41edo: Difference between revisions

| de = 41-EDO

| en = 41edo

{{Wikipedia| 41 equal temperament }}

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.

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 used by the [[Kite Guitar]], see below in [[#Instruments]].

{{Harmonics in equal|41|columns=9}}

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

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]].

 

 

 

 

 

 

{| class="wikitable center-1 right-2 center-5 center-6 center-8 center-9"

! #

! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v⁴A1 and ^d2)

! 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

| [[1/1]]

| perfect unison

| P1

| D

| perfect unison

| P1

| D

| Da

| Do

| 1

| 29.3

| [[49/48]], [[50/49]], [[64/63]], [[81/80]]

| up-unison

| ^1

| ^D

| comma-wide unison, super unison

| K1/S1

| Du

| Di

| 2

| [[25/24]], [[28/27]], [[33/32]], [[36/35]]

| dup-unison, downminor 2nd

| ^^1, vm2

| ^^D, vEb

| subminor 2nd, classic aug unison, uber unison

| sm2, kkA1, U1

| sEb, kkD#, UD

| Fro

| Ro

| 3

| 87.8

| [[19/18]], [[20/19]], [[21/20]], [[22/21]]

| down-aug 1sn, minor 2nd

| vA1, m2

| vD#, Eb

| minor 2nd, comma-narrow augmented unison

| m2, kA1

| Eb, kD#

| Fra

| Rih

| 4

| 117.1

| [[14/13]], [[15/14]], [[16/15]]

| augmented 1sn, upminor 2nd

| A1, ^m2

| D#, ^Eb

| classic minor 2nd, augmented unison

| Km2, A1

| KEb, D#

| Fru

| Ra

| 5

| 146.3

| [[12/11]], [[13/12]]

| mid 2nd

| ~2

| ^D#, vvE

| neutral second, super augmented unison

| N2, SA1

| UEb/uE, sD#

| Ri

| Ru

| 6

| 175.6

| [[10/9]], [[11/10]], [[21/19]]

| downmajor 2nd

| vM2

| vE

| classic/comma-wide major 2nd

| kM2

| kE

| Ro

| Reh

| 7

| 204.9

| [[9/8]]

| major 2nd

| M2

| E

| major 2nd

| M2

| E

| Ra

| Re

| 8

| 234.1

| [[8/7]], [[15/13]]

| upmajor 2nd

| ^M2

| ^E

| supermajor 2nd

| SM2

| SE

| Ru

| Ri

| 9

| [[7/6]], [[22/19]]

| downminor 3rd

| vm3

| vF

| subminor 3rd

| sm3

| sF

| No

| Ma

| 10

| [[13/11]], [[19/16]], [[32/27]]

| minor 3rd

| m3

| F

| minor 3rd

| m3

| F

| Na

| Meh

| 11

| 322.0

| [[6/5]]

| upminor 3rd

| ^m3

| ^F

| classicminor 3rd

| Km3

| KF

| Nu

| Me

| 12

| 351.2

| [[11/9]], [[16/13]]

| ~3

| ^^F, vGb

| neutral 3rd, sub diminished 4th

| N3, sd4

| UF/uF#, sGb

| Mi

| Mu

| 13

| 380.5

| [[5/4]], [[26/21]]

| downmajor 3rd

| vM3

| vF#, Gb

| classic major 3rd, diminished 4th

| kM3, d4

| kF#, Gb

| Mo

| Mi

| 14

| 409.8

| [[14/11]], [[19/15]], [[24/19]]

| major 3rd

| M3

| F#, ^Gb

| major 3rd, comma-wide diminished 4th

| M3, Kd4

| F#, KGb

| Ma

| Maa

| 15

| 439.0

| [[9/7]], [[32/25]]

| upmajor 3rd

| ^M3

| ^F#, vvG

| supermajor 3rd, classic diminished 4th

| SM3, KKd4

| SF#, KKGb

| Mu

| Mo

| 16

| 468.3

| [[21/16]], [[13/10]]

| down-4th

| v4

| vG

| sub 4th

| s4

| sG

| Fo

| Fe

| 17

| 497.6

| [[4/3]]

| perfect 4th

| P4

| G

| perfect 4th

| P4

| G

| Fa

| Fa

| 18

| 526.8

| [[15/11]], [[19/14]], [[27/20]]

| up-4th

| ^4

| ^G

| comma-wide 4th

| K4

| KG

| Fu

| Fih

| 19

| 556.1

| [[11/8]], [[18/13]], [[26/19]]

| mid-4th, downdim 5th

| ~4, vd5

| ^^G, vAb

| uber/neutral 4th, classic augmented 4th

| U4/N4, kkA4

| UG, kkG#

| Fi/Sho

| Fu

| 20

| 585.4

| [[7/5]], [[45/32]]

| downaug 4th, dim 5th

| vA4, d5

| vG#, Ab

| comma-narrow augmented 4th, diminished 5th

| kA4/d5

| kG#, Ab

| Po/Sha

| Fi

| 21

| 614.6

| [[10/7]], [[64/45]]

| aug 4th, updim 5th

| A4, ^d5

| G#, ^Ab

| augmented 4th, comma-wide diminished 5th

| A4/Kd5

| G#, KAb

| Pa/Shu

| Se

| 22

| [[13/9]], [[16/11]], [[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

| [[22/15]], [[28/19]], [[40/27]]

| down-5th

| v5

| vA

| comma-narrow 5th

| k5

| kA

| So

| Sih

| 24

| 702.4

| [[3/2]]

| perfect 5th

| P5

| A

| perfect 5th

| P5

| A

| Sa

| Sol

| 25

| [[20/13]], [[32/21]]

| up-5th

| ^5

| ^A

| super 5th

| S5

| SA

| Su

| Si

| 26

| 761.0

| [[14/9]], [[25/16]]

| downminor 6th

| vm6

| ^^A, vBb

| subminor 6th, classic augmented 5th

| sm6

| sBb, kkA#

| Flo

| Lo

| 27

| 790.2

| [[11/7]], [[19/12]], [[30/19]]

| minor 6th

| m6

| vA#, Bb

| minor 6th, comma-narrow augmented 5th

| m6

| Bb, kA#

| Fla

| Leh

| 28

| 819.5

| [[8/5]], [[21/13]]

| upminor 6th

| ^m6

| A#, ^Bb

| classic minor 6th, augmented 5th

| Km6, A5

| KBb, A#

| Flu

| Le

| 29

| [[13/8]], [[18/11]]

| mid 6th

| ~6

| ^A#, vvB

| neutral 6th, super augmented 5th

| N6

| UBb/uB, sA#

| Li

| Lu

| 30

| 878.0

| [[5/3]]

| downmajor 6th

| vM6

| vB

| classic major 6th

| kM6

| kB

| Lo

| La

| 31

| 907.3

| [[22/13]], [[27/16]], [[32/19]]

| major 6th

| M6

| B

| major 6th

| M6

| B

| La

| Laa

| 32

| 936.6

| [[12/7]], [[19/11]]

| ^M6

| ^B

| supermajor 6th

| SM6

| SB

| Lu

| Li

| 33

| [[7/4]], [[26/15]]

| downminor 7th

| vm7

| vC

| subminor 7th

| sm7

| sC

| Tho

| Ta

| 34

| 995.1

| [[16/9]]

| minor 7th

| m7

| C

| minor 7th

| m7

| C

| Tha

| Teh

| 35

| 1024.4

| [[9/5]], [[20/11]], [[38/21]]

| upminor 7th

| ^m7

| ^C

| classic/comma-wide minor seventh

| Km7

| KC

| Thu

| Te

| 36

| 1053.7

| [[11/6]], [[24/13]]

| mid 7th

| ~7

| ^^C, vDb

| neutral 7th, sub diminished 8ve

| N7

| UC/uC#, sDb

| Ti

| Tu

| 37

| [[13/7]], [[15/8]], [[28/15]]

| downmajor 7th

| vM7

| vC#, Db

| classic major 7th, diminished 8ve

| kM7, d8

| kC#, Db

| To

| Ti

| 38

| 1112.2

| [[19/10]], [[21/11]], [[36/19]], [[40/21]]

| major 7th

| M7

| C#, ^Db

| major 7th, comma-wide diminished 8ve

| M7, Kd8

| C#, KDb

| Ta

| Taa

| 39

| 1141.5

| [[27/14]], [[35/18]], [[48/25]], [[64/33]]

| upmajor 7th

| ^M7

| ^C#, vvD

| supermajor 7th, classic dim 8ve, unter 8ve

| SM7, KKd8, U8

| SC#, KKDb, u8

| Tu

| To

| 40

| 1170.7

| [[49/25]], [[63/32]], [[96/49]], [[160/81]]

| dim 8ve

| v8

| vD

| comma-narrow 8ve, sub 8ve

| k8/s8

| kD, sD

| Do

| Da

| 41

| 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.

 

 

{| class="wikitable center-all"

! [[Color notation|Color]]

| downminor

| zo

| (a, b, 0, 1)

| 7/6, 7/4

| minor

| fourthward wa

| (a, b) with b < -1

| 32/27, 16/9

| upminor

| gu

| (a, b, -1)

| 6/5, 9/5

| mid

| ilo

| (a, b, 0, 0, 1)

| 11/9, 11/6

| "

| lu

| (a, b, 0, 0, -1)

| 12/11, 18/11

| downmajor

| yo

| (a, b, 1)

| 5/4, 5/3

| major

| fifthward wa

| (a, b) with b > 1

| 9/8, 27/16

| upmajor

| ru

| (a, b, 0, -1)

| 9/7, 12/7

|}

 

All 41edo chords can be named using ups and downs. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:

 

{| class="wikitable center-all"

! [[Color notation|Color of the 3rd]]

| zo (7-over)

| 6:7:9

| 0-9-24

| C vEb G

| Cvm

| C downminor

| gu (5-under)

| 10:12:15

| 0-11-24

| C ^Eb G

| C^m

| C upminor

| ilo (11-over)

| 18:22:27

| 0-12-24

| C vvE G

| C~

| C mid

| yo (5-over)

| 4:5:6

| 0-13-24

| C vE G

| Cv

| C downmajor or C down

| ru (7-under)

| 14:18:21

| 0-15-24

| C ^E G

| C^

| C upmajor or C up

* 0-10-24 = D F A = Dm = D minor

* 0-14-24 = D F# A = D = D or D major

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

| 2.3

| {{monzo| 65 -41 }}

| {{mapping| 41 65 }}

| −0.153

| 0.15

| 0.52

| 2.3.5

| 3125/3072, 20000/19683

| {{mapping| 41 65 95 }}

| +0.734

| 1.26

| 4.31

| 2.3.5.7

|-

| 2.3.5.7.11

| {{mapping| 41 65 95 115 142 }}

| +0.375

| 1.32

| 4.51

| 2.3.5.7.11.13

| 100/99, 105/104, 144/143, 196/195, 243/242

| {{mapping| 41 65 95 115 142 152 }}

| 1.55

| 5.29

|-

| 100/99, 105/104, 133/132, 144/143, 171/169, 196/195

| {{mapping| 41 65 95 115 142 152 174 }}

| +0.111

| 1.49

* 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.

 

| [[20000/19683|(10 digits)]]

| [[Tetracot comma]]

|-

| ssy³

| 5

| [[32805/32768|(10 digits)]]

| 1.95

| {{monzo| -15 8 1 }}

| Layo

| Ly

| [[Schisma]]

| 7

| [[15625/15309|(10 digits)]]

| 35.37

| {{monzo| 0 -7 6 -1 }}

| Rutribiyo

| ry⁶

| Arcturus comma, great BP diesis

| 7

| <abbr title="854296875/843308032">(18 digits)</abbr>

| 22.41

| {{monzo| -10 7 8 -7 }}

| Lasepru-aquadbiyo

| Lr⁷y⁸

| [[Blackjackisma]]

| 7

| [[875/864]]

| 21.90

| {{monzo| -5 -3 3 1 }}

| Zotriyo

| zy³

| Keema

| 7

| [[3125/3087]]

| 21.18

| {{monzo| 0 -2 5 -3 }}

| Triru-aquinyo

| r³y⁵

| Gariboh comma

| 7

| <abbr title="179200/177147">(12 digits)</abbr>

| 19.95

| {{monzo| 10 -11 2 1 }}

| Sazoyoyo

| szyy

| [[Tolerma]]

| 7

| [[33075/32768|(10 digits)]]

| 16.14

| {{monzo| -15 3 2 2 }}

| Labizoyo

| Lzzyy

| [[Mirwomo comma]]

| 7

| [[245/243]]

| 14.19

| {{monzo| 0 -5 1 2 }}

| Zozoyo

| zzy

| Sensamagic comma

| 7

| [[4000/3969]]

| 13.47

| {{monzo| 5 -4 3 -2 }}

| Rurutriyo

| rry³

| Octagar comma

| 7

| <abbr title="823543/819200">(12 digits)</abbr>