluzozoyo
| 1uuzzy
| Frostma
|-
|-
| 11
| major
| [[100/99]]
| fifthward wa
| 17.40
| (a, b) with b > 1
| {{monzo| 2 -2 2 0 -1 }}
| 9/8, 27/16
| Luyoyo
| 1uyy
| Ptolemisma
|-
|-
| 11
| upmajor
| [[1344/1331]]
| ru
| 16.83
| (a, b, 0, -1)
| {{monzo| 6 1 0 1 -3 }}
| 9/7, 12/7
| Trilu-azo
|}
| 1u<sup>3</sup>z
 
| Hemimin comma
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"
|-
|-
| 11
! [[Color notation|Color of the 3rd]]
| [[896/891]]
! JI chord
| 9.69
! Notes as edosteps
| {{monzo| 7 -4 0 1 -1 }}
! Notes of C chord
| Saluzo
! Written name
| s1uz
! Spoken name
| [[Pentacircle comma]]
|-
|-
| 11
| zo (7-over)
| [[65536/65219|(10 digits)]]
| 6:7:9
| 8.39
| 0-9-24
| {{monzo| 16 0 0 -2 -3 }}
| C vEb G
| Satrilu-aruru
| Cvm
| s1u<sup>3</sup>rr
| C downminor
| [[Orgonisma]]
|-
|-
| 11
| gu (5-under)
| [[243/242]]
| 10:12:15
| 7.14
| 0-11-24
| {{monzo| -1 5 0 0 -2 }}
| C ^Eb G
| Lulu
| C^m
| 1uu
| C upminor
| Rastma
|-
|-
| 11
| ilo (11-over)
| [[385/384]]
| 18:22:27
| 4.50
| 0-12-24
| {{monzo| -7 -1 1 1 1 }}
| C vvE G
| Lozoyo
| C~
| 1ozg
| C mid
| Keenanisma
|-
|-
| 11
| yo (5-over)
| [[441/440]]
| 4:5:6
| 3.93
| 0-13-24
| {{monzo| -3 2 -1 2 -1 }}
| C vE G
| Luzozogu
| Cv
| 1uzzg
| C downmajor or C down
| Werckisma
|-
|-
| 11
| ru (7-under)
| [[1375/1372]]
| 14:18:21
| 3.78
| 0-15-24
| {{monzo| -2 0 3 -3 1 }}
| C ^E G
| Lotriruyo
| C^
| 1or<sup>3</sup>y
| C upmajor or C up
| Moctdel comma
|}
|-
 
| 11
Other common triads are
| [[540/539]]
* 0-10-20 = D F Ab = Dd = D dim
| 3.21
* 0-10-21 = D F ^Ab = Dd(^5) = D dim up-five
| {{monzo| 2 3 1 -2 -1 }}
* 0-10-22 = D F vvA = Dm(~5) = D minor mid-five
| Lururuyo
* 0-10-23 = D F vA = Dm(v5) = D minor down-five
| 1urry
* 0-10-24 = D F A = Dm = D minor
| Swetisma
* 0-14-24 = D F# A = D = D or D major
|-
* 0-14-25 = D F# ^A = D(^5) = D up-five
| 11
* 0-14-26 = D F# ^^A = D(^^5) = D half-aug
| [[3025/3024]]
* 0-14-27 = D F# vA# = Da(v5) = D aug down-five or perhaps D(v#5) = D downsharp-five
| 0.57
* 0-14-28 = D F# A# = Da = D aug
| {{monzo| -4 -3 2 -1 2 }}
 
| Loloruyoyo
For a more complete list, see [[41edo chord names]] and [[Ups and downs notation #Chords and chord progressions]].
| 1ooryy
 
| Lehmerisma
== Notations ==
|-
=== Stein–Zimmermann–Gould notation ===
| 11
[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
| [[151263/151250|<abbr title="151263/151250">(12 digits)</abbr>]]
{{Sharpness-sharp4-szg}}
| 0.15
 
| {{monzo| -1 2 -4 5 -2 }}
The notes within an octave from A are thus:
| Luluquinzo-aquadgu
 
| 1uuz<sup>5</sup>g<sup>4</sup>
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
| [[Odiheim comma]]
 
=== 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>
|-
|-
| 13
! Revo
| [[343/338]]
| <big>{{sagittal| ||\ }}</big>
| 25.42
| <big>{{sagittal| /||\ }}</big>
| {{monzo| -1 0 0 3 0 -2 }}
|}
| Thuthutrizo
The following enharmonics from the Spartan set are present (comma tempered out):
| 3uuz<sup>3</sup>
* {{Sagittal| //| }} = {{sagittal| /|) }} = {{sagittal| /|\ }} ([[325/324]], [[352/351]])
|  
* {{Sagittal| /| }} = {{sagittal| |) }} ([[225/224]])
|-
* {{Sagittal| |( }} = {{sagittal| |//| }} ([[5120/5103]])
| 13
 
| [[105/104]]
See [[Sagittal notation #Revo|apotome complements]] for equivalent accidental pairs.
| 16.57
 
| {{monzo| -3 1 1 1 0 -1 }}
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]].
| Thuzoyo
 
| 3uzy
==== Evo flavor ====
| Animist comma
{{Sagittal chart|Evo}}
|-
 
| 13
==== Evo-SZ flavor ====
| [[28672/28431|(10 digits)]]
{{Sagittal chart|Evo-SZ}}
| 14.61
 
| {{monzo| 12 -7 0 1 0 -1 }}
==== Revo flavor ====
| Sathuzo
{{Sagittal chart}}
| s3uz
 
| [[Secorian comma]]
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"
|-
|-
| 13
! rowspan="2" | [[Subgroup]]
| [[275/273]]
! rowspan="2" | [[Comma list]]
| 12.64
! rowspan="2" | [[Mapping]]
| {{monzo| 0 -1 2 -1 1 -1 }}
! rowspan="2" | Optimal<br>8ve stretch (¢)
| Thuloruyoyo
! colspan="2" | Tuning error
| 3u1oryy
| Gassorma
|-
|-
| 13
! [[TE error|Absolute]] (¢)
| [[144/143]]
! [[TE simple badness|Relative]] (%)
| 12.06
| {{monzo| 4 2 0 0 -1 -1 }}
| Thulu
| 3u1u
| Grossma
|-
|-
| 13
| 2.3
| [[196/195]]
| {{Monzo| 65 -41 }}
| 8.86
| {{Mapping| 41 65 }}
| {{monzo| 2 -1 -1 2 0 -1 }}
| −0.153
| Thuzozogu
| 0.15
| 3uzzg
| 0.52
| Mynucuma
|-
|-
| 13
| 2.3.5
| [[640/637]]
| 3125/3072, 20000/19683
| 8.13
| {{Mapping| 41 65 95 }}
| {{monzo| 7 0 1 -2 0 -1 }}
| +0.734
| Thururuyo
| 1.26
| 3urry
| 4.31
| Huntma
|-
|-
| 13
| 2.3.5.7
| [[1188/1183]]
| 225/224, 245/243, 1029/1024
| 7.30
| {{Mapping| 41 65 95 115 }}
| {{monzo| 2 3 0 -1 1 -2 }}
| +0.815
| Thuthuloru
| 1.10
| 3uu1or
| 3.76
| Kestrel comma
|-
|-
| 13
| 2.3.5.7.11
| [[31213/31104]]
| 100/99, 225/224, 243/242, 245/242
| 6.06
| {{Mapping| 41 65 95 115 142 }}
| {{monzo| -7 -5 0 4 0 1 }}
| +0.375
| Thoquadzo
| 1.32
| 3oz<sup>4</sup>3
| 4.51
| Praveensma
|-
|-
| 13
| 2.3.5.7.11.13
| [[325/324]]
| 100/99, 105/104, 144/143, 196/195, 243/242
| 5.34
| {{Mapping| 41 65 95 115 142 152 }}
| {{monzo| -2 -4 2 0 0 1 }}
| −0.060
| Thoyoyo
| 1.55
| 3oyy
| 5.29
| Marveltwin comma
|-
|-
| 13
| 2.3.5.7.11.13.19
| [[352/351]]
| 100/99, 105/104, 133/132, 144/143, 171/169, 196/195
| 4.93
| {{Mapping| 41 65 95 115 142 152 174 }}
| {{monzo| 5 -3 0 0 1 -1 }}
| +0.111
| Thulo
| 1.49
| 3u1o
| 5.10
| Major minthma
|}
* 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 ===
41et [[tempering out|tempers out]] the following [[comma]]s using its patent [[val]], {{val| 41 65 95 115 142 152 168 174 185 199 203 }}.
 
{| class="commatable wikitable center-1 center-2 right-3 center-6"
|-
|-
| 13
! [[Harmonic limit|Prime<br>limit]]
| [[364/363]]
! [[Ratio]]<ref>Ratios with more than 8 digits are presented by placeholders with informative hints</ref>
| 4.76
! [[Cents]]
| {{monzo| 2 -1 0 1 -2 1 }}
! [[Monzo]]
| Tholuluzo
! colspan="2" | [[Color name]]
| 3o1uuz
! Name(s)
| Minor minthma
|-
|-
| 13
| 3
| [[847/845]]
| <abbr title="36893488147419103232/36472996377170786403">(40 digits)</abbr>
| 4.09
| 19.84
| {{monzo| 0 0 -1 1 2 -2 }}
| {{Monzo| 65 -41 }}
| Thuthulolozogu
| Wa-41
| 3uu1oozg
| 41-edo
| Cuthbert comma
| [[41-comma]]
|-
|-
| 13
| 5
| [[729/728]]
| <abbr title="1953125/1889568">(14 digits)</abbr>
| 2.38
| 57.27
| {{monzo| -3 6 0 -1 0 -1 }}
| {{Monzo| -5 -10 9 }}
| Lathuru
| Tritriyo
| L3ur
| y<sup>9</sup>
| Squbema
| [[Shibboleth comma]]
|-
|-
| 13
| 5
| [[2080/2079]]
| [[34171875/33554432|(16 digits)]]
| 0.83
| 31.57
| {{monzo| 5 -3 1 -1 -1 1 }}
| {{Monzo| -25 7 6 }}
| Tholuruyo
| Lala-tribiyo
| 3o1ury
| LLy<sup>3</sup>
| Ibnsinma
| [[Ampersand comma]]
|-
|-
| 13
| 5
| [[4096/4095]]
| [[3125/3072]]
| 0.42
| 29.61
| {{monzo| 12 -2 -1 -1 0 -1 }}
| {{Monzo| -10 -1 5 }}
| Sathurugu
| Laquinyo
| s3urg
| Ly<sup>5</sup>
| Schismina
| Magic comma
|-
|-
| 13
| 5
| [[6656/6655]]
| [[20000/19683|(10 digits)]]
| 0.26
| 27.66
| {{monzo| 9 0 -1 0 -3 1 }}
| {{Monzo| 5 -9 4 }}
| Thotrilo-agu
| Saquadyo
| 3u1o<sup>3</sup>g2
| sy<sup>4</sup>
| Jacobin comma
| [[Tetracot comma]]
|-
|-
| 13
| 5
| [[10648/10647|(10 digits)]]
| <abbr title="131072000/129140163">(18 digits)</abbr>
| 0.16
| 25.71
| {{monzo| 3 -2 0 -1 3 -2 }}
| {{Monzo| 20 -17 3 }}
| Thuthutrilo-aru
| Sasa-triyo
| 3uu1o<sup>3</sup>r
| ssy<sup>3</sup>
| [[Harmonisma]]
| [[Roda]]
|-
|-
| 17
| 5
| [[2187/2176]]
| [[32805/32768|(10 digits)]]
| 8.73
| 1.95
| {{monzo| -7 7 0 0 0 0 -1 }}
| {{Monzo| -15 8 1 }}
| Lasu
| Layo
| L17u
| Ly
| Septendecimal schisma
| [[Schisma]]
|-
|-
| 17
| 7
| [[256/255]]
| [[15625/15309|(10 digits)]]
| 6.78
| 35.37
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| {{Monzo| 0 -7 6 -1 }}
| Sugu
| Rutribiyo
| 17ug
| ry<sup>6</sup>
| Charisma
| Arcturus comma, great BP diesis
|-
|-
| 17
| 7
| [[715/714]]
| <abbr title="854296875/843308032">(18 digits)</abbr>
| 2.42
| 22.41
| {{monzo| -1 -1 1 -1 1 1 -1 }}
| {{Monzo| -10 7 8 -7 }}
| Sutholoruyo
| Lasepru-aquadbiyo
| 17u3o1ory
| Lr<sup>7</sup>y<sup>8</sup>
| Septendecimal bridge comma
| [[Blackjackisma]]
|-
|-
| 19
| 7
| [[210/209]]
| [[875/864]]
| 8.26
| 21.90
| {{monzo| 1 1 1 1 -1 0 0 -1 }}
| {{Monzo| -5 -3 3 1 }}
| Nuluzoyo
| Zotriyo
| 19u1uzy
| zy<sup>3</sup>
| Spleen comma
| Keema
|-
|-
| 19
| 7
| [[361/360]]
| [[3125/3087]]
| 4.80
| 21.18
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| {{Monzo| 0 -2 5 -3 }}
| Nonogu
| Triru-aquinyo
| 19oog2
| r<sup>3</sup>y<sup>5</sup>
| Go comma
| Gariboh comma
|-
|-
| 19
| 7
| [[513/512]]
| <abbr title="179200/177147">(12 digits)</abbr>
| 3.38
| 19.95
| {{monzo| -9 3 0 0 0 0 0 1 }}
| {{Monzo| 10 -11 2 1 }}
| Lano
| Sazoyoyo
| L19o
| szyy
| Boethius' comma
| [[Tolerma]]
|-
|-
| 19
| 7
| [[1216/1215]]
| [[33075/32768|(10 digits)]]
| 1.42
| 16.14
| {{monzo| 6 -5 -1 0 0 0 0 1 }}
| {{Monzo| -15 3 2 2 }}
| Sanogu
| Labizoyo
| s19og
| Lzzyy
| Eratosthenes' comma
| [[Mirwomo comma]]
|-
|-
| 23
| 7
| [[736/729]]
| [[245/243]]
| 16.54
| 14.19
| {{monzo| 5 -6 0 0 0 0 0 0 1 }}
| {{Monzo| 0 -5 1 2 }}
| Satwetho
| Zozoyo
| s23o
| zzy
| Vicesimotertial comma
| Sensamagic comma
|-
|-
| 29
| 7
| [[145/144]]
| [[4000/3969]]
| 11.98
| 13.47
| {{monzo| -4 -2 1 0 0 0 0 0 0 1 }}
| {{Monzo| 5 -4 3 -2 }}
| Twenoyo
| Rurutriyo
| 29oy
| rry<sup>3</sup>
| 29th-partial chroma
| Octagar comma
|}
 
=== Rank-2 temperaments ===
* [[List of edo-distinct 41et rank two temperaments]]
* [[Schismic–countercommatic equivalence continuum]]
 
{| class="wikitable right-1 right-2"
|+ Table of temperaments by generator
|-
|-
! Degree
| 7
! Cents
| <abbr title="823543/819200">(12 digits)</abbr>
! Temperament(s)
| 9.15
! [[Pergen]]
| {{Monzo| -15 0 -2 7 }}
! Mos scales
| Lasepzo-agugu
| Lz<sup>7</sup>gg
| [[Quince comma]]
|-
|-
| 1
| 7
| 29.27
| [[1029/1024]]
| [[Slendi]]
| 8.43
| (P8, P4/17)
| {{Monzo| -10 1 0 3 }}
| Pathological 38-tone mos
| Latrizo
| Lz<sup>3</sup>
| Gamelisma
|-
|-
| 2
| 7
| 58.54
| [[225/224]]
| [[Hemimiracle]]<br>[[Dodecacot]]
| 7.71
| (P8, P5/12)
| {{Monzo| -5 2 2 -1 }}
| 21-tone mos
| Ruyoyo
| ryy
| Marvel comma
|-
|-
| 3
| 7
| 87.80
| [[16875/16807|(10 digits)]]
| [[Octacot]]
| 6.99
| (P8, P5/8)
| {{Monzo| 0 3 4 -5 }}
| 14-tone mos: 3 3 3 3 3 3 3 3 3 3 3 3 3 2
| Quinru-aquadyo
| r<sup>5</sup>y<sup>4</sup>
| [[Mirkwai comma]]
|-
|-
| 4
| 7
| 117.07
| [[10976/10935|(10 digits)]]
| [[Miracle]]
| 6.48
| (P8, P5/6)
| {{Monzo| 5 -7 -1 3 }}
| 11-tone mos: 4 4 4 4 4 4 4 4 4 4 1
| Satrizo-agu
| sz<sup>3</sup>g
| [[Hemimage comma]]
|-
|-
| 5
| 7
| 146.34
| [[5120/5103]]
| [[BPS]] / [[bohpier]]
| 5.76
| (P8, P12/13)
| {{Monzo| 10 -6 1 -1 }}
| 20-tone mos
| Saruyo
| sry
| Hemifamity comma
|-
|-
| 6
| 7
| 175.61
| [[33554432/33480783|(16 digits)]]
| [[Tetracot]] / [[bunya]] / [[monkey]]<br>[[Sesquiquartififths]] / [[sesquart]]
| 3.80
| (P8, P5/4)
| {{Monzo| 25 -14 0 -1 }}
| 13-tone mos: 1 5 1 5 1 5 1 5 5 1 5 1 5
| Sasaru
| ssr
| [[Garischisma]]
|-
|-
| 7
| 7
| 204.88
| [[2401/2400]]
| [[Baldy]]<br>[[Quadrimage]]
| 0.72
| (P8, c<sup>3</sup>P4/20)
| {{Monzo| -5 -1 -2 4 }}
| 11-tone mos: 6 1 6 6 1 6 1 6 1 6 1
| Bizozogu
| z<sup>4</sup>gg
| Breedsma
|-
|-
| 8
| 11
| 234.15
| <abbr title="163840/161051">(12 digits)</abbr>
| [[Slendric]] / [[rodan]] / [[guiron]]
| 29.72
| (P8, P5/3)
| {{Monzo| 15 0 1 0 -5 }}
| 11-tone mos: 7 1 7 1 7 1 7 1 1 7 1
| Saquinlu-ayo
| s1u<sup>5</sup>y
| [[Thuja comma]]
|-
|-
| 9
| 11
| 263.41
| [[245/242]]
| [[Septimin]]
| 21.33
| (P8, ccP4/11)
| {{Monzo| -1 0 1 2 -2 }}
| 9-tone mos: 5 4 5 5 4 5 4 5 4
| Luluzozoyo
| 1uuzzy
| Frostma
|-
|-
| 10
| 11
| 292.68
| [[100/99]]
| [[Quasitemp]]
| 17.40
| (P8, c<sup>3</sup>P4/14)
| {{Monzo| 2 -2 2 0 -1 }}
| 29-tone mos
| Luyoyo
| 1uyy
| Ptolemisma
|-
|-
| 11
| 11
| 321.95
| [[1344/1331]]
| [[Superkleismic]]
| 16.83
| (P8, ccP4/9)
| {{Monzo| 6 1 0 1 -3 }}
| 11-tone mos: 5 3 5 3 3 5 3 3 5 3 3
| Trilu-azo
| 1u<sup>3</sup>z
| Hemimin comma
|-
|-
| 12
| 11
| 351.22
| [[896/891]]
| [[Hemif]] / [[hemififths]] / [[salsa]]<br>[[Karadeniz]]
| 9.69
| (P8, P5/2)
| {{Monzo| 7 -4 0 1 -1 }}
| 10-tone mos: 5 2 5 5 2 5 5 5 2 5
| Saluzo
| s1uz
| [[Pentacircle comma]]
|-
|-
| 13
| 11
| 380.49
| [[65536/65219|(10 digits)]]
| [[Magic]] / [[witchcraft]]<br>[[Quanharuk]]
| 8.39
| (P8, P12/5)
| {{Monzo| 16 0 0 -2 -3 }}
| 10-tone mos: 2 9 2 2 9 2 2 9 2 2
| Satrilu-aruru
| s1u<sup>3</sup>rr
| [[Orgonisma]]
|-
|-
| 14
| 11
| 409.76
| [[243/242]]
| [[Hocum]]<br>[[Hocus]]
| 7.14
| (P8, c<sup>3</sup>P4/10)
| {{Monzo| -1 5 0 0 -2 }}
| 32-tone mos
| Lulu
| 1uu
| Rastma
|-
|-
| 15
| 11
| 439.02
| [[385/384]]
| [[Superthird]]
| 4.50
| (P8, c<sup>6</sup>P5/18)
| {{Monzo| -7 -1 1 1 1 }}
| 11-tone mos: 4 3 4 4 4 3 4 4 3 4 4
| Lozoyo
| 1ozg
| Keenanisma
|-
|-
| 16
| 11
| 468.29
| [[441/440]]
| [[Barbad]]
| 3.93
| (P8, c<sup>7</sup>P4/19)
| {{Monzo| -3 2 -1 2 -1 }}
| 8-tone mos: 7 2 7 7 2 7 7 2
| Luzozogu
| 1uzzg
| Werckisma
|-
|-
| 17
| 11
| 497.56
| [[1375/1372]]
| [[Helmholtz]] / [[garibaldi]] / [[cassandra]] / [[andromeda]]<br>[[Kwai]]
| 3.78
| (P8, P5)
| {{Monzo| -2 0 3 -3 1 }}
| 12-tone mos: 4 3 4 3 3 4 3 4 3 4 3 4 3 3
| Lotriruyo
| 1or<sup>3</sup>y
| Moctdel comma
|-
|-
| 18
| 11
| 526.83
| [[540/539]]
| [[Trismegistus]]
| 3.21
| (P8, c<sup>6</sup>P5/15)
| {{Monzo| 2 3 1 -2 -1 }}
| 9-tone mos: 5 5 3 5 5 5 5 3 5
| Lururuyo
| 1urry
| Swetisma
|-
| 11
| [[3025/3024]]
| 0.57
| {{Monzo| -4 -3 2 -1 2 }}
| Loloruyoyo
| 1ooryy
| Lehmerisma
|-
|-
| 19
| 11
| 556.10
| [[151263/151250|<abbr title="151263/151250">(12 digits)</abbr>]]
| [[Alphorn]]
| 0.15
| (P8, c<sup>7</sup>P4/16)
| {{Monzo| -1 2 -4 5 -2 }}
| 9-tone mos: 3 3 3 10 3 3 3 3 10
| Luluquinzo-aquadgu
| 1uuz<sup>5</sup>g<sup>4</sup>
| [[Odiheim comma]]
|-
|-
| 20
| 13
| 585.37
| [[343/338]]
| [[Pluto]]<br>[[Merman]]
| 25.42
| (P8, c<sup>3</sup>P4/7)
| {{Monzo| -1 0 0 3 0 -2 }}
| Pathological 35-tone mos
| Thuthutrizo
|}
| 3uuz<sup>3</sup>
 
|  
== Scales and modes ==
=== Lists of 41edo scales ===
* [[41edo modes]]
* [[List of MOS scales in 41edo]]
* [[The Kite Guitar Scales]]
* [[Kite Giedraitis's Categorizations of 41edo Scales]]
 
=== Harmonic scale ===
41edo is the first edo to do some justice to Mode 8 of the [[harmonic series]], which Dante Rosati calls the "[[overtone scale|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).
 
{| class="wikitable" style="text-align: center;"
|-
|-
! Overtones in "Mode 8":
| 8
| 9
| 10
| 11
| 12
| 13
| 13
| 14
| [[105/104]]
| 15
| 16.57
| 16
| {{Monzo| -3 1 1 1 0 -1 }}
| Thuzoyo
| 3uzy
| Animist comma
|-
|-
! … as JI Ratio from 1/1:
| 13
| 1/1
| [[28672/28431|(10 digits)]]
| 9/8
| 14.61
| 5/4
| {{Monzo| 12 -7 0 1 0 -1 }}
| 11/8
| Sathuzo
| 3/2
| s3uz
| 13/8
| [[Secorian comma]]
| 7/4
| 15/8
| 2/1
|-
! … in cents:
| 0
| 203.9
| 386.3
| 551.3
| 702.0
| 840.5
| 968.8
| 1088.3
| 1200.0
|-
|-
! Nearest degree of 41edo:
| 0
| 7
| 13
| 13
| 19
| [[275/273]]
| 24
| 12.64
| 29
| {{Monzo| 0 -1 2 -1 1 -1 }}
| 33
| Thuloruyoyo
| 37
| 3u1oryy
| 41
| Gassorma
|-
| 13
| [[144/143]]
| 12.06
| {{Monzo| 4 2 0 0 -1 -1 }}
| Thulu
| 3u1u
| Grossma
|-
|-
! … in cents:
| 13
| 0
| [[196/195]]
| 204.9
| 8.86
| 380.5
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 556.1
| Thuzozogu
| 702.4
| 3uzzg
| 848.8
| Mynucuma
| 965.9
| 1082.9
| 1200.0
|}
 
While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)
 
* 7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) – a close match.
* 6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
* 5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
* 4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).
 
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.
 
=== Nonoctave temperaments ===
Taking every third degree of 41edo produces a scale extremely close to [[88cET]] or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <span style="">[[BP|Bohlen–Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See [[Relationship between Bohlen–Pierce and octave-ful temperaments]], and see this chart:
 
{| class="wikitable center-all right-3 right-4 right-5 mw-collapsible mw-collapsed"
|-
|-
! colspan="3" | 3 degrees of 41edo near 88cET
| 13
! overlap
| [[640/637]]
! colspan="3" | 5 degrees of 41edo near BP
| 8.13
| {{Monzo| 7 0 1 -2 0 -1 }}
| Thururuyo
| 3urry
| Huntma
|-
|-
! 41edo
| 13
! 88cET
| [[1188/1183]]
! cents
| 7.30
! cents
| {{Monzo| 2 3 0 -1 1 -2 }}
! cents
| Thuthuloru
! BP
| 3uu1or
! 41edo
| Kestrel comma
|-
|-
| 0
| 13
| 0
| [[31213/31104]]
|  
| 6.06
| 0
| {{Monzo| -7 -5 0 4 0 1 }}
|  
| Thoquadzo
| 0
| 3oz<sup>4</sup>3
| 0
| Praveensma
|-
|-
| 3
| 13
| 1
| [[325/324]]
| 87.8
| 5.34
|  
| {{Monzo| -2 -4 2 0 0 1 }}
|  
| Thoyoyo
|  
| 3oyy
|  
| Marveltwin comma
|-
|-
|  
| 13
|  
| [[352/351]]
|  
| 4.93
|  
| {{Monzo| 5 -3 0 0 1 -1 }}
| 146.3
| Thulo
| 1
| 3u1o
| 5
| Major minthma
|-
|-
| 6
| 13
| 2
| [[364/363]]
| 175.6
| 4.76
|  
| {{Monzo| 2 -1 0 1 -2 1 }}
|  
| Tholuluzo
|  
| 3o1uuz
|  
| Minor minthma
|-
|-
| 9
| 13
| 3
| [[847/845]]
| 263.4
| 4.09
|  
| {{Monzo| 0 0 -1 1 2 -2 }}
|  
| Thuthulolozogu
|  
| 3uu1oozg
|  
| Cuthbert comma
|-
|-
|  
| 13
|  
| [[729/728]]
|  
| 2.38
|  
| {{Monzo| -3 6 0 -1 0 -1 }}
| 292.7
| Lathuru
| 2
| L3ur
| 10
| Squbema
|-
|-
| 12
| 13
| 4
| [[2080/2079]]
| 351.2
| 0.83
|  
| {{Monzo| 5 -3 1 -1 -1 1 }}
|  
| Tholuruyo
|  
| 3o1ury
|  
| Ibnsinma, sinaisma
|-
|-
| 15
| 13
| 5
| [[4096/4095]]
|  
| 0.42
| 439.0
| {{Monzo| 12 -2 -1 -1 0 -1 }}
|  
| Sathurugu
| 3
| s3urg
| 15
| Minisma
|-
|-
| 18
| 13
| 6
| [[6656/6655]]
| 526.8
| 0.26
|  
| {{Monzo| 9 0 -1 0 -3 1 }}
|  
| Thotrilo-agu
|  
| 3u1o<sup>3</sup>g2
|  
| Jacobin comma
|-
|-
|  
| 13
|  
| [[10648/10647|(10 digits)]]
|  
| 0.16
|  
| {{Monzo| 3 -2 0 -1 3 -2 }}
| 585.4
| Thuthutrilo-aru
| 4
| 3uu1o<sup>3</sup>r
| 20
| [[Harmonisma]]
|-
|-
| 21
| 17
| 7
| [[2187/2176]]
| 614.6
| 8.73
|  
| {{Monzo| -7 7 0 0 0 0 -1 }}
|  
| Lasu
|  
| L17u
|  
| Septendecimal schisma
|-
|-
| 24
| 17
| 8
| [[256/255]]
| 702.4
| 6.78
|  
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
|  
| Sugu
|  
| 17ug
|  
| Charisma
|-
|-
|  
| 17
|  
| [[715/714]]
|  
| 2.42
|  
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| 731.7
| Sutholoruyo
| 5
| 17u3o1ory
| 25
| Septendecimal bridge comma
|-
|-
| 27
| 19
| 9
| [[210/209]]
| 790.2
| 8.26
|  
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
|  
| Nuluzoyo
|  
| 19u1uzy
|  
| Spleen comma
|-
|-
| 30
| 19
| 10
| [[361/360]]
|  
| 4.80
| 878.0
| {{Monzo| -3 -2 -1 0 0 0 0 2 }}
|  
| Nonogu
| 6
| 19oog2
| 30
| Go comma
|-
|-
| 33
| 19
| 11
| [[513/512]]
| 965.9
| 3.38
|  
| {{Monzo| -9 3 0 0 0 0 0 1 }}
|  
| Lano
|  
| L19o
|  
| Boethius' comma
|-
|-
|  
| 19
|  
| [[1216/1215]]
|  
| 1.42
|  
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| 1024.4
| Sanogu
| 7
| s19og
| 35
| Eratosthenes' comma
|-
|-
| 36
| 23
| 12
| [[736/729]]
| 1053.7
| 16.54
|  
| {{Monzo| 5 -6 0 0 0 0 0 0 1 }}
|  
| Satwetho
|  
| s23o
|  
| Vicesimotertial comma
|-
| 29
| [[145/144]]
| 11.98
| {{Monzo| -4 -2 1 0 0 0 0 0 0 1 }}
| Twenoyo
| 29oy
| 29th-partial chroma
|}
 
=== Rank-2 temperaments ===
* [[List of edo-distinct 41et rank two temperaments]]
* [[Schismic–countercommatic equivalence continuum]]
 
{| class="wikitable right-1 right-2"
|+ Table of temperaments by generator
|-
|-
| 39
! Degree
| 13
! Cents
| 1141.5
! Temperament(s)
|
! [[Pergen]]
|
! Mos scales
|
|
|-
|-
| 1
| 29.27
| [[Slendi]]
| (P8, P4/17)
|  
|  
|
|
|
| 1170.7
| 8
| 40
|-
|-
! colspan="7" | [ second octave ]
| 2
| 58.54
| [[Hemimiracle]]<br>[[Dodecacot]]
| (P8, P5/12)
| 21-tone mos
|-
|-
| 1
| 3
| 14
| 87.80
| 29.2
| [[Octacot]]
|
| (P8, P5/8)
|  
| 14-tone mos: 3 3 3 3 3 3 3 3 3 3 3 3 3 2
|  
|  
|-
|-
| 4
| 4
| 15
| 117.07
|  
| [[Miracle]]
| 117.1
| (P8, P5/6)
|  
| 11-tone mos: 4 4 4 4 4 4 4 4 4 4 1
| 9
|-
| 4
| 5
| 146.34
| [[BPS]] / [[bohpier]]
| (P8, P12/13)
| 20-tone mos
|-
| 6
| 175.61
| [[Tetracot]] / [[bunya]] / [[monkey]]<br>[[Sesquiquartififths]] / [[sesquart]]
| (P8, P5/4)
| 13-tone mos: 1 5 1 5 1 5 1 5 5 1 5 1 5
|-
|-
| 7
| 7
| 16
| 204.88
| 204.9
| [[Baldy]]<br>[[Quadrimage]]
|  
| (P8, c<sup>3</sup>P4/20)
|  
| 11-tone mos: 6 1 6 6 1 6 1 6 1 6 1
|  
|-
|  
| 8
| 234.15
| [[Slendric]] / [[rodan]] / [[guiron]]
| (P8, P5/3)
| 11-tone mos: 7 1 7 1 7 1 7 1 1 7 1
|-
|-
|
|
|
|
| 263.4
| 10
| 9
| 9
| 263.41
| [[Septimin]]
| (P8, ccP4/11)
| 9-tone mos: 5 4 5 5 4 5 4 5 4
|-
|-
| 10
| 10
| 17
| 292.68
| 292.7
| [[Quasitemp]]
|
| (P8, c<sup>3</sup>P4/14)
|  
| 29-tone mos
|  
|  
|-
|-
| 13
| 11
| 18
| 321.95
| 380.5
| [[Superkleismic]]
|  
| (P8, ccP4/9)
|  
| 11-tone mos: 5 3 5 3 3 5 3 3 5 3 3
|  
|-
|  
| 12
| 351.22
| [[Hemif]] / [[hemififths]] / [[salsa]]<br>[[Karadeniz]]
| (P8, P5/2)
| 10-tone mos: 5 2 5 5 2 5 5 5 2 5
|-
| 13
| 380.49
| [[Magic]] / [[witchcraft]]<br>[[Quanharuk]]
| (P8, P12/5)
| 10-tone mos: 2 9 2 2 9 2 2 9 2 2
|-
|-
|
|
|
|
| 409.8
| 11
| 14
| 14
| 409.76
| [[Hocum]]<br>[[Hocus]]
| (P8, c<sup>3</sup>P4/10)
| 32-tone mos
|-
|-
| 16
| 15
| 19
| 439.02
| 468.3
| [[Superthird]]
|  
| (P8, c<sup>6</sup>P5/18)
|  
| 11-tone mos: 4 3 4 4 4 3 4 4 3 4 4
|  
|-
|  
| 16
| 468.29
| [[Barbad]]
| (P8, c<sup>7</sup>P4/19)
| 8-tone mos: 7 2 7 7 2 7 7 2
|-
|-
| 19
| 17
| 20
| 497.56
|  
| [[Helmholtz (temperament)|Helmholtz]] / [[garibaldi]] / [[cassandra]] / [[andromeda]]<br>[[Kwai]]
| 556.1
| (P8, P5)
|  
| 12-tone mos: 4 3 4 3 3 4 3 4 3 4 3 4 3 3
| 12
| 19
|-
|-
| 22
| 18
| 21
| 526.83
| 643.9
| [[Trismegistus]]
|
| (P8, c<sup>6</sup>P5/15)
|  
| 9-tone mos: 5 5 3 5 5 5 5 3 5
|  
|  
|-
|-
|  
| 19
|  
| 556.10
|  
| [[Alphorn]]
|  
| (P8, c<sup>7</sup>P4/16)
| 702.4
| 9-tone mos: 3 3 3 10 3 3 3 3 10
| 13
| 24
|-
|-
| 25
| 20
| 22
| 585.37
| 731.7
| [[Pluto]]<br>[[Merman]]
|
| (P8, c<sup>3</sup>P4/7)
|  
|  
|  
|  
|}
== 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 ==
=== Lists of 41edo scales ===
* [[41edo modes]]
* [[List of MOS scales in 41edo]]
* [[The Kite Guitar Scales]]
* [[Kite Giedraitis's Categorizations of 41edo Scales]]
=== Harmonic scale ===
41edo is the first edo to do some justice to Mode 8 of the [[harmonic series]], which Dante Rosati calls the "[[overtone scale|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).
{| class="wikitable" style="text-align: center;"
|-
|-
| 28
! Overtones in "Mode 8":
| 23
| 8
| 819.5
| 9
|
| 10
|
| 11
|
| 12
|
| 13
|-
|  
|  
|  
|  
| 848.8
| 14
| 14
| 29
|-
| 31
| 24
| 907.3
|
|
|
|
|-
| 34
| 25
|
| 995.1
|
| 15
| 15
| 34
| 16
|-
|-
| 37
! … as JI Ratio from 1/1:
| 26
| 1/1
| 1082.9
| 9/8
|  
| 5/4
|  
| 11/8
|  
| 3/2
|  
| 13/8
| 7/4
| 15/8
| 2/1
|-
|-
|  
! … in cents:
|  
| 0
|  
| 203.9
|  
| 386.3
| 1141.5
| 551.3
| 16
| 702.0
| 39
| 840.5
| 968.8
| 1088.3
| 1200.0
|-
|-
| 40
! Nearest degree of 41edo:
| 27
| 0
| 1170.7
| 7
|  
| 13
|  
| 19
|  
| 24
|  
| 29
| 33
| 37
| 41
|-
|-
! colspan="7" | [ third octave ]
! … in cents:
|-
| 0
| 2
| 204.9
| 28
| 380.5
| 58.5
| 556.1
|  
| 702.4
|  
| 848.8
|  
| 965.9
|  
| 1082.9
| 1200.0
|}
 
While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)
 
* 7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) – a close match.
* 6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
* 5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
* 4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).
 
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.
 
=== Nonoctave temperaments ===
Taking every third degree of 41edo produces a scale extremely close to [[88cET]] or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <span style="">[[BP|Bohlen–Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See [[Relationship between Bohlen–Pierce and octave-ful temperaments]], and see this chart:
 
{| class="wikitable center-all right-3 right-4 right-5 mw-collapsible mw-collapsed"
|-
|-
|  
! colspan="3" | 3 degrees of 41edo near 88cET
|
! overlap
|  
! colspan="3" | 5 degrees of 41edo near BP
|
| 87.8
| 17
| 3
|-
|-
| 5
! 41edo
| 29
! 88cET
| 146.3
! cents
|
! cents
|
! cents
|
! BP
|
! 41edo
|-
|-
| 8
| 0
| 30
| 0
|  
|  
| 234.1
| 0
|  
|  
| 18
| 0
| 8
| 0
|-
|-
| 11
| 3
| 31
| 1
| 322.0
| 87.8
|  
|  
|  
|  
Line 1,937: Line 1,770:
|  
|  
|  
|  
| 380.5
| 146.3
| 19
| 1
| 13
| 5
|-
|-
| 14
| 6
| 32
| 2
| 409.8
| 175.6
|  
|  
|  
|  
Line 1,949: Line 1,782:
|  
|  
|-
|-
| 17
| 9
| 33
| 3
| 497.6
| 263.4
|  
|  
|  
|  
Line 1,961: Line 1,794:
|  
|  
|  
|  
| 526.8
| 292.7
| 20
| 2
| 18
| 10
|-
|-
| 20
| 12
| 34
| 4
| 585.3
| 351.2
|  
|  
|  
|  
Line 1,973: Line 1,806:
|  
|  
|-
|-
| 23
| 15
| 35
| 5
|  
|  
| 673.2
| 439.0
|  
|  
| 21
| 3
| 23
| 15
|-
|-
| 26
| 18
| 36
| 6
| 761.0
| 526.8
|  
|  
|  
|  
Line 1,993: Line 1,826:
|  
|  
|  
|  
| 819.5
| 585.4
| 22
| 4
| 28
| 20
|-
|-
| 29
| 21
| 37
| 7
| 848.8
| 614.6
|  
|  
|  
|  
Line 2,005: Line 1,838:
|  
|  
|-
|-
| 32
| 24
| 38
| 8
| 936.6
| 702.4
|  
|  
|  
|  
Line 2,017: Line 1,850:
|  
|  
|  
|  
| 965.9
| 731.7
| 23
| 5
| 33
| 25
|-
|-
| 35
| 27
| 39
| 9
| 1024.4
| 790.2
|  
|  
|  
|  
Line 2,029: Line 1,862:
|  
|  
|-
|-
| 38
| 30
| 40
| 10
|  
|  
| 1112.2
| 878.0
|  
|  
| 24
| 6
| 38
| 30
|}
|-
 
| 33
=== More scales ===
| 11
* [[Bohpier8]]
| 965.9
* [[Bohpier9]]
|
* [[Bohpier17]]
|
* [[Bohpier25]]
|
* [[Bohpier33]]
|
* [[Compdye]]
|-
 
|
== Instruments ==
|
=== Guitars ===
|
The first 41edo guitar was probably this one, built by [[Erv Wilson]] in the 1960's:
|
 
| 1024.4
[[File:Erv Wilson's full-41 guitar 2.jpg|none|thumb]]
| 7
 
| 35
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.
|-
 
| 36
[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|838x838px]]
| 12
 
| 1053.7
Several more modern guitars:
|
 
|
[[File:Melleweijters.com 41edo.jpg|frameless|832x832px]]
|
 
|
''[[Melle Weijters]]' 10-string guitar ([https://melleweijters.com Melleweijters.com])''
|-
 
| 39
[[File:41-EDD_elektrische_gitaar.jpg|alt=41-EDD elektrische gitaar.jpg|560x745px|41-EDD elektrische gitaar.jpg]]
| 13
 
| 1141.5
''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]]
|
 
| 1170.7
For more photos of Kite guitars, see [[Kite Guitar Photographs]].
| 8
 
| 40
=== 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 [[User:Richie|Richie]]]]
! colspan="7" | [ second octave ]
 
|-
 
| 1
 
| 14
 
| 29.2
 
|
 
|
 
|
 
|
 
|-
 
| 4
 
| 15
 
|
=== Keyboards ===
| 117.1
A possible 41edo keyboard design:
|
[[File:41edo keyboard layout.png|none|thumb|484x484px]]
| 9
[[File:Xenachord with 41edo layout.png|left|thumb|[https://richiegreene.com/instruments/ Xenachord] with 41edo layout by [[User:Richie|Richie]]
| 4
 
|-
128-key isomorphic controller with snap-fit key caps.]]
| 7
 
| 16
 
| 204.9
 
|
 
|
 
|
 
|
 
|-
 
|
 
|
 
|
 
|
See also [[41-edo Keyboards]] for Lumatone, Linnstrument and Harpejji options, as well as DIY options.
| 263.4
 
| 10
== 41edo as a Universal Tuning ==
| 9
41's claim to fame as a "universal tuning" is the fact that it approximates scales present in many important world music traditions, and thus is good for both combining and exploring cultural playstyles. It makes no claim to perfectly and faithfully represent the musical cultures listed, as doing so would require far more notes and small details than are present in 41. That being said, it has certain attributes that allow it to approximate common scales in these cultures with far more accuracy than most comparable EDOs.
|-
 
| 10
=== Western ===
| 17
Due to 41edo's extremely accurate perfect fifth, it makes a good tuning for [[schismatic]] temperament and the 12-note MOS, which in turn is a good approximation of the standard [[12edo]] scale, and when arranged as a Bbb-D gamut, approximates the 12-note roughly [[Pythagorean tuning]] known as [[Kirnberger I]]. This extends the Ptolemy Diatonic Scale ('''7 6 4 7 6 7 4'''), which 41 approximates excellently, by completing the circle of fifths with pure 3/2s. By using this system and occasionally substituting in alternate major seconds and sixths when necessary, it becomes quite reminiscent of (and can improve on) 12edo harmony. Additionally, the Pythagorean Pentatonic scale can be used for melodies overtop due to the strong quartal nature of the scale. The Pythagorean diatonic scale exists as an option as well, but use may be limited unless [[Gentle chords|Gentle triads]] are ideal. An alternate option is approximating a Just Intonation scale such as the [[Duodene|Asymmetric scale]], a common option for a 5-limit JI scale, or [[Centaur]], a 7-limit JI scale using "blue" or subminor intervals for the accidental notes. There exist other options for 5-limit JI scales, all of which have some reasonable approximation in 41 due to its relative excellence in the 5-limit.
| 292.7
 
|
=== Middle Eastern ===
|
{{See also| Arabic, Turkish, Persian }}
|
 
|
While the [[Hemif|Hemif[7]]] scale itself and MODMOSes related to it give the middle eastern sound well, 41 has other interesting properties that make it an ideal system for Arabic and Turkish music. It is considered a "Level 2 EDO" due to the fact that it has neutral seconds and thirds as well as submajor and supraminor ones added to a Pythagorean skeleton, with small semitones as minor seconds and major whole tones as major seconds. The submajor third is great for Turkish Rast, around [[Ozan Yarman]]’s ideal size, and is sharp enough to sound close to a [[5/4]], while the neutral third exists as half of a [[3/2]] and works well for Arabic Rast and some Persian scales. Additionally, a large [[apotome]] exists for the Hijaz maqam.
|-
| 13
| 18
| 380.5
|
|
|
|
|-
|
|
|
|
| 409.8
| 11
| 14
|-
| 16
| 19
| 468.3
|
|
|
|
|-
| 19
| 20
|
| 556.1
|
| 12
| 19
|-
| 22
| 21
| 643.9
|
|
|
|
|-
|
|
|
|
| 702.4
| 13
| 24
|-
| 25
| 22
| 731.7
|
|
|
|
|-
| 28
| 23
| 819.5
|
|
|
|
|-
|
|
|
|
| 848.8
| 14
| 29
|-
| 31
| 24
| 907.3
|
|
|
|
|-
| 34
| 25
|
| 995.1
|
| 15
| 34
|-
| 37
| 26
| 1082.9
|
|
|
|
|-
|
|
|
|
| 1141.5
| 16
| 39
|-
| 40
| 27
| 1170.7
|
|
|
|
|-
! colspan="7" | [ third octave ]
|-
| 2
| 28
| 58.5
|
|
|
|
|-
|
|
|
|
| 87.8
| 17
| 3
|-
| 5
| 29
| 146.3
|
|
|
|
|-
| 8
| 30
|
| 234.1
|
| 18
| 8
|-
| 11
| 31
| 322.0
|
|
|
|
|-
|
|
|
|
| 380.5
| 19
| 13
|-
| 14
| 32
| 409.8
|
|
|
|
|-
| 17
| 33
| 497.6
|
|
|
|
|-
|
|
|
|
| 526.8
| 20
| 18
|-
| 20
| 34
| 585.3
|
|
|
|
|-
| 23
| 35
|
| 673.2
|
| 21
| 23
|-
| 26
| 36
| 761.0
|
|
|
|
|-
|
|
|
|
| 819.5
| 22
| 28
|-
| 29
| 37
| 848.8
|
|
|
|
|-
| 32
| 38
| 936.6
|
|
|
|
|-
|
|
|
|
| 965.9
| 23
| 33
|-
| 35
| 39
| 1024.4
|
|
|
|
|-
| 38
| 40
|
| 1112.2
|
| 24
| 38
|}
 
=== More scales ===
* [[Bohpier8]]
* [[Bohpier9]]
* [[Bohpier17]]
* [[Bohpier25]]
* [[Bohpier33]]
* [[Compdye]]
 
== Instruments ==
=== Guitars ===
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|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.
 
[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|500px]]
 
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>
 
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.  
 
[[File:Caleb's Kite guitar.jpg|none|thumb|200px|Kite guitar]]
 
For more photos of Kite guitars, see [[Kite Guitar Photographs]].
{{clear}}
 
=== 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 ===
A possible 41edo keyboard design:
<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}}
 
=== 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 ==
41's claim to fame as a "universal tuning" is the fact that it approximates scales present in many important world music traditions, and thus is good for both combining and exploring cultural playstyles. It makes no claim to perfectly and faithfully represent the musical cultures listed, as doing so would require far more notes and small details than are present in 41. That being said, it has certain attributes that allow it to approximate common scales in these cultures with far more accuracy than most comparable EDOs.
 
=== Western ===
Due to 41edo's extremely accurate perfect fifth, it makes a good tuning for [[schismatic]] temperament and the 12-note MOS, which in turn is a good approximation of the standard [[12edo]] scale, and when arranged as a Bbb-D gamut, approximates the 12-note roughly [[Pythagorean tuning]] known as [[Kirnberger I]]. This extends the Ptolemy Diatonic Scale ('''7 6 4 7 6 7 4'''), which 41 approximates excellently, by completing the circle of fifths with pure 3/2s. By using this system and occasionally substituting in alternate major seconds and sixths when necessary, it becomes quite reminiscent of (and can improve on) 12edo harmony. Additionally, the Pythagorean Pentatonic scale can be used for melodies overtop due to the strong quartal nature of the scale. The Pythagorean diatonic scale exists as an option as well, but use may be limited unless [[Gentle chords|Gentle triads]] are ideal. An alternate option is approximating a Just Intonation scale such as the [[Duodene|Asymmetric scale]], a common option for a 5-limit JI scale, or [[Centaur]], a 7-limit JI scale using "blue" or subminor intervals for the accidental notes. There exist other options for 5-limit JI scales, all of which have some reasonable approximation in 41 due to its relative excellence in the 5-limit.
 
=== Middle Eastern ===
{{See also| Arabic, Turkish, Persian }}
 
While the [[Hemif|Hemif[7]]] scale itself and MODMOSes related to it give the middle eastern sound well, 41 has other interesting properties that make it an ideal system for Arabic and Turkish music. It is considered a "Level 2 EDO" due to the fact that it has neutral seconds and thirds as well as submajor and supraminor ones added to a Pythagorean skeleton, with small semitones as minor seconds and major whole tones as major seconds. The submajor third is great for Turkish Rast, around [[Ozan Yarman]]’s ideal size, and is sharp enough to sound close to a [[5/4]], while the neutral third exists as half of a [[3/2]] and works well for Arabic Rast and some Persian scales. Additionally, a large [[apotome]] exists for the Hijaz maqam.


=== Indonesian ===
=== Indonesian ===
Line 2,142: 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 2,242: 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"