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<span style="display: block; text-align: right;">[[:de:41edo|Deutsch]]</span>
{{interwiki
| de = 41-EDO
| en = 41edo
| es =
| ja =
}}
{{Infobox ET}}
{{Wikipedia| 41 equal temperament }}
{{ED intro}}


__FORCETOC__
== Theory ==
=Theory=
41edo is the second smallest equal division (after [[29edo]]) whose [[3/2|perfect fifth]] is closer to just intonation than that of [[12edo]], and is the seventh [[zeta integral edo]], after [[31edo|31]]; it is not, however, a [[zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit]] fairly well, and perhaps the [[13-limit]]. In fact, it is [[consistent]] to the [[15-odd-limit]], or the no-17's [[21-odd-limit]]. ''All'' of its intervals between 100 and 1100 cents in size are 15-odd-limit [[consonance]]s, although its [[~]][[13/10]] is 14 cents sharp and arguably manifests itself as [[21/16]] rather than 13/10.  
The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cent|cent]]s, an [[interval|interval]] close in size to [[64/63|64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&amp;26) temperament. It is the second smallest equal temperament (after [[29edo|29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], and is the seventh [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit|11-limit]] fairly well, and the [[13-limit|13-limit]] perhaps close enough for government work, though its [[13/10|13/10]] is 14 cents sharp. Various 13-limit [[Magic_extensions|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.


41edo is consistent in the 15 odd limit. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 is debatable. (In comparison, [[31edo|31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).
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.  


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


41edo is the 13th [[prime_numbers|prime]] edo, following [[37edo|37edo]] and coming before [[43edo|43edo]].
=== Prime harmonics ===
{{Harmonics in equal|41|columns=9}}
{{Harmonics in equal|41|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 41edo (continued)}}


=Notation=
=== As a tuning of other temperaments ===
{| class="wikitable"
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]].
 
Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]].
 
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.
 
A step of 41edo is close and consistently mapped to [[64/63]], the septimal comma.
 
=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup 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 way milder, whereas 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 by tunings such as [[65edt]], [[106ed6]], and [[147ed12]]. Primes 19, 29, and 31 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.
 
=== Subsets and supersets ===
41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it contains [[41ed4]].
 
[[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.
 
== Intervals ==
{{See also| 41edo solfege }}
 
{| class="wikitable center-1 right-2 center-5 center-6 center-8 center-9"
|-
|-
! |
! #
! | Cents Value
! Cents
! | Approximate Ratios
! Approximate ratios*
in the [[11-limit|11-limit]]
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^d2)
! colspan="3" |[[Ups_and_Downs_Notation|Ups and Downs Notation]]
! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)
! | Andrew's  
! [[41edo solfege|Kite's<br>solfege]]
Solfege Syllables
! [[41edo solfege|Andrew's<br>solfege]]
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0.00
| 0.0
| | [[1/1|1/1]]
| [[1/1]]
|perfect unison
| perfect unison
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | D
| D
| | do
| perfect unison
| P1
| D
| Da
| Do
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 29.27
| 29.3
| | [[81/80|81/80]], [[64/63]]
| [[49/48]], [[50/49]], [[64/63]], [[81/80]]
|up-unison
| up-unison
| style="text-align:center;" | ^1
| ^1
| style="text-align:center;" | ^D
| ^D
| | di
| comma-wide unison, super unison
| K1/S1
| KD, SD
| Du
| Di
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 58.54
| 58.5
| | [[25/24|25/24]], [[28/27|28/27]], [[33/32|33/32]]
| [[25/24]], [[28/27]], [[33/32]], [[36/35]]
|double-up unison,
| dup-unison, downminor 2nd
downminor 2nd
| ^^1, vm2
| style="text-align:center;" | ^^1, vm2
| ^^D, vEb
| style="text-align:center;" | ^^D, vEb
| subminor 2nd, classic aug unison, uber unison
| | ro
| sm2, kkA1, U1
| sEb, kkD#, UD
| Fro
| Ro
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 87.805
| 87.8
| | [[21/20|21/20]], [[22/21|22/21]]
| [[19/18]], [[20/19]], [[21/20]], [[22/21]]
|down-aug 1sn,
| down-aug 1sn, minor 2nd
minor 2nd
| vA1, m2
| style="text-align:center;" | vA1, m2
| vD#, Eb
| style="text-align:center;" | vD#, Eb
| minor 2nd, comma-narrow augmented unison
| | rih
| m2, kA1
| Eb, kD#
| Fra
| Rih
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 117.07
| 117.1
| | [[16/15|16/15]], [[15/14|15/14]]
| [[14/13]], [[15/14]], [[16/15]]
|augmented 1sn,
| augmented 1sn, upminor 2nd
 
| A1, ^m2
upminor 2nd
| D#, ^Eb
| style="text-align:center;" | A1, ^m2
| classic minor 2nd, augmented unison
| style="text-align:center;" | D#, ^Eb
| Km2, A1
| | ra
| KEb, D#
| Fru
| Ra
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 146.34
| 146.3
| | [[12/11|12/11]]
| [[12/11]], [[13/12]]
|mid 2nd
| mid 2nd
| style="text-align:center;" | ~2
| ~2
| style="text-align:center;" | ^D#, vvE
| ^D#, vvE
| | ru
| neutral second, super augmented unison
| N2, SA1
| UEb/uE, sD#
| Ri
| Ru
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 175.61
| 175.6
| | [[10/9|10/9]], [[11/10|11/10]]
| [[10/9]], [[11/10]], [[21/19]]
|downmajor 2nd
| downmajor 2nd
| style="text-align:center;" | vM2
| vM2
| style="text-align:center;" | vE
| vE
| | reh
| classic/comma-wide major 2nd
| kM2
| kE
| Ro
| Reh
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 204.88
| 204.9
| | [[9/8|9/8]]
| [[9/8]]
|major 2nd
| major 2nd
| style="text-align:center;" | M2
| M2
| style="text-align:center;" | E
| E
| | re
| major 2nd
| M2
| E
| Ra
| Re
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 234.15
| 234.1
| | [[8/7|8/7]]
| [[8/7]], [[15/13]]
|upmajor 2nd
| upmajor 2nd
| style="text-align:center;" | ^M2
| ^M2
| style="text-align:center;" | ^E
| ^E
| | ri
| supermajor 2nd
| SM2
| SE
| Ru
| Ri
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 263.415
| 263.4
| | [[7/6|7/6]]
| [[7/6]], [[22/19]]
|downminor 3rd
| downminor 3rd
| style="text-align:center;" | vm3
| vm3
| style="text-align:center;" | vF
| vF
| | ma
| subminor 3rd
| sm3
| sF
| No
| Ma
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | 292.68
| 292.7
| | [[32/27|32/27]], [[13/11]]
| [[13/11]], [[19/16]], [[32/27]]
|minor 3rd
| minor 3rd
| style="text-align:center;" | m3
| m3
| style="text-align:center;" | F
| F
| | meh
| minor 3rd
| m3
| F
| Na
| Meh
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 321.95
| 322.0
| | [[6/5|6/5]]
| [[6/5]]
|upminor 3rd
| upminor 3rd
| style="text-align:center;" | ^m3
| ^m3
| style="text-align:center;" | ^F
| ^F
| | me
| classicminor 3rd
| Km3
| KF
| Nu
| Me
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 351.22
| 351.2
| | [[11/9|11/9]], [[27/22|27/22]]
| [[11/9]], [[16/13]]
|mid 3rd
| mid 3rd
| style="text-align:center;" | ~3
| ~3
| style="text-align:center;" | ^^F, vGb
| ^^F, vGb
| | mu
| neutral 3rd, sub diminished 4th
| N3, sd4
| UF/uF#, sGb
| Mi
| Mu
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 380.49
| 380.5
| | [[5/4|5/4]]
| [[5/4]], [[26/21]]
|downmajor 3rd
| downmajor 3rd
| style="text-align:center;" | vM3
| vM3
| style="text-align:center;" | vF#, Gb
| vF#, Gb
| | mi
| classic major 3rd, diminished 4th
| kM3, d4
| kF#, Gb
| Mo
| Mi
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 409.76
| 409.8
| | [[14/11|14/11]], [[81/64|81/64]]
| [[14/11]], [[19/15]], [[24/19]]
|major 3rd
| major 3rd
| style="text-align:center;" | M3
| M3
| style="text-align:center;" | F#, ^Gb
| F#, ^Gb
| | maa
| major 3rd, comma-wide diminished 4th
| M3, Kd4
| F#, KGb
| Ma
| Maa
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 439.02
| 439.0
| | [[9/7|9/7,]] [[32/25|32/25]]
| [[9/7]], [[32/25]]
|upmajor 3rd
| upmajor 3rd
| style="text-align:center;" | ^M3
| ^M3
| style="text-align:center;" | ^F#, vvG
| ^F#, vvG
| | mo
| supermajor 3rd, classic diminished 4th
| SM3, KKd4
| SF#, KKGb
| Mu
| Mo
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | 468.29
| 468.3
| | [[21/16|21/16]], [[13/10]]
| [[21/16]], [[13/10]]
|down-4th
| down-4th
| style="text-align:center;" | v4
| v4
| style="text-align:center;" | vG
| vG
| | fe
| sub 4th
| s4
| sG
| Fo
| Fe
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | 497.56
| 497.6
| | [[4/3|4/3]]
| [[4/3]]
|perfect 4th
| perfect 4th
| style="text-align:center;" | P4
| P4
| style="text-align:center;" | G
| G
| | fa
| perfect 4th
| P4
| G
| Fa
| Fa
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 526.83
| 526.8
| | [[15/11|15/11]], [[27/20|27/20]]
| [[15/11]], [[19/14]], [[27/20]]
|up-4th
| up-4th
| style="text-align:center;" | ^4
| ^4
| style="text-align:center;" | ^G
| ^G
| | fih
| comma-wide 4th
| K4
| KG
| Fu
| Fih
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | 556.1
| 556.1
| | [[11/8|11/8]]
| [[11/8]], [[18/13]], [[26/19]]
|mid-4th
| mid-4th, downdim 5th
| style="text-align:center;" | ~4
| ~4, vd5
| style="text-align:center;" | ^^G, vAb
| ^^G, vAb
| | fu
| uber/neutral 4th, classic augmented 4th
| U4/N4, kkA4
| UG, kkG#
| Fi/Sho
| Fu
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" | 585.37
| 585.4
| | [[7/5|7/5]]
| [[7/5]], [[45/32]]
|downaug 4th, dim 5th
| downaug 4th, dim 5th
| style="text-align:center;" | vA4, d5
| vA4, d5
| style="text-align:center;" | vG#, Ab
| vG#, Ab
| | fi
| comma-narrow augmented 4th, diminished 5th
| kA4/d5
| kG#, Ab
| Po/Sha
| Fi
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 614.63
| 614.6
| | [[10/7|10/7]]
| [[10/7]], [[64/45]]
|aug 4th, updim 5th
| aug 4th, updim 5th
| style="text-align:center;" | A4, ^d5
| A4, ^d5
| style="text-align:center;" | G#, ^Ab
| G#, ^Ab
| | se
| augmented 4th, comma-wide diminished 5th
| A4/Kd5
| G#, KAb
| Pa/Shu
| Se
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:center;" | 643.90
| 643.9
| | [[16/11|16/11]], [[13/9]]
| [[13/9]], [[16/11]], [[19/13]]
|mid-5th
| mid-5th, upaug 4th
| style="text-align:center;" | ~5
| ~5, ^A4
| style="text-align:center;" | vvA
| ^G#, vvA
| | su
| unter/neutral 5th, classic diminished 5th
| u5/N5, KKd5
| uA, KKAb
| Pu/Si
| Su
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:center;" | 673.17
| 673.2
| | [[22/15|22/15]], [[40/27|40/27]]
| [[22/15]], [[28/19]], [[40/27]]
|down-5th
| down-5th
| style="text-align:center;" | v5
| v5
| style="text-align:center;" | vA
| vA
| | sih
| comma-narrow 5th
| k5
| kA
| So
| Sih
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" | 702.44
| 702.4
| | [[3/2|3/2]]
| [[3/2]]
|perfect 5th
| perfect 5th
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | A
| A
| | sol
| perfect 5th
| P5
| A
| Sa
| Sol
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:center;" | 731.71
| 731.7
| | [[32/21|32/21]], [[20/13]]
| [[20/13]], [[32/21]]
|up-5th
| up-5th
| style="text-align:center;" | ^5
| ^5
| style="text-align:center;" | ^A
| ^A
| | si
| super 5th
| S5
| SA
| Su
| Si
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:center;" | 760.98
| 761.0
| | [[14/9|14/9]], [[25/16|25/16]]
| [[14/9]], [[25/16]]
|downminor 6th
| downminor 6th
| style="text-align:center;" | vm6
| vm6
| style="text-align:center;" | ^^A, vBb
| ^^A, vBb
| | lo
| subminor 6th, classic augmented 5th
| sm6
| sBb, kkA#
| Flo
| Lo
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:center;" | 790.24
| 790.2
| | [[11/7|11/7]], [[128/81|128/81]]
| [[11/7]], [[19/12]], [[30/19]]
|minor 6th
| minor 6th
| style="text-align:center;" | m6
| m6
| style="text-align:center;" | vA#, Bb
| vA#, Bb
| | leh
| minor 6th, comma-narrow augmented 5th
| m6
| Bb, kA#
| Fla
| Leh
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:center;" | 819.51
| 819.5
| | [[8/5|8/5]]
| [[8/5]], [[21/13]]
|upminor 6th
| upminor 6th
| style="text-align:center;" | ^m6
| ^m6
| style="text-align:center;" | A#, ^Bb
| A#, ^Bb
| | le
| classic minor 6th, augmented 5th
| Km6, A5
| KBb, A#
| Flu
| Le
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:center;" | 848.78
| 848.8
| | [[18/11|18/11]], [[44/27|44/27]]
| [[13/8]], [[18/11]]
|mid 6th
| mid 6th
| style="text-align:center;" | ~6
| ~6
| style="text-align:center;" | ^A#, vvB
| ^A#, vvB
| | lu
| neutral 6th, super augmented 5th
| N6
| UBb/uB, sA#
| Li
| Lu
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:center;" | 878.05
| 878.0
| | [[5/3|5/3]]
| [[5/3]]
|downmajor 6th
| downmajor 6th
| style="text-align:center;" | vM6
| vM6
| style="text-align:center;" | vB
| vB
| | la
| classic major 6th
| kM6
| kB
| Lo
| La
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:center;" | 907.32
| 907.3
| | [[27/16|27/16]]
| [[22/13]], [[27/16]], [[32/19]]
|major 6th
| major 6th
| style="text-align:center;" | M6
| M6
| style="text-align:center;" | B
| B
| | laa
| major 6th
| M6
| B
| La
| Laa
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:center;" | 936.59
| 936.6
| | [[12/7|12/7]]
| [[12/7]], [[19/11]]
|upmajor 6th
| upmajor 6th
| style="text-align:center;" | ^M6
| ^M6
| style="text-align:center;" | ^B
| ^B
| | li
| supermajor 6th
| SM6
| SB
| Lu
| Li
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:center;" | 965.85
| 965.9
| | [[7/4|7/4]]
| [[7/4]], [[26/15]]
|downminor 7th
| downminor 7th
| style="text-align:center;" | vm7
| vm7
| style="text-align:center;" | vC
| vC
| | ta
| subminor 7th
| sm7
| sC
| Tho
| Ta
|-
|-
| style="text-align:center;" | 34
| 34
| style="text-align:center;" | 995.12
| 995.1
| | [[16/9|16/9]]
| [[16/9]]
|minor 7th
| minor 7th
| style="text-align:center;" | m7
| m7
| style="text-align:center;" | C
| C
| | teh
| minor 7th
| m7
| C
| Tha
| Teh
|-
|-
| style="text-align:center;" | 35
| 35
| style="text-align:center;" | 1024.39
| 1024.4
| | [[9/5|9/5]], [[20/11|20/11]]
| [[9/5]], [[20/11]], [[38/21]]
|upminor 7th
| upminor 7th
| style="text-align:center;" | ^m7
| ^m7
| style="text-align:center;" | ^C
| ^C
| | te
| classic/comma-wide minor seventh
| Km7
| KC
| Thu
| Te
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:center;" | 1053.66
| 1053.7
| | [[11/6|11/6]]
| [[11/6]], [[24/13]]
|mid 7th
| mid 7th
| style="text-align:center;" | ~7
| ~7
| style="text-align:center;" | ^^C, vDb
| ^^C, vDb
| | tu
| neutral 7th, sub diminished 8ve
| N7
| UC/uC#, sDb
| Ti
| Tu
|-
|-
| style="text-align:center;" | 37
| 37
| style="text-align:center;" | 1082.93
| 1082.9
| | [[15/8|15/8]]
| [[13/7]], [[15/8]], [[28/15]]
|downmajor 7th
| downmajor 7th
| style="text-align:center;" | vM7
| vM7
| style="text-align:center;" | vC#, Db
| vC#, Db
| | ti
| classic major 7th, diminished 8ve
| kM7, d8
| kC#, Db
| To
| Ti
|-
|-
| style="text-align:center;" | 38
| 38
| style="text-align:center;" | 1112.195
| 1112.2
| | [[40/21|40/21]], [[21/11|21/11]]
| [[19/10]], [[21/11]], [[36/19]], [[40/21]]
|major 7th
| major 7th
| style="text-align:center;" | M7
| M7
| style="text-align:center;" | C#, ^Db
| C#, ^Db
| | taa
| major 7th, comma-wide diminished 8ve
| M7, Kd8
| C#, KDb
| Ta
| Taa
|-
|-
| style="text-align:center;" | 39
| 39
| style="text-align:center;" | 1141.46
| 1141.5
| | [[48/25|48/25]], [[27/14|27/14]], [[64/33|64/33]]
| [[27/14]], [[35/18]], [[48/25]], [[64/33]]
|upmajor 7th
| upmajor 7th
| style="text-align:center;" | ^M7
| ^M7
| style="text-align:center;" | C#^, vvD
| ^C#, vvD
| | to
| supermajor 7th, classic dim 8ve, unter 8ve
| SM7, KKd8, U8
| SC#, KKDb, u8
| Tu
| To
|-
|-
| style="text-align:center;" | 40
| 40
| style="text-align:center;" | 1170.73
| 1170.7
| | [[160/81|160/81]], [[63/32]]
| [[49/25]], [[63/32]], [[96/49]], [[160/81]]
|
| dim 8ve
| style="text-align:center;" | v8
| v8
| style="text-align:center;" | vD
| vD
| | da
| comma-narrow 8ve, sub 8ve
| k8/s8
| kD, sD
| Do
| Da
|-
|-
| style="text-align:center;" | 41
| 41
| style="text-align:center;" | 1200
| 1200.0
| | 2/1
| [[2/1]]
|
| perfect 8ve
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | D
| D
| | do
| 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.


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
=== 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"
{| class="wikitable center-all"
|-
|-
! | quality
! Quality
! | [[Kite's color notation|color]]
! [[Color notation|Color]]
! | monzo format
! Monzo format
! | examples
! Examples
|-
|-
| style="text-align:center;" | downminor
| downminor
| style="text-align:center;" | zo
| zo
| style="text-align:center;" | (a, b, 0, 1)
| (a, b, 0, 1)
| style="text-align:center;" | 7/6, 7/4
| 7/6, 7/4
|-
|-
| style="text-align:center;" | minor
| minor
| style="text-align:center;" | fourthward wa
| fourthward wa
| style="text-align:center;" | (a, b) with b &lt; -1
| (a, b) with b < -1
| style="text-align:center;" | 32/27, 16/9
| 32/27, 16/9
|-
|-
| style="text-align:center;" | upminor
| upminor
| style="text-align:center;" | gu
| gu
| style="text-align:center;" | (a, b, -1)
| (a, b, -1)
| style="text-align:center;" | 6/5, 9/5
| 6/5, 9/5
|-
|-
| style="text-align:center;" | mid
| mid
| style="text-align:center;" | ilo
| ilo
| style="text-align:center;" | (a, b, 0, 0, 1)
| (a, b, 0, 0, 1)
| style="text-align:center;" | 11/9, 11/6
| 11/9, 11/6
|-
|-
| style="text-align:center;" | "
| "
| style="text-align:center;" | lu
| lu
| style="text-align:center;" | (a, b, 0, 0, -1)
| (a, b, 0, 0, -1)
| style="text-align:center;" | 12/11, 18/11
| 12/11, 18/11
|-
|-
| style="text-align:center;" | downmajor
| downmajor
| style="text-align:center;" | yo
| yo
| style="text-align:center;" | (a, b, 1)
| (a, b, 1)
| style="text-align:center;" | 5/4, 5/3
| 5/4, 5/3
|-
|-
| style="text-align:center;" | major
| major
| style="text-align:center;" | fifthward wa
| fifthward wa
| style="text-align:center;" | (a, b) with b &gt; 1
| (a, b) with b > 1
| style="text-align:center;" | 9/8, 27/16
| 9/8, 27/16
|-
|-
| style="text-align:center;" | upmajor
| upmajor
| style="text-align:center;" | ru
| ru
| style="text-align:center;" | (a, b, 0, -1)
| (a, b, 0, -1)
| style="text-align:center;" | 9/7, 12/7
| 9/7, 12/7
|}
|}


== Chord Names ==
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:
All 41edo chords can be named using ups and downs. An up, down or mid 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"
{| class="wikitable center-all"
|-
|-
! | [[Kite's color notation|color of the 3rd]]
! [[Color notation|Color of the 3rd]]
! | JI chord
! JI chord
! | notes as edosteps
! Notes as edosteps
! | notes of C chord
! Notes of C chord
! | written name
! Written name
! | spoken name
! Spoken name
|-
|-
| style="text-align:center;" | zo
| zo (7-over)
| style="text-align:center;" | 6:7:9
| 6:7:9
| style="text-align:center;" | 0-9-24
| 0-9-24
| style="text-align:center;" | C vEb G
| C vEb G
| style="text-align:center;" | Cvm
| Cvm
| style="text-align:center;" | C downminor
| C downminor
|-
|-
| style="text-align:center;" | gu
| gu (5-under)
| style="text-align:center;" | 10:12:15
| 10:12:15
| style="text-align:center;" | 0-11-24
| 0-11-24
| style="text-align:center;" | C ^Eb G
| C ^Eb G
| style="text-align:center;" | C^m
| C^m
| style="text-align:center;" | C upminor
| C upminor
|-
|-
| style="text-align:center;" | ilo
| ilo (11-over)
| style="text-align:center;" | 18:22:27
| 18:22:27
| style="text-align:center;" | 0-12-24
| 0-12-24
| style="text-align:center;" | C vvE G
| C vvE G
| style="text-align:center;" | C~
| C~
| style="text-align:center;" | C mid
| C mid
|-
|-
| style="text-align:center;" | yo
| yo (5-over)
| style="text-align:center;" | 4:5:6
| 4:5:6
| style="text-align:center;" | 0-13-24
| 0-13-24
| style="text-align:center;" | C vE G
| C vE G
| style="text-align:center;" | Cv
| Cv
| style="text-align:center;" | C downmajor or C down
| C downmajor or C down
|-
|-
| style="text-align:center;" | ru
| ru (7-under)
| style="text-align:center;" | 14:18:27
| 14:18:21
| style="text-align:center;" | 0-15-24
| 0-15-24
| style="text-align:center;" | C ^E G
| C ^E G
| style="text-align:center;" | C^
| C^
| style="text-align:center;" | C upmajor or C up
| C upmajor or C up
|}
|}
0-10-20 = D F Ab = Ddim = D dim


0-10-21 = D F ^Ab = Ddim(^5) = D dim up-five
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]].


0-10-22 = D F vvA = Dm(~5) = D minor mid-five
== Notations ==
=== Ups and downs notation ===
41edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}


0-10-23 = D F vA = Dm(v5) = D minor down-five
41edo can also be notated with quarter-tone accidentals and [[Alternative symbols for ups and downs notation#Sharp-3|ups and downs]]. This can be done by combining sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:


0-10-24 = D F A = Dm = D minor
{{Sharpness-sharp4}}


0-14-24 = D F# A = D = D or D major
The notes within an octave from A are thus:


0-14-25 = D F# ^A = D(^5) = D up-five
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


0-14-26 = D F# ^^A = D(^^5) = D double-up-five, or possibly Daug(vv5)
=== 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:


0-14-27 = D F# vA# = Daug(v5) = D aug down-five
{{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}}


0-14-28 = D F# A# = Daug = D aug
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.


For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
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.


==Red-Blue Notation==
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.


A red-note/blue-note system, similar to the one proposed for [[36edo|36edo]], is one option for notating 41edo. (This is separate from and not compatible with Kite's [[Kite's_color_notation|color notation]].) 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:
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[34edo #Sagittal notation|34edo]].


A, red A, blue Bb, Bb, A#, red A#, blue B, B, red B, blue C, C, red C, blue Db, Db, C#, red C#, blue D, D, red D, blue Eb, Eb, D#, red D#, blue E, E, red E, blue F, F, red F, blue Gb, Gb, F#, red F#, blue G, G, red G, blue Ab, Ab, G#, red G#, blue A, A.
==== 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>


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.
==== 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}}
 
== 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.


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.
[[File:41-edo spiral.png|579x579px]]


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.
The same spiral, but with notes not intervals:


=Selected just intervals by error=
[[File:41-edo spiral with notes.png|549x549px]]
The following table shows how [[Just-24|some prominent just intervals]] are represented in 41edo (ordered by absolute error).


{| class="wikitable"
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| | '''Interval, complement'''
! rowspan="2" | [[Subgroup]]
| | '''Error (abs., in [[cent|cents]])'''
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
! [[TE error|Absolute]] (¢)
| style="text-align:center;" | '''0.484'''
! [[TE simple badness|Relative]] (%)
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| 2.3
| style="text-align:center;" | 0.968
| {{monzo| 65 -41 }}
| {{mapping| 41 65 }}
| −0.153
| 0.15
| 0.52
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| 2.3.5
| style="text-align:center;" | 2.370
| 3125/3072, 20000/19683
| {{mapping| 41 65 95 }}
| +0.734
| 1.26
| 4.31
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| 2.3.5.7
| style="text-align:center;" | 2.854
| 225/224, 245/243, 1029/1024
| {{mapping| 41 65 95 115 }}
| +0.815
| 1.10
| 3.76
|-
|-
| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| 2.3.5.7.11
| style="text-align:center;" | '''2.972'''
| 100/99, 225/224, 243/242, 245/242
| {{mapping| 41 65 95 115 142 }}
| +0.375
| 1.32
| 4.51
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 2.3.5.7.11.13
| style="text-align:center;" | 3.456
| 100/99, 105/104, 144/143, 196/195, 243/242
| {{mapping| 41 65 95 115 142 152 }}
| −0.060
| 1.55
| 5.29
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| 2.3.5.7.11.13.19
| style="text-align:center;" | 3.473
| 100/99, 105/104, 133/132, 144/143, 171/169, 196/195
|-
| {{mapping| 41 65 95 115 142 152 174 }}
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| +0.111
| style="text-align:center;" | 3.812
| 1.49
|-
| 5.10
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
|}
| style="text-align:center;" | 3.940
* 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.  
|-
 
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
=== Commas ===
| style="text-align:center;" | 4.296
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 }}.
|-
 
| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
{| class="commatable wikitable center-1 center-2 right-3 center-6"
| style="text-align:center;" | '''4.780'''
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 5.342
|-
| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
| style="text-align:center;" | '''5.826'''
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 6.310
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 6.794
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 7.285
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 7.752
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 7.769
|-
| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| style="text-align:center;" | '''8.253'''
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
! [[Harmonic limit|Prime<br>limit]]
| style="text-align:center;" | 10.122
! [[Ratio]]<ref>Ratios with more than 8 digits are presented by placeholders with informative hints</ref>
! [[Cents]]
! [[Monzo]]
! colspan="2" | [[Color name]]
! Name(s)
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| 3
| style="text-align:center;" | 10.606
| <abbr title="36893488147419103232/36472996377170786403">(40 digits)</abbr>
| 19.84
| {{monzo| 65 -41 }}
| Wa-41
| 41-edo
| [[41-comma]]
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| 5
| style="text-align:center;" | 11.225
| <abbr title="1953125/1889568">(14 digits)</abbr>
| 57.27
| {{monzo| -5 -10 9 }}
| Tritriyo
| y<sup>9</sup>
| [[Shibboleth comma]]
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| 5
| style="text-align:center;" | 13.595
| [[34171875/33554432|(16 digits)]]
| 31.57
| {{monzo| -25 7 6 }}
| Lala-tribiyo
| LLy<sup>3</sup>
| [[Ampersand comma]]
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 5
| style="text-align:center;" | 14.079
| [[3125/3072]]
|}
| 29.61
 
| {{monzo| -10 -1 5 }}
=Relationship to 12-edo=
| Laquinyo
Whereas 12-edo has a circle of twelve 5ths, 41-edo has a spiral of twelve 5ths (since 24\41 is on the 7\12 kite in the scale tree). This spiral of 5th shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with microtonal music. 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.
| Ly<sup>5</sup>
 
| Magic comma
[[File:41-edo spiral.png|673x673px]]
 
The same spiral, but with notes not intervals:
 
[[File:41-edo spiral with notes.png|730x730px]]
 
=Commas=
41 EDO tempers out the following commas using its patent val, &lt; 41 65 95 115 142 152 168 174 185 199 203 |.
 
{| class="wikitable"
|-
|-
![[Prime limit|Prime]]
| 5
[[Prime limit|Limit]]
| [[20000/19683|(10 digits)]]
! | [[Ratio]]
| 27.66
! | [[Cents]]
| {{monzo| 5 -9 4 }}
! | [[Monzo]]
| Saquadyo
! colspan="2" |[[Color notation/Temperament Names|color name]]
| sy<sup>4</sup>
! | Name
| [[Tetracot comma]]
|-
|-
|3
| 5
| style="text-align:center;" |  
| <abbr title="131072000/129140163">(18 digits)</abbr>
| | 19.84
| 25.71
| |<nowiki> | 65 -41 </nowiki>&gt;
| {{monzo| 20 -17 3 }}
|Wa-41
| Sasa-triyo
|41-edo
| ssy<sup>3</sup>
| | '41-tone' comma
| [[Roda]]
|-
|-
|5
| 5
| style="text-align:center;" |  
| [[32805/32768|(10 digits)]]
| | 57.27
| 1.95
| |<nowiki> | -5 -10 9 </nowiki>&gt;
| {{monzo| -15 8 1 }}
|Tritriyo
| Layo
|y<sup>9</sup>
| Ly
| | shibboleth
| [[Schisma]]
|-
|-
|"
| 7
|
| [[15625/15309|(10 digits)]]
|31.57
| 35.37
|<nowiki> | -25 7 6 </nowiki>&gt;
| {{monzo| 0 -7 6 -1 }}
|Lala-tribiyo
| Rutribiyo
|LLy<sup>3</sup>
| ry<sup>6</sup>
|Ampersand's comma
| Arcturus comma, great BP diesis
|-
|-
|"
| 7
| style="text-align:center;" | 3125/3072
| <abbr title="854296875/843308032">(18 digits)</abbr>
| | 29.61
| 22.41
| |<nowiki> | -10 -1 5 </nowiki>&gt;
| {{monzo| -10 7 8 -7 }}
|Laquinyo
| Lasepru-aquadbiyo
|Ly<sup>5</sup>
| Lr<sup>7</sup>y<sup>8</sup>
| | small diesis, magic comma
| [[Blackjackisma]]
|-
|-
|"
| 7
| style="text-align:center;" | 20000/19683
| [[875/864]]
| | 27.66
| 21.90
| |<nowiki> | 5 -9 4 </nowiki>&gt;
| {{monzo| -5 -3 3 1 }}
|Saquadyo
| Zotriyo
|sy<sup>4</sup>
| zy<sup>3</sup>
| | minimal diesis, tetracot comma
| Keema
|-
|-
|"
| 7
| style="text-align:center;" |
| [[3125/3087]]
| | 25.71
| 21.18
| |<nowiki> | 20 -17 3 </nowiki>&gt;
| {{monzo| 0 -2 5 -3 }}
|Sasa-triyo
| Triru-aquinyo
|ssy<sup>3</sup>
| r<sup>3</sup>y<sup>5</sup>
| | roda
| Gariboh comma
|-
|-
|"
| 7
|32805/32768
| <abbr title="179200/177147">(12 digits)</abbr>
|1.95
| 19.95
|<nowiki> | -15 8 1 </nowiki>&gt;
| {{monzo| 10 -11 2 1 }}
|Layo
| Sazoyoyo
|Ly
| szyy
|schisma
| [[Tolerma]]
|-
|-
|7
| 7
| style="text-align:center;" | 15625/15309
| [[33075/32768|(10 digits)]]
| | 35.37
| 16.14
| |<nowiki> | 0 -7 6 -1 </nowiki>&gt;
| {{monzo| -15 3 2 2 }}
|Rutribiyo
| Labizoyo
|ry<sup>6</sup>
| Lzzyy
| | great BP diesis
| [[Mirwomo comma]]
|-
|-
|"
| 7
|
| [[245/243]]
|22.41
| 14.19
|<nowiki> | -10 7 8 -7 </nowiki>&gt;
| {{monzo| 0 -5 1 2 }}
|Lasepru-aquadbiyo
| Zozoyo
|Lr<sup>7</sup>y<sup>8</sup>
| zzy
|blackjackisma
| Sensamagic comma
|-
|-
|"
| 7
| style="text-align:center;" | 875/864
| [[4000/3969]]
| | 21.90
| 13.47
| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| {{monzo| 5 -4 3 -2 }}
|Zotriyo
| Rurutriyo
|zy<sup>3</sup>
| rry<sup>3</sup>
| | keema
| Octagar comma
|-
|-
|"
| 7
|3125/3087
| <abbr title="823543/819200">(12 digits)</abbr>
|21.18
| 9.15
|<nowiki> | 0 -2 5 -3 </nowiki>&gt;
| {{monzo| -15 0 -2 7 }}
|Triru-aquinyo
| Lasepzo-agugu
|r<sup>3</sup>y<sup>5</sup>
| Lz<sup>7</sup>gg
|major BP diesis, gariboh
| [[Quince comma]]
|-
|-
|"
| 7
|
| [[1029/1024]]
|19.95
| 8.43
|<nowiki> | 10 -11 2 1 </nowiki>&gt;
| {{monzo| -10 1 0 3 }}
|Sazoyoyo
| Latrizo
|szyy
| Lz<sup>3</sup>
|tolerma
| Gamelisma
|-
|-
|"
| 7
|33075/32768
| [[225/224]]
|16.14
| 7.71
|<nowiki> | -15 3 2 2 </nowiki>&gt;
| {{monzo| -5 2 2 -1 }}
|Labizoyo
| Ruyoyo
|Lzzyy
| ryy
|mirwomo comma
| Marvel comma
|-
|-
|"
| 7
|245/243
| [[16875/16807|(10 digits)]]
|14.19
| 6.99
|<nowiki> | 0 -5 1 2 </nowiki>&gt;
| {{monzo| 0 3 4 -5 }}
|Zozoyo
| Quinru-aquadyo
|zzy
| r<sup>5</sup>y<sup>4</sup>
|minor BP diesis, sensamagic
| [[Mirkwai comma]]
|-
|-
|"
| 7
|4000/3969
| [[10976/10935|(10 digits)]]
|13.47
| 6.48
|<nowiki> | 5 -4 3 -2 </nowiki>&gt;
| {{monzo| 5 -7 -1 3 }}
|Rurutriyo
| Satrizo-agu
|rry<sup>3</sup>
| sz<sup>3</sup>g
|septimal semicomma, octagar
| [[Hemimage comma]]
|-
|-
|"
| 7
|
| [[5120/5103]]
|9.15
| 5.76
|<nowiki> | -15 0 -2 7 </nowiki>&gt;
| {{monzo| 10 -6 1 -1 }}
|Lasepzo-agugu
| Saruyo
|Lz<sup>7</sup>gg
| sry
|quince
| Hemifamity comma
|-
|-
|"
| 7
|1029/1024
| [[33554432/33480783|(16 digits)]]
|8.43
| 3.80
|<nowiki> | -10 1 0 3 </nowiki>&gt;
| {{monzo| 25 -14 0 -1 }}
|Latrizo
| Sasaru
|Lz<sup>3</sup>
| ssr
|gamelan residue, gamelisma
| [[Garischisma]]
|-
|-
|"
| 7
|225/224
| [[2401/2400]]
|7.71
| 0.72
|<nowiki> | -5 2 2 -1 </nowiki>&gt;
| {{monzo| -5 -1 -2 4 }}
|Ruyoyo
| Bizozogu
|ryy
| z<sup>4</sup>gg
|septimal kleisma, marvel comma
| Breedsma
|-
|-
|"
| 11
|16875/16807
| <abbr title="163840/161051">(12 digits)</abbr>
|6.99
| 29.72
|<nowiki> | 0 3 4 -5 </nowiki>&gt;
| {{monzo| 15 0 1 0 -5 }}
|Quinru-aquadyo
| Saquinlu-ayo
|r<sup>5</sup>y<sup>4</sup>
| s1u<sup>5</sup>y
|small BP diesis, mirkwai
| [[Thuja comma]]
|-
|-
|"
| 11
|10976/10935
| [[245/242]]
|6.48
| 21.33
|<nowiki> | 5 -7 -1 3 </nowiki>&gt;
| {{monzo| -1 0 1 2 -2 }}
|Satrizo-agu
| Luluzozoyo
|sz<sup>3</sup>g
| 1uuzzy
|hemimage
| Frostma
|-
|-
|"
| 11
|5120/5103
| [[100/99]]
|5.76
| 17.40
|<nowiki> | 10 -6 1 -1 </nowiki>&gt;
| {{monzo| 2 -2 2 0 -1 }}
|Saruyo
| Luyoyo
|sry
| 1uyy
|Beta 5, Garibaldi comma, hemifamity
| Ptolemisma
|-
|-
|"
| 11
|
| [[1344/1331]]
|3.80
| 16.83
|<nowiki> | 25 -14 0 -1 </nowiki>&gt;
| {{monzo| 6 1 0 1 -3 }}
|Sasaru
| Trilu-azo
|ssr
| 1u<sup>3</sup>z
|Beta 2, septimal schisma, garischisma
| Hemimin comma
|-
|-
|"
| 11
|2401/2400
| [[896/891]]
|0.72
| 9.69
|<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| {{monzo| 7 -4 0 1 -1 }}
|Bizozogu
| Saluzo
|z<sup>4</sup>gg
| s1uz
|Breedsma
| [[Pentacircle comma]]
|-
|-
|11
| 11
|
| [[65536/65219|(10 digits)]]
|29.72
| 8.39
|<nowiki> | 15 0 1 0 -5 </nowiki>&gt;
| {{monzo| 16 0 0 -2 -3 }}
|Saquinlu-ayo
| Satrilu-aruru
|s1u<sup>5</sup>y
| s1u<sup>3</sup>rr
|thuja comma
| [[Orgonisma]]
|-
|-
|"
| 11
|245/242
| [[243/242]]
|21.33
| 7.14
|<nowiki> | -1 0 1 2 -2 </nowiki>&gt;
| {{monzo| -1 5 0 0 -2 }}
|Luluzozoyo
| Lulu
|1uuzzy
| 1uu
|cassacot
| Rastma
|-
|-
|"
| 11
|100/99
| [[385/384]]
|17.40
| 4.50
|<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
| {{monzo| -7 -1 1 1 1 }}
|Luyoyo
| Lozoyo
|1uyy
| 1ozg
|Ptolemy's comma, ptolemisma
| Keenanisma
|-
|-
|"
| 11
|1344/1331
| [[441/440]]
|16.83
| 3.93
|<nowiki> | 6 1 0 1 -3 </nowiki>&gt;
| {{monzo| -3 2 -1 2 -1 }}
|Trilu-azo
| Luzozogu
|1u<sup>3</sup>z
| 1uzzg
|hemimin
| Werckisma
|-
|-
|"
| 11
|896/891
| [[1375/1372]]
|9.69
| 3.78
|<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
| {{monzo| -2 0 3 -3 1 }}
|Saluzo
| Lotriruyo
|s1uz
| 1or<sup>3</sup>y
|undecimal semicomma, pentacircle (minthma * gentle)
| Moctdel comma
|-
|-
|"
| 11
|65536/65219
| [[540/539]]
|8.39
| 3.21
|<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
| {{monzo| 2 3 1 -2 -1 }}
|Satrilu-aruru
| Lururuyo
|s1u<sup>3</sup>rr
| 1urry
|orgonisma
| Swetisma
|-
|-
|"
| 11
|243/242
| [[3025/3024]]
|7.14
| 0.57
|<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| {{monzo| -4 -3 2 -1 2 }}
|Lulu
| Loloruyoyo
|1uu
| 1ooryy
|neutral third comma, rastma
| Lehmerisma
|-
|-
|"
| 11
|385/384
| [[151263/151250|<abbr title="151263/151250">(12 digits)</abbr>]]
|4.50
| 0.15
|<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| {{monzo| -1 2 -4 5 -2 }}
|Lozoyo
| Luluquinzo-aquadgu
|1ozg
| 1uuz<sup>5</sup>g<sup>4</sup>
|undecimal kleisma, keenanisma
| [[Odiheim comma]]
|-
|-
|"
| 13
| style="text-align:center;" | 441/440
| [[343/338]]
| | 3.93
| 25.42
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
| {{monzo| -1 0 0 3 0 -2 }}
|Luzozogu
| Thuthutrizo
|1uzzg
| 3uuz<sup>3</sup>
| | Werckmeister's undecimal septenarian schisma, werckisma
|  
|-
|-
|"
| 13
| style="text-align:center;" | 1375/1372
| [[105/104]]
| | 3.78
| 16.57
| |<nowiki> | -2 0 3 -3 1 </nowiki>&gt;
| {{monzo| -3 1 1 1 0 -1 }}
|Lotriruyo
| Thuzoyo
|1or<sup>3</sup>y
| 3uzy
| | moctdel
| Animist comma
|-
|-
|"
| 13
| style="text-align:center;" | 540/539
| [[28672/28431|(10 digits)]]
| | 3.21
| 14.61
| |<nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
| {{monzo| 12 -7 0 1 0 -1 }}
|Lururuyo
| Sathuzo
|1urry
| s3uz
| | Swets' comma, swetisma
| [[Secorian comma]]
|-
|-
|"
| 13
| style="text-align:center;" | 3025/3024
| [[275/273]]
| | 0.57
| 12.64
| |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
| {{monzo| 0 -1 2 -1 1 -1 }}
|Loloruyoyo
| Thuloruyoyo
|1ooryy
| 3u1oryy
| | Lehmerisma
| Gassorma
|-
|-
|"
| 13
| style="text-align:center;" |
| [[144/143]]
| | 0.15
| 12.06
| |<nowiki> | -1 2 -4 5 -2 </nowiki>&gt;
| {{monzo| 4 2 0 0 -1 -1 }}
|Luluquinzo-aquadgu
| Thulu
|1uuz<sup>5</sup>g<sup>4</sup>
| 3u1u
| | odiheim
| Grossma
|-
|-
|13
| 13
| style="text-align:center;" | 105/104
| [[196/195]]
| | 16.57
| 8.86
| |<nowiki> | -3 1 1 1 0 -1 </nowiki>&gt;
| {{monzo| 2 -1 -1 2 0 -1 }}
|Thuzoyo
| Thuzozogu
|3uzy
| 3uzzg
| | small tridecimal comma, animist
| Mynucuma
|-
|-
|"
| 13
| style="text-align:center;" | 28672/28431
| [[640/637]]
| | 14.61
| 8.13
| |<nowiki> | 12 -7 0 1 0 -1 </nowiki>&gt;
| {{monzo| 7 0 1 -2 0 -1 }}
|Sathuzo
| Thururuyo
|s3uz
| 3urry
| | secorian
| Huntma
|-
|-
|"
| 13
| style="text-align:center;" | 275/273
| [[1188/1183]]
| | 12.64
| 7.30
| |<nowiki> | 0 -1 2 -1 1 -1 </nowiki>&gt;
| {{monzo| 2 3 0 -1 1 -2 }}
|Thuloruyoyo
| Thuthuloru
|3u1oryy
| 3uu1or
| | gassorma
| Kestrel comma
|-
|-
|"
| 13
| style="text-align:center;" | 144/143
| [[31213/31104]]
| | 12.06
| 6.06
| |<nowiki> | 4 2 0 0 -1 -1 </nowiki>&gt;
| {{monzo| -7 -5 0 4 0 1 }}
|Thulu
| Thoquadzo
|3u1u
| 3oz<sup>4</sup>3
| | grossma
| Praveensma
|-
|-
|"
| 13
| style="text-align:center;" | 196/195
| [[325/324]]
| | 8.86
| 5.34
| |<nowiki> | 2 -1 -1 2 0 -1 </nowiki>&gt;
| {{monzo| -2 -4 2 0 0 1 }}
|Thuzozogu
| Thoyoyo
|3uzzg
| 3oyy
| | mynucuma
| Marveltwin comma
|-
|-
|"
| 13
| style="text-align:center;" | 640/637
| [[352/351]]
| | 8.13
| 4.93
| |<nowiki> | 7 0 1 -2 0 -1 </nowiki>&gt;
| {{monzo| 5 -3 0 0 1 -1 }}
|Thururuyo
| Thulo
|3urry
| 3u1o
| | huntma
| Major minthma
|-
|-
|"
| 13
| style="text-align:center;" | 1188/1183
| [[364/363]]
| | 7.30
| 4.76
| |<nowiki> | 2 3 0 -1 1 -2 </nowiki>&gt;
| {{monzo| 2 -1 0 1 -2 1 }}
|Thuthuloru
| Tholuluzo
|3uu1or
| 3o1uuz
| | kestrel comma
| Minor minthma
|-
|-
|"
| 13
| style="text-align:center;" | 325/324
| [[847/845]]
| | 5.34
| 4.09
| |<nowiki> | -2 -4 2 0 0 1 </nowiki>&gt;
| {{monzo| 0 0 -1 1 2 -2 }}
|Thoyoyo
| Thuthulolozogu
|3oyy
| 3uu1oozg
| | marveltwin
| Cuthbert comma
|-
|-
|"
| 13
| style="text-align:center;" | 352/351
| [[729/728]]
| | 4.93
| 2.38
| |<nowiki> | 5 -3 0 0 1 -1 </nowiki>&gt;
| {{monzo| -3 6 0 -1 0 -1 }}
|Thulo
| Lathuru
|3u1o
| L3ur
| | minthma
| Squbema
|-
|-
|"
| 13
| style="text-align:center;" | 364/363
| [[2080/2079]]
| | 4.76
| 0.83
| |<nowiki> | 2 -1 0 1 -2 1 </nowiki>&gt;
| {{monzo| 5 -3 1 -1 -1 1 }}
|Tholuluzo
| Tholuruyo
|3o1uuz
| 3o1ury
| | gentle comma
| Ibnsinma
|-
|-
|"
| 13
| style="text-align:center;" | 847/845
| [[4096/4095]]
| | 4.09
| 0.42
| |<nowiki> | 0 0 -1 1 2 -2 </nowiki>&gt;
| {{monzo| 12 -2 -1 -1 0 -1 }}
|Thuthulolozogu
| Sathurugu
|3uu1oozg
| s3urg
| | cuthbert
| Schismina
|-
|-
|"
| 13
| style="text-align:center;" | 729/728
| [[6656/6655]]
| | 2.38
| 0.26
| |<nowiki> | -3 6 0 -1 0 -1 </nowiki>&gt;
| {{monzo| 9 0 -1 0 -3 1 }}
|Lathuru
| Thotrilo-agu
|L3ur
| 3u1o<sup>3</sup>g2
| | squbema
| Jacobin comma
|-
|-
|"
| 13
| style="text-align:center;" | 4096/4095
| [[10648/10647|(10 digits)]]
| | 0.42
| 0.16
| |<nowiki> | 12 -2 -1 -1 0 -1 </nowiki>&gt;
| {{monzo| 3 -2 0 -1 3 -2 }}
|Sathurugu
| Thuthutrilo-aru
|s3urg
| 3uu1o<sup>3</sup>r
| | tridecimal schisma, Sagittal schismina
| [[Harmonisma]]
|-
|-
|"
| 17
| style="text-align:center;" | 10648/10647
| [[2187/2176]]
| | 0.16
| 8.73
| |<nowiki> | 3 -2 0 -1 3 -2 </nowiki>&gt;
| {{monzo| -7 7 0 0 0 0 -1 }}
|Thuthutrilo-aru
| Lasu
|3uu1o<sup>3</sup>r
| L17u
| | harmonisma
| Septendecimal schisma
|-
|-
|17
| 17
| style="text-align:center;" | 2187/2176
| [[256/255]]
| | 8.73
| 6.78
| |<nowiki> | -7 7 0 0 0 0 -1 </nowiki>&gt;
| {{monzo| 8 -1 -1 0 0 0 -1 }}
|Lasu
| Sugu
|L17u
| 17ug
| | septendecimal comma
| Charisma
|-
|-
|"
| 17
|256/255
| [[715/714]]
|6.78
| 2.42
|<nowiki> | 8 -1 -1 0 0 0 -1 </nowiki>&gt;
| {{monzo| -1 -1 1 -1 1 1 -1 }}
|Sugu
| Sutholoruyo
|17ug
| 17u3o1ory
|septendecimal kleisma
| Septendecimal bridge comma
|-
|-
|"
| 19
|715/714
| [[210/209]]
|2.42
| 8.26
|<nowiki> | -1 -1 1 -1 1 1 -1 </nowiki>&gt;
| {{monzo| 1 1 1 1 -1 0 0 -1 }}
|Sutholoruyo
| Nuluzoyo
|17u3o1ory
| 19u1uzy
|septendecimal bridge comma
| Spleen comma
|-
|-
|19
| 19
| style="text-align:center;" | 210/209
| [[361/360]]
| | 8.26
| 4.80
| |<nowiki> | 1 1 1 1 -1 0 0 -1 </nowiki>&gt;
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
|Nuluzoyo
| Nonogu
|19u1uzy
| 19oog2
| | spleen comma
| Go comma
|-
|-
|"
| 19
|1216/1215
| [[513/512]]
|1.42
| 3.38
|<nowiki> | 6 -5 -1 0 0 0 0 1 </nowiki>&gt;
| {{monzo| -9 3 0 0 0 0 0 1 }}
|Sanogu
| Lano
|s19og
| L19o
|Eratosthenes' comma
| Boethius' comma
|-
|-
|"
| 19
| style="text-align:center;" | 513/512
| [[1216/1215]]
| | 3.38
| 1.42
| |<nowiki> | -9 3 0 0 0 0 0 1 </nowiki>&gt;
| {{monzo| 6 -5 -1 0 0 0 0 1 }}
|Lano
| Sanogu
|L19o
| s19og
| | undevicesimal comma, Boethius' comma
| Eratosthenes' comma
|-
|-
|23
| 23
|736/729
| [[736/729]]
|16.54
| 16.54
|<nowiki> | 5 -6 0 0 0 0 0 0 1 </nowiki>&gt;
| {{monzo| 5 -6 0 0 0 0 0 0 1 }}
|Sa-twenty-tho
| Satwetho
|s23o
| s23o
|vicesimotertial comma
| Vicesimotertial comma
|-
|-
|29
| 29
|145/144
| [[145/144]]
|11.98
| 11.98
|<nowiki> | -4 -2 1 0 0 0 0 0 0 1 </nowiki>&gt;
| {{monzo| -4 -2 1 0 0 0 0 0 0 1 }}
|Twenty-noyo
| Twenoyo
|29oy
| 29oy
|29th-partial chroma
| 29th-partial chroma
|}
|}


=Temperaments=
=== Rank-2 temperaments ===
[[List_of_edo-distinct_41et_rank_two_temperaments|List of edo-distinct 41et rank two temperaments]]
* [[List of edo-distinct 41et rank two temperaments]]
 
* [[Schismic–countercommatic equivalence continuum]]
Table of Temperaments by generator


{| class="wikitable"
{| class="wikitable right-1 right-2"
|+ Table of temperaments by generator
|-
|-
! |Degree
! Degree
! | Cents
! Cents
! | Generator
! Temperament(s)
! | Some MOS and MODMOS Scales Available
! [[Pergen]]
! Mos scales
|-
|-
| style="text-align:center;" | 0
| 1
| style="text-align:center;" | 0.00
| 29.27
| |  
| [[Slendi]]
| |  
| (P8, P4/17)
|  
|-
|-
| style="text-align:center;" | 1
| 2
| style="text-align:center;" | 29.27
| 58.54
| |  
| [[Hemimiracle]]<br>[[Dodecacot]]
| |
| (P8, P5/12)
| 21-tone mos
|-
|-
| style="text-align:center;" | 2
| 3
| style="text-align:center;" | 58.54
| 87.80
| | [[Hemimiracle|Hemimiracle]]
| [[Octacot]]
| |  
| (P8, P5/8)
| 14-tone mos: 3 3 3 3 3 3 3 3 3 3 3 3 3 2
|-
|-
| style="text-align:center;" | 3
| 4
| style="text-align:center;" | 87.805
| 117.07
| | 88cET (approx),
| [[Miracle]]
 
| (P8, P5/6)
[[Octacot|octacot]]
| 11-tone mos: 4 4 4 4 4 4 4 4 4 4 1
| |  
|-
|-
| style="text-align:center;" | 4
| 5
| style="text-align:center;" | 117.07
| 146.34
| | [[Miracle|Miracle]]
| [[BPS]] / [[bohpier]]
| |  
| (P8, P12/13)
| 20-tone mos
|-
|-
| style="text-align:center;" | 5
| 6
| style="text-align:center;" | 146.34
| 175.61
| | [[Bohlen-Pierce|Bohlen-Pierce]]/[[bohpier|bohpier]]
| [[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
|-
|-
| style="text-align:center;" | 6
| 7
| style="text-align:center;" | 175.61
| 204.88
| | [[Tetracot|Tetracot]]/[[bunya|bunya]]/[[Monkey|monkey]]
| [[Baldy]]<br>[[Quadrimage]]
| | 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
| (P8, c<sup>3</sup>P4/20)
| 11-tone mos: 6 1 6 6 1 6 1 6 1 6 1
|-
|-
| style="text-align:center;" | 7
| 8
| style="text-align:center;" | 204.88
| 234.15
| | [[Baldy|Baldy]]
| [[Slendric]] / [[rodan]] / [[guiron]]
| | 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
| (P8, P5/3)
| 11-tone mos: 7 1 7 1 7 1 7 1 1 7 1
|-
|-
| style="text-align:center;" | 8
| 9
| style="text-align:center;" | 234.15
| 263.41
| | [[Rodan|Rodan]]/[[guiron|guiron]]
| [[Septimin]]
| | 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
| (P8, ccP4/11)
| 9-tone mos: 5 4 5 5 4 5 4 5 4
|-
|-
| style="text-align:center;" | 9
| 10
| style="text-align:center;" | 263.415
| 292.68
| | [[Septimin|Septimin]]
| [[Quasitemp]]
| | 9-tone MOS: 5 4 5 5 4 5 4 5 4
| (P8, c<sup>3</sup>P4/14)
| 29-tone mos
|-
|-
| style="text-align:center;" | 10
| 11
| style="text-align:center;" | 292.68
| 321.95
| | [[Quasitemp|Quasitemp]]
| [[Superkleismic]]
| |  
| (P8, ccP4/9)
| 11-tone mos: 5 3 5 3 3 5 3 3 5 3 3
|-
|-
| style="text-align:center;" | 11
| 12
| style="text-align:center;" | 321.95
| 351.22
| | [[Superkleismic|Superkleismic]]
| [[Hemif]] / [[hemififths]] / [[salsa]]<br>[[Karadeniz]]
| | 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
| (P8, P5/2)
| 10-tone mos: 5 2 5 5 2 5 5 5 2 5
|-
|-
| style="text-align:center;" | 12
| 13
| style="text-align:center;" | 351.22
| 380.49
| | [[Hemififths|Hemififths]]/[[karadeniz|karadeniz]]
| [[Magic]] / [[witchcraft]]<br>[[Quanharuk]]
| | 10-tone MOS: 5 2 5 5 2 5 5 5 2 5
| (P8, P12/5)
| 10-tone mos: 2 9 2 2 9 2 2 9 2 2
|-
|-
| style="text-align:center;" | 13
| 14
| style="text-align:center;" | 380.49
| 409.76
| | [[Magic|Magic]]/[[witchcraft|witchcraft]]
| [[Hocum]]<br>[[Hocus]]
| | 10-tone MOS: 2 9 2 2 9 2 2 9 2 2
| (P8, c<sup>3</sup>P4/10)
| 32-tone mos
|-
|-
| style="text-align:center;" | 14
| 15
| style="text-align:center;" | 409.76
| 439.02
| | [[Hocus|Hocus]]
| [[Superthird]]
| |  
| (P8, c<sup>6</sup>P5/18)
| 11-tone mos: 4 3 4 4 4 3 4 4 3 4 4
|-
|-
| style="text-align:center;" | 15
| 16
| style="text-align:center;" | 439.02
| 468.29
| |  
| [[Barbad]]
| | 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
| (P8, c<sup>7</sup>P4/19)
| 8-tone mos: 7 2 7 7 2 7 7 2
|-
|-
| style="text-align:center;" | 16
| 17
| style="text-align:center;" | 468.29
| 497.56
| | [[Barbad|Barbad]]
| [[Helmholtz (temperament)|Helmholtz]] / [[garibaldi]] / [[cassandra]] / [[andromeda]]<br>[[Kwai]]
| |  
| (P8, P5)
| 12-tone mos: 4 3 4 3 3 4 3 4 3 4 3 4 3 3
|-
|-
| style="text-align:center;" | 17
| 18
| style="text-align:center;" | 497.56
| 526.83
| | [[Schismatic|Schismatic]] ([[Helmholtz|helmholtz]], [[Garibaldi_temperament|garibaldi]], [[cassandra|cassandra]])
| [[Trismegistus]]
| |
| (P8, c<sup>6</sup>P5/15)
| 9-tone mos: 5 5 3 5 5 5 5 3 5
|-
|-
| style="text-align:center;" | 18
| 19
| style="text-align:center;" | 526.83
| 556.10
| | [[Trismegistus|Trismegistus]]
| [[Alphorn]]
| | 9-tone MOS: 5 5 3 5 5 5 5 3 5
| (P8, c<sup>7</sup>P4/16)
| 9-tone mos: 3 3 3 10 3 3 3 3 10
|-
|-
| style="text-align:center;" | 19
| 20
| style="text-align:center;" | 556.1
| 585.37
| |
| [[Pluto]]<br>[[Merman]]
| |
| (P8, c<sup>3</sup>P4/7)
|-
|  
| style="text-align:center;" | 20
| style="text-align:center;" | 585.37
| | [[Pluto|Pluto]]
| |  
|}
|}


=Scales and modes=
== Scales and modes ==
A list of [[41edo_modes|41edo modes]] (MOS and others).
=== 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===
=== Harmonic scale ===
41edo is the first edo to do some justice to Mode 8 of the [[OverToneSeries|harmonic series]], which Dante Rosati calls the "[[overtone_scales|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).
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"
{| class="wikitable" style="text-align: center;"
|-
|-
| | Overtones in "Mode 8":
! Overtones in "Mode 8":
| | 8
| 8
| | 9
| 9
| | 10
| 10
| | 11
| 11
| | 12
| 12
| | 13
| 13
| | 14
| 14
| | 15
| 15
| | 16
| 16
|-
|-
| | ...as JI Ratio from 1/1:
! … as JI Ratio from 1/1:
| | 1/1
| 1/1
| | 9/8
| 9/8
| | 5/4
| 5/4
| | 11/8
| 11/8
| | 3/2
| 3/2
| | 13/8
| 13/8
| | 7/4
| 7/4
| | 15/8
| 15/8
| | 2/1
| 2/1
|-
|-
| | ...in cents:
! … in cents:
| | 0
| 0
| | 203.9
| 203.9
| | 386.3
| 386.3
| | 551.3
| 551.3
| | 702.0
| 702.0
| | 840.5
| 840.5
| | 968.8
| 968.8
| | 1088.3
| 1088.3
| | 1200.0
| 1200.0
|-
|-
| | Nearest degree of 41edo:
! Nearest degree of 41edo:
| | 0
| 0
| | 7
| 7
| | 13
| 13
| | 19
| 19
| | 24
| 24
| | 29
| 29
| | 33
| 33
| | 37
| 37
| | 41
| 41
|-
|-
| | ...in cents:
! … in cents:
| | 0
| 0
| | 204.9
| 204.9
| | 380.5
| 380.5
| | 556.1
| 556.1
| | 702.4
| 702.4
| | 848.8
| 848.8
| | 965.9
| 965.9
| | 1082.9
| 1082.9
| | 1200.0
| 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.)
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.
* 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).
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).
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.
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.


=Nonoctave Temperaments=
=== Nonoctave temperaments ===
Taking every third degree of 41edo produces a scale extremely close to [[88cET|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 chart:
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"
{| class="wikitable center-all right-3 right-4 right-5 mw-collapsible mw-collapsed"
|-
|-
| colspan="3" style="text-align:center;" | 3 degrees of 41edo (near 88cET)
! colspan="3" | 3 degrees of 41edo near 88cET
| style="text-align:center;" | overlap
! overlap
| colspan="3" style="text-align:center;" | 5 degrees of 41edo (near BP)
! colspan="3" | 5 degrees of 41edo near BP
|-
|-
! | deg of 41edo
! 41edo
! | deg of 88cET
! 88cET
! | cents
! cents
! | cents
! cents
! | cents
! cents
! | deg of BP
! BP
! | deg of 41edo
! 41edo
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0
| 0
| style="text-align:center;" |  
|  
| style="text-align:center;" | 0
| 0
| style="text-align:center;" |  
|  
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | 0
| 0
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 87.8
| 87.8
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" | 146.3
| 146.3
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 5
| 5
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 175.6
| 175.6
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 263.4
| 263.4
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" | 292.7
| 292.7
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | 10
| 10
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 351.2
| 351.2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | 5
| 5
| style="text-align:center;" |  
|  
| style="text-align:center;" | 439.0
| 439.0
| style="text-align:center;" |  
|  
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | 15
| 15
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 526.8
| 526.8
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" | 585.4
| 585.4
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 20
| 20
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 614.6
| 614.6
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 702.4
| 702.4
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" | 731.7
| 731.7
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | 25
| 25
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 790.2
| 790.2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:center;" | 10
| 10
| style="text-align:center;" |  
|  
| style="text-align:center;" | 878.0
| 878.0
| style="text-align:center;" |  
|  
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | 30
| 30
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 965.9
| 965.9
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" | 1024.4
| 1024.4
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 35
| 35
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | 1053.7
| 1053.7
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 39
| 39
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 1141.5
| 1141.5
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" | 1170.7
| 1170.7
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | 40
| 40
|-
|-
! colspan="7" | [ second octave ]
! colspan="7" | [ second octave ]
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | 29.2
| 29.2
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | 15
| 15
| style="text-align:center;" |  
|  
| style="text-align:center;" | 117.1
| 117.1
| style="text-align:center;" |  
|  
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | 4
| 4
|-
|-
| style="text-align:center;" | 7
| 7
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==Instruments==
=== More scales ===
[[File:41-EDD_elektrische_gitaar.jpg|alt=41-EDD elektrische gitaar.jpg|560x745px|41-EDD elektrische gitaar.jpg]]
* [[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 [[User:Richie|Richie]]]]
{{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 [[User:Richie|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 ===
Gamelan music is mainly based on two scales, the older [[Slendro]] and newer [[Pelog]], though these scales are expanded on extensively through [[octave stretching]], extensions and combination of the scales, and more. Slendro is excellently [https://gamelan-slendro-just-tuning.tiiny.site approximated] by the 8\41 generator. Pelog is approximated quite well also, this time by [[mavila]] temperament, using the "grave" fifth of 41 as the generator (23\41).
 
=== Indian ===
{{See also| Magic22 as srutis }}
Carnatic music, which is normally based on a 22-note unequal scale, has found some use from [[22edo]] as a good approximation, but 41 offers another option with [[Magic|Magic[22]]], which not only represents 22edo closely, but preserves accurate perfect fifths and the unequal quality of a more typical carnatic scale. Like any EDO system with an accurate 5-limit and essentially pure fifth, 41 can also approximate a system of [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104546.html#104549 Just Shrutis].
 
=== Japanese ===
Japanese classical music known as Gagaku is largely built around winds, strings, and percussion, and the melodies, like many Asian cultures, are built around Pythagorean pentatonic scales, alongside chromaticism with narrow semitones, which are well represented by Pythagorean limmas.
 
=== Blues ===
Due to its pure sounding major thirds and approximations of standard western harmony, 41 naturally is good for jazz and blues music, though a great strength of this system as opposed to many others is its excellent harmonic seventh, alongside MOS scales that supply them, [[Magic]] and [[Miracle]] in particular.
 
Coltrane changes can be represented with two Pythagorean major thirds and a pental one, or a temperament like Magic, whose MOSes are characterized by circles of major thirds, giving options for rotating major and minor triads within one scale. Similarly, the Whole Tone scale is represented by [[Baldy6|Baldy[6]]], with two pental major thirds and four Pythagorean. This scale can be extended to an 11-note MOS, including a single 4:5:7:9:11 chord and numerous subsets.
 
[[Superkleismic]] presents another option for this purpose, featuring circles of minor thirds, and generating harmonic sevenths with very low complexity.
 
Blue notes, rather than being considered inflections, can be notated as accidentals instead, such as the "blue third" which is represented by a neutral third, or any number of septimal intervals that are useful in a blues context.
 
=== Other ===
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 ==
=== Modern renderings ===
; {{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)
* [https://www.youtube.com/watch?v=LWd3ZOaAZlY "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
 
; {{W|Nicolaus Bruhns}}
* [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"
 
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/kLMuRP82bZw ''microtonal dance in 41edo ''] (2023)
* [https://www.youtube.com/shorts/m8X-IqH8tok ''Waltz in 41edo''] (2025)
 
; [[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] – in Tetracot[13], 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) – in gary, 41edo tuning
 
; [[Jake Freivald]]
* [https://soundcloud.com/jdfreivald/little-magical-object ''Little Magical Object''] – in Magic[19], 41edo tuning
 
; [[L4MPLIGHT]]
* ''Caftaphata'' (2024) – [https://www.youtube.com/watch?v=cMnuMjXeHrY YouTube] | [https://soundcloud.com/l4mplight/caftaphata-microtones-conlang SoundCloud] – also partially in just intonation and 12edo
* ''Yxeni'' (2025) - [https://www.youtube.com/watch?v=PrfAz8V4WNc YouTube]
 
; [[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)
 
; [[Tristan Bay]]
* [https://www.youtube.com/watch?v=sD2-2z85YEI ''Chasing Dusk''] (2025)
 
; [[Chris Vaisvil]] ([https://www.chrisvaisvil.com/ site])
* [https://web.archive.org/web/20230610075457/http://micro.soonlabel.com/41edo/20130910_magic%5b19%5dor_41_the_magic_of_belief.mp3 ''The Magic of Belief''] (2013) – in Magic[19], 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'']
 
; [[John Platter]]
* [https://johnplatter.bandcamp.com/album/in-the-know In the Know] - Full album recorded using kite guitar & bass.
 
; [[Pixel Archipelago]]
* [https://pixelarchipelago.bandcamp.com/album/intervallic-prism ''Intervallic Prism''] (2020) – a 7-track album
: "Red" · "Orange" · "Yellow" · "Green" · "Blue" · "Indigo" · "Violet"


''41-EDO Electric guitar, by Gregory Sanchez.''
; [[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


[[File:Ron_Sword_with_a_41ET_Guitar.jpg|alt=Ron_Sword_with_a_41ET_Guitar.jpg|Ron_Sword_with_a_41ET_Guitar.jpg]]
=== Kite Guitar videos ===
; [[Timmy Barnett]]
* [https://TallKite.com/KiteGuitar/Downminor&#x20;Etude.m4v ''Downminor Etude''] {{dead link}}


''41-EDO Classical guitar, by Ron Sword.''
; [[Wilckerson Ganda]]
* [https://www.youtube.com/watch?v=gQERKtbkMCE ''Vintage Rock'']


The [[The Kite Guitar|Kite Guitar]] (see also [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf Kite Tuning]) is a guitar fretting using every other step of 41-edo, i.e. 41-ED4 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 41-edo, but the full edo can be found on every pair of adjacent strings.The Kite Tuning makes 41-edo about as playable as 19-edo or 22-edo, although there are certain trade-offs.
; [[Travis Johnson]]
* [https://www.youtube.com/watch?v=eAPzZ9oJYyY ''Evening Rondo'']


[[File:Caleb's Kite guitar.jpg|480x640px]]
=== Kite Guitar scores ===
; [[Kite Guitar originals]]
; [[Kite Guitar translations]]


A possible system to tune keyboards in 41EDO is discussed in http://launch.groups.yahoo.com/group/tuning/message/74155.
== See also ==
* [https://KiteGuitar.com KiteGuitar.com] for recordings, videos, etc.
* [[Magic22 as srutis]] describes a possible use of 41edo for [[indian]] music.  


=Music=
== External links ==
[http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro EveningHorizon] [http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 play] by Cameron Bobro
* [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.


=Links=
== Notes ==
<ul><li>[http://en.wikipedia.org/wiki/41_equal_temperament Wikipedia article on 41edo]</li><li>[[Magic22_as_srutis#magic22assrutis|Magic22 as srutis]] describes a possible use of 41edo for [[Indian|indian]] music.</li><li>see also [[Magic_family|Magic family]]</li><li>Sword, Ron. [http://www.ronsword.com "Tetracontamonophonic Scales for Guitar"]</li><li>Taylor, Cam. [https://drive.google.com/open?id=0B3wIGTmjY_VZYllwcHI0d3hEc3M Intervals, Scales and Chords in 41EDO], a work in progress using just intonation concepts and simplified Sagittal notation.</li></ul>
<references/>
-----
<ol><li>[[#cite_ref-1|^]] [http://x31eq.com/schismic.htm "Schismic Temperaments"] at x31eq.com the website of [[Graham_Breed|Graham Breed]]</li><li>[[#cite_ref-2|^]] [http://x31eq.com/decimal_lattice.htm "Lattices with Decimal Notation"] at x31eq.com</li><li>[[#cite_ref-3|^]] [http://en.wikipedia.org/wiki/Schismatic_temperament Schismatic temperament]</li><li>[[#cite_ref-4|^]] [http://en.wikipedia.org/wiki/Magic_temperament Magic temperament]</li></ol>    


[[Category:41edo]]
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[[Category:Edo]]
[[Category:Magic]]
[[Category:Magic]]
[[Category:Prime EDO]]
[[Category:Superkleismic]]
[[Category:Pythagorean]]
[[Category:Supermagic]]
[[Category:Zeta]]
[[Category:Tetracot]]
[[Category:Octacot]]
[[Category:Listen]]
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