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__FORCETOC__
__FORCETOC__
-----
=Theory=
 
=Introduction=
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&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.
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&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.


Line 13: Line 11:
41edo is the 13th [[prime_numbers|prime]] edo, following [[37edo|37edo]] and coming before [[43edo|43edo]].
41edo is the 13th [[prime_numbers|prime]] edo, following [[37edo|37edo]] and coming before [[43edo|43edo]].


=Commas=
=Notation=
41 EDO tempers out the following commas using its patent val, < 41 65 95 115 142 152 168 174 185 199 203 |.
 
{| class="wikitable"
{| class="wikitable"
|-
|-
! | Name
! |  
! colspan="2" |[[Color notation/Temperament Names|color name]]
! | Cents Value
! | Monzo
! | Approximate Ratios
! | Ratio
in the [[11-limit|11-limit]]
! | Cents
! colspan="3" |[[Ups_and_Downs_Notation|Ups and Downs Notation]]
! | Andrew's
Solfege Syllables
|-
|-
| | odiheim
| style="text-align:center;" | 0
|Luluquinzo-aquadgu
| style="text-align:center;" | 0.00
|1uuz<sup>5</sup>g<sup>4</sup>
| | [[1/1|1/1]]
| |<nowiki> | -1 2 -4 5 -2 </nowiki>&gt;
|perfect unison
| style="text-align:center;" |  
| style="text-align:center;" | P1
| | 0.15
| style="text-align:center;" | D
| | do
|-
|-
| | harmonisma
| style="text-align:center;" | 1
|Thuthutrilo-aru
| style="text-align:center;" | 29.27
|3uu1o<sup>3</sup>r
| | [[81/80|81/80]], [[64/63]]
| |<nowiki> | 3 -2 0 -1 3 -2 </nowiki>&gt;
|up-unison
| style="text-align:center;" | 10648/10647
| style="text-align:center;" | ^1
| | 0.16
| style="text-align:center;" | ^D
| | di
|-
|-
| | tridecimal schisma, Sagittal schismina
| style="text-align:center;" | 2
|Sathurugu
| style="text-align:center;" | 58.54
|s3urg
| | [[25/24|25/24]], [[28/27|28/27]], [[33/32|33/32]]
| |<nowiki> | 12 -2 -1 -1 0 -1 </nowiki>&gt;
|double-up unison,
| style="text-align:center;" | 4096/4095
downminor 2nd
| | 0.42
| style="text-align:center;" | ^^1, vm2
| style="text-align:center;" | ^^D, vEb
| | ro
|-
|-
| | Lehmerisma
| style="text-align:center;" | 3
|Loloruyoyo
| style="text-align:center;" | 87.805
|1ooryy
| | [[21/20|21/20]], [[22/21|22/21]]
| |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
|down-aug 1sn,
| style="text-align:center;" | 3025/3024
minor 2nd
| | 0.57
| style="text-align:center;" | vA1, m2
| style="text-align:center;" | vD#, Eb
| | rih
|-
|-
| | Breedsma
| style="text-align:center;" | 4
|Bizozogu
| style="text-align:center;" | 117.07
|z<sup>4</sup>gg
| | [[16/15|16/15]], [[15/14|15/14]]
| |<nowiki> | -5 -1 -2 4 </nowiki>&gt;
|augmented 1sn,
| style="text-align:center;" | 2401/2400
 
| | 0.72
upminor 2nd
| style="text-align:center;" | A1, ^m2
| style="text-align:center;" | D#, ^Eb
| | ra
|-
|-
| | Eratosthenes' comma
| style="text-align:center;" | 5
|Sanogu
| style="text-align:center;" | 146.34
|s19og
| | [[12/11|12/11]]
| |<nowiki> | 6 -5 -1 0 0 0 0 1 </nowiki>&gt;
|mid 2nd
| style="text-align:center;" | 1216/1215
| style="text-align:center;" | ~2
| | 1.42
| style="text-align:center;" | ^D#, vvE
| | ru
|-
|-
| | schisma
| style="text-align:center;" | 6
|Layo
| style="text-align:center;" | 175.61
|Ly
| | [[10/9|10/9]], [[11/10|11/10]]
| |<nowiki> | -15 8 1 </nowiki>&gt;
|downmajor 2nd
| style="text-align:center;" | 32805/32768
| style="text-align:center;" | vM2
| | 1.95
| style="text-align:center;" | vE
| | reh
|-
|-
| | squbema
| style="text-align:center;" | 7
|Lathuru
| style="text-align:center;" | 204.88
|L3ur
| | [[9/8|9/8]]
| |<nowiki> | -3 6 0 -1 0 -1 </nowiki>&gt;
|major 2nd
| style="text-align:center;" | 729/728
| style="text-align:center;" | M2
| | 2.38
| style="text-align:center;" | E
| | re
|-
|-
| | septendecimal bridge comma
| style="text-align:center;" | 8
|Sutholoruyo
| style="text-align:center;" | 234.15
|17u3o1ory
| | [[8/7|8/7]]
| |<nowiki> | -1 -1 1 -1 1 1 -1 </nowiki>&gt;
|upmajor 2nd
| style="text-align:center;" | 715/714
| style="text-align:center;" | ^M2
| | 2.42
| style="text-align:center;" | ^E
|-
| | ri
| | Swets' comma, swetisma
|Lururuyo
|1urry
| |<nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
| style="text-align:center;" | 540/539
| | 3.21
|-
|-
| | undevicesimal comma, Boethius' comma
| style="text-align:center;" | 9
|Lano
| style="text-align:center;" | 263.415
|L19o
| | [[7/6|7/6]]
| |<nowiki> | -9 3 0 0 0 0 0 1 </nowiki>&gt;
|downminor 3rd
| style="text-align:center;" | 513/512
| style="text-align:center;" | vm3
| | 3.38
| style="text-align:center;" | vF
| | ma
|-
|-
| | moctdel
| style="text-align:center;" | 10
|Lotriruyo
| style="text-align:center;" | 292.68
|1or<sup>3</sup>y
| | [[32/27|32/27]], [[13/11]]
| |<nowiki> | -2 0 3 -3 1 </nowiki>&gt;
|minor 3rd
| style="text-align:center;" | 1375/1372
| style="text-align:center;" | m3
| | 3.78
| style="text-align:center;" | F
| | meh
|-
|-
| | Beta 2, septimal schisma, garischisma
| style="text-align:center;" | 11
|Sasaru
| style="text-align:center;" | 321.95
|ssr
| | [[6/5|6/5]]
| |<nowiki> | 25 -14 0 -1 </nowiki>&gt;
|upminor 3rd
| style="text-align:center;" |  
| style="text-align:center;" | ^m3
| | 3.80
| style="text-align:center;" | ^F
| | me
|-
|-
| | Werckmeister's undecimal septenarian schisma, werckisma
| style="text-align:center;" | 12
|Luzozogu
| style="text-align:center;" | 351.22
|1uzzg
| | [[11/9|11/9]], [[27/22|27/22]]
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
|mid 3rd
| style="text-align:center;" | 441/440
| style="text-align:center;" | ~3
| | 3.93
| style="text-align:center;" | ^^F, vGb
| | mu
|-
|-
| | cuthbert
| style="text-align:center;" | 13
|Thuthulolozogu
| style="text-align:center;" | 380.49
|3uu1oozg
| | [[5/4|5/4]]
| |<nowiki> | 0 0 -1 1 2 -2 </nowiki>&gt;
|downmajor 3rd
| style="text-align:center;" | 847/845
| style="text-align:center;" | vM3
| | 4.09
| style="text-align:center;" | vF#, Gb
| | mi
|-
|-
| | undecimal kleisma, keenanisma
| style="text-align:center;" | 14
|Lozoyo
| style="text-align:center;" | 409.76
|1ozg
| | [[14/11|14/11]], [[81/64|81/64]]
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
|major 3rd
| style="text-align:center;" | 385/384
| style="text-align:center;" | M3
| | 4.50
| style="text-align:center;" | F#, ^Gb
| | maa
|-
|-
| | gentle comma
| style="text-align:center;" | 15
|Tholuluzo
| style="text-align:center;" | 439.02
|3o1uuz
| | [[9/7|9/7,]] [[32/25|32/25]]
| |<nowiki> | 2 -1 0 1 -2 1 </nowiki>&gt;
|upmajor 3rd
| style="text-align:center;" | 364/363
| style="text-align:center;" | ^M3
| | 4.76
| style="text-align:center;" | ^F#, vvG
| | mo
|-
|-
| | minthma
| style="text-align:center;" | 16
|Thulo
| style="text-align:center;" | 468.29
|3u1o
| | [[21/16|21/16]], [[13/10]]
| |<nowiki> | 5 -3 0 0 1 -1 </nowiki>&gt;
|down-4th
| style="text-align:center;" | 352/351
| style="text-align:center;" | v4
| | 4.93
| style="text-align:center;" | vG
| | fe
|-
|-
| | marveltwin
| style="text-align:center;" | 17
|Thoyoyo
| style="text-align:center;" | 497.56
|3oyy
| | [[4/3|4/3]]
| |<nowiki> | -2 -4 2 0 0 1 </nowiki>&gt;
|perfect 4th
| style="text-align:center;" | 325/324
| style="text-align:center;" | P4
| | 5.34
| style="text-align:center;" | G
| | fa
|-
|-
| | Beta 5, Garibaldi comma, hemifamity
| style="text-align:center;" | 18
|Saruyo
| style="text-align:center;" | 526.83
|sry
| | [[15/11|15/11]], [[27/20|27/20]]
| |<nowiki> | 10 -6 1 -1 </nowiki>&gt;
|up-4th
| style="text-align:center;" | 5120/5103
| style="text-align:center;" | ^4
| | 5.76
| style="text-align:center;" | ^G
| | fih
|-
|-
| | hemimage
| style="text-align:center;" | 19
|Satrizo-agu
| style="text-align:center;" | 556.1
|sz<sup>3</sup>g
| | [[11/8|11/8]]
| |<nowiki> | 5 -7 -1 3 </nowiki>&gt;
|mid-4th
| style="text-align:center;" | 10976/10935
| style="text-align:center;" | ~4
| | 6.48
| style="text-align:center;" | ^^G, vAb
| | fu
|-
|-
| | septendecimal kleisma
| style="text-align:center;" | 20
|Sugu
| style="text-align:center;" | 585.37
|17ug
| | [[7/5|7/5]]
| |<nowiki> | 8 -1 -1 0 0 0 -1 </nowiki>&gt;
|downaug 4th, dim 5th
| style="text-align:center;" | 256/255
| style="text-align:center;" | vA4, d5
| | 6.78
| style="text-align:center;" | vG#, Ab
| | fi
|-
|-
| | small BP diesis, mirkwai
| style="text-align:center;" | 21
|Quinru-aquadyo
| style="text-align:center;" | 614.63
|r<sup>5</sup>y<sup>4</sup>
| | [[10/7|10/7]]
| |<nowiki> | 0 3 4 -5 </nowiki>&gt;
|aug 4th, updim 5th
| style="text-align:center;" | 16875/16807
| style="text-align:center;" | A4, ^d5
| | 6.99
| style="text-align:center;" | G#, ^Ab
| | se
|-
|-
| | neutral third comma, rastma
| style="text-align:center;" | 22
|Lulu
| style="text-align:center;" | 643.90
|1uu
| | [[16/11|16/11]], [[13/9]]
| |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
|mid-5th
| style="text-align:center;" | 243/242
| style="text-align:center;" | ~5
| | 7.14
| style="text-align:center;" | vvA
| | su
|-
|-
| | kestrel comma
| style="text-align:center;" | 23
|Thuthuloru
| style="text-align:center;" | 673.17
|3uu1or
| | [[22/15|22/15]], [[40/27|40/27]]
| |<nowiki> | 2 3 0 -1 1 -2 </nowiki>&gt;
|down-5th
| style="text-align:center;" | 1188/1183
| style="text-align:center;" | v5
| | 7.30
| style="text-align:center;" | vA
| | sih
|-
|-
| | septimal kleisma, marvel comma
| style="text-align:center;" | 24
|Ruyoyo
| style="text-align:center;" | 702.44
|ryy
| | [[3/2|3/2]]
| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
|perfect 5th
| style="text-align:center;" | 225/224
| style="text-align:center;" | P5
| | 7.71
| style="text-align:center;" | A
| | sol
|-
|-
| | huntma
| style="text-align:center;" | 25
|Thururuyo
| style="text-align:center;" | 731.71
|3urry
| | [[32/21|32/21]], [[20/13]]
| |<nowiki> | 7 0 1 -2 0 -1 </nowiki>&gt;
|up-5th
| style="text-align:center;" | 640/637
| style="text-align:center;" | ^5
| | 8.13
| style="text-align:center;" | ^A
| | si
|-
|-
| | spleen comma
| style="text-align:center;" | 26
|Nuluzoyo
| style="text-align:center;" | 760.98
|19u1uzy
| | [[14/9|14/9]], [[25/16|25/16]]
| |<nowiki> | 1 1 1 1 -1 0 0 -1 </nowiki>&gt;
|downminor 6th
| style="text-align:center;" | 210/209
| style="text-align:center;" | vm6
| | 8.26
| style="text-align:center;" | ^^A, vBb
| | lo
|-
|-
| | orgonisma
| style="text-align:center;" | 27
|Satrilu-aruru
| style="text-align:center;" | 790.24
|s1u<sup>3</sup>rr
| | [[11/7|11/7]], [[128/81|128/81]]
| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
|minor 6th
| style="text-align:center;" | 65536/65219
| style="text-align:center;" | m6
| | 8.39
| style="text-align:center;" | vA#, Bb
| | leh
|-
|-
| | gamelan residue, gamelisma
| style="text-align:center;" | 28
|Latrizo
| style="text-align:center;" | 819.51
|Lz<sup>3</sup>
| | [[8/5|8/5]]
| |<nowiki> | -10 1 0 3 </nowiki>&gt;
|upminor 6th
| style="text-align:center;" | 1029/1024
| style="text-align:center;" | ^m6
| | 8.43
| style="text-align:center;" | A#, ^Bb
|-
| | le
| | septendecimal comma
|Lasu
|L17u
| |<nowiki> | -7 7 0 0 0 0 -1 </nowiki>&gt;
| style="text-align:center;" | 2187/2176
| | 8.73
|-
|-
| | mynucuma
| style="text-align:center;" | 29
|Thuzozogu
| style="text-align:center;" | 848.78
|3uzzg
| | [[18/11|18/11]], [[44/27|44/27]]
| |<nowiki> | 2 -1 -1 2 0 -1 </nowiki>&gt;
|mid 6th
| style="text-align:center;" | 196/195
| style="text-align:center;" | ~6
| | 8.86
| style="text-align:center;" | ^A#, vvB
| | lu
|-
|-
| | quince
| style="text-align:center;" | 30
|Lasepzo-agugu
| style="text-align:center;" | 878.05
|Lz<sup>7</sup>gg
| | [[5/3|5/3]]
| |<nowiki> | -15 0 -2 7 </nowiki>&gt;
|downmajor 6th
| style="text-align:center;" |  
| style="text-align:center;" | vM6
| | 9.15
| style="text-align:center;" | vB
| | la
|-
|-
| | undecimal semicomma, pentacircle (minthma * gentle)
| style="text-align:center;" | 31
|Saluzo
| style="text-align:center;" | 907.32
|s1uz
| | [[27/16|27/16]]
| |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
|major 6th
| style="text-align:center;" | 896/891
| style="text-align:center;" | M6
| | 9.69
| style="text-align:center;" | B
| | laa
|-
|-
| | 29th-partial chroma
| style="text-align:center;" | 32
|Twenty-noyo
| style="text-align:center;" | 936.59
|29oy
| | [[12/7|12/7]]
| |<nowiki> | -4 -2 1 0 0 0 0 0 0 1 </nowiki>&gt;
|upmajor 6th
| style="text-align:center;" | 145/144
| style="text-align:center;" | ^M6
| | 11.98
| style="text-align:center;" | ^B
| | li
|-
|-
| | grossma
| style="text-align:center;" | 33
|Thulu
| style="text-align:center;" | 965.85
|3u1u
| | [[7/4|7/4]]
| |<nowiki> | 4 2 0 0 -1 -1 </nowiki>&gt;
|downminor 7th
| style="text-align:center;" | 144/143
| style="text-align:center;" | vm7
| | 12.06
| style="text-align:center;" | vC
| | ta
|-
|-
| | gassorma
| style="text-align:center;" | 34
|Thuloruyoyo
| style="text-align:center;" | 995.12
|3u1oryy
| | [[16/9|16/9]]
| |<nowiki> | 0 -1 2 -1 1 -1 </nowiki>&gt;
|minor 7th
| style="text-align:center;" | 275/273
| style="text-align:center;" | m7
| | 12.64
| style="text-align:center;" | C
| | teh
|-
|-
| | septimal semicomma, octagar
| style="text-align:center;" | 35
|Rurutriyo
| style="text-align:center;" | 1024.39
|rry<sup>3</sup>
| | [[9/5|9/5]], [[20/11|20/11]]
| |<nowiki> | 5 -4 3 -2 </nowiki>&gt;
|upminor 7th
| style="text-align:center;" | 4000/3969
| style="text-align:center;" | ^m7
| | 13.47
| style="text-align:center;" | ^C
| | te
|-
|-
| | minor BP diesis, sensamagic
| style="text-align:center;" | 36
|Zozoyo
| style="text-align:center;" | 1053.66
|zzy
| | [[11/6|11/6]]
| |<nowiki> | 0 -5 1 2 </nowiki>&gt;
|mid 7th
| style="text-align:center;" | 245/243
| style="text-align:center;" | ~7
| | 14.19
| style="text-align:center;" | ^^C, vDb
| | tu
|-
|-
| | secorian
| style="text-align:center;" | 37
|Sathuzo
| style="text-align:center;" | 1082.93
|s3uz
| | [[15/8|15/8]]
| |<nowiki> | 12 -7 0 1 0 -1 </nowiki>&gt;
|downmajor 7th
| style="text-align:center;" | 28672/28431
| style="text-align:center;" | vM7
| | 14.61
| style="text-align:center;" | vC#, Db
| | ti
|-
|-
| | mirwomo comma
| style="text-align:center;" | 38
|Labizoyo
| style="text-align:center;" | 1112.195
|Lzzyy
| | [[40/21|40/21]], [[21/11|21/11]]
| |<nowiki> | -15 3 2 2 </nowiki>&gt;
|major 7th
| style="text-align:center;" | 33075/32768
| style="text-align:center;" | M7
| | 16.14
| style="text-align:center;" | C#, ^Db
|-
| | taa
| | vicesimotertial comma
|Sa-twenty-tho
|s23o
| |<nowiki> | 5 -6 0 0 0 0 0 0 1 </nowiki>&gt;
| style="text-align:center;" | 736/729
| | 16.54
|-
|-
| | small tridecimal comma, animist
| style="text-align:center;" | 39
|Thuzoyo
| style="text-align:center;" | 1141.46
|3uzy
| | [[48/25|48/25]], [[27/14|27/14]], [[64/33|64/33]]
| |<nowiki> | -3 1 1 1 0 -1 </nowiki>&gt;
|upmajor 7th
| style="text-align:center;" | 105/104
| style="text-align:center;" | ^M7
| | 16.57
| style="text-align:center;" | C#^, vvD
| | to
|-
|-
| | hemimin
| style="text-align:center;" | 40
|Trilu-azo
| style="text-align:center;" | 1170.73
|1u<sup>3</sup>z
| | [[160/81|160/81]], [[63/32]]
| |<nowiki> | 6 1 0 1 -3 </nowiki>&gt;
|
| style="text-align:center;" | 1344/1331
| style="text-align:center;" | v8
| | 16.83
| style="text-align:center;" | vD
| | da
|-
|-
| | Ptolemy's comma, ptolemisma
| style="text-align:center;" | 41
|Luyoyo
| style="text-align:center;" | 1200
|1uyy
| | 2/1
| |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
|
| style="text-align:center;" | 100/99
| style="text-align:center;" | P8
| | 17.40
| style="text-align:center;" | D
| | do
|}
 
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
 
{| class="wikitable"
|-
|-
| | '41-tone' comma
! | quality
|Wa-41
! | [[Kite's color notation|color]]
|41-edo
! | monzo format
| |<nowiki> | 65 -41 </nowiki>&gt;
! | examples
| style="text-align:center;" |
| | 19.84
|-
|-
| | tolerma
| style="text-align:center;" | downminor
|Sazoyoyo
| style="text-align:center;" | zo
|szyy
| style="text-align:center;" | (a, b, 0, 1)
| |<nowiki> | 10 -11 2 1 </nowiki>&gt;
| style="text-align:center;" | 7/6, 7/4
| style="text-align:center;" |  
| | 19.95
|-
|-
| | major BP diesis, gariboh
| style="text-align:center;" | minor
|Triru-aquinyo
| style="text-align:center;" | fourthward wa
|r<sup>3</sup>y<sup>5</sup>
| style="text-align:center;" | (a, b) with b &lt; -1
| |<nowiki> | 0 -2 5 -3 </nowiki>&gt;
| style="text-align:center;" | 32/27, 16/9
| style="text-align:center;" | 3125/3087
| | 21.18
|-
|-
| | cassacot
| style="text-align:center;" | upminor
|Luluzozoyo
| style="text-align:center;" | gu
|1uuzzy
| style="text-align:center;" | (a, b, -1)
| |<nowiki> | -1 0 1 2 -2 </nowiki>&gt;
| style="text-align:center;" | 6/5, 9/5
| style="text-align:center;" | 245/242
| | 21.33
|-
|-
| | keema
| style="text-align:center;" | mid
|Zotriyo
| style="text-align:center;" | ilo
|zy<sup>3</sup>
| style="text-align:center;" | (a, b, 0, 0, 1)
| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| style="text-align:center;" | 11/9, 11/6
| style="text-align:center;" | 875/864
| | 21.90
|-
|-
| | blackjackisma
| style="text-align:center;" | "
|Lasepru-aquadbiyo
| style="text-align:center;" | lu
|Lr<sup>7</sup>y<sup>8</sup>
| style="text-align:center;" | (a, b, 0, 0, -1)
| |<nowiki> | -10 7 8 -7 </nowiki>&gt;
| style="text-align:center;" | 12/11, 18/11
| style="text-align:center;" |  
| | 22.41
|-
|-
| | roda
| style="text-align:center;" | downmajor
|Sasa-triyo
| style="text-align:center;" | yo
|ssy<sup>3</sup>
| style="text-align:center;" | (a, b, 1)
| |<nowiki> | 20 -17 3 </nowiki>&gt;
| style="text-align:center;" | 5/4, 5/3
| style="text-align:center;" |  
| | 25.71
|-
|-
| | minimal diesis, tetracot comma
| style="text-align:center;" | major
|Saquadyo
| style="text-align:center;" | fifthward wa
|sy<sup>4</sup>
| style="text-align:center;" | (a, b) with b &gt; 1
| |<nowiki> | 5 -9 4 </nowiki>&gt;
| style="text-align:center;" | 9/8, 27/16
| style="text-align:center;" | 20000/19683
| | 27.66
|-
|-
| | small diesis, magic comma
| style="text-align:center;" | upmajor
|Laquinyo
| style="text-align:center;" | ru
|Ly<sup>5</sup>
| style="text-align:center;" | (a, b, 0, -1)
| |<nowiki> | -10 -1 5 </nowiki>&gt;
| style="text-align:center;" | 9/7, 12/7
| style="text-align:center;" | 3125/3072
|}
| | 29.61
 
== Chord Names ==
All 41edo chords can be named using ups and downs. Here are the zo, gu, ilo, yo and ru triads:
 
{| class="wikitable"
|-
|-
| | thuja comma
! | [[Kite's color notation|color of the 3rd]]
|Saquinlu-ayo
! | JI chord
|s1u<sup>5</sup>y
! | notes as edosteps
| |<nowiki> | 15 0 1 0 -5 </nowiki>&gt;
! | notes of C chord
| style="text-align:center;" |
! | written name
| | 29.72
! | spoken name
|-
|-
| | Ampersand's comma
| style="text-align:center;" | zo
|Lala-tribiyo
| style="text-align:center;" | 6:7:9
|LLy<sup>3</sup>
| style="text-align:center;" | 0-9-24
| |<nowiki> | -25 7 6 </nowiki>&gt;
| style="text-align:center;" | C vEb G
| style="text-align:center;" |  
| style="text-align:center;" | Cvm
| | 31.57
| style="text-align:center;" | C downminor
|-
|-
| | great BP diesis
| style="text-align:center;" | gu
|Rutribiyo
| style="text-align:center;" | 10:12:15
|ry<sup>6</sup>
| style="text-align:center;" | 0-11-24
| |<nowiki> | 0 -7 6 -1 </nowiki>&gt;
| style="text-align:center;" | C ^Eb G
| style="text-align:center;" | 15625/15309
| style="text-align:center;" | C^m
| | 35.37
| style="text-align:center;" | C upminor
|-
|-
| | shibboleth
| style="text-align:center;" | ilo
|Tritriyo
| style="text-align:center;" | 18:22:27
|y<sup>9</sup>
| style="text-align:center;" | 0-12-24
| |<nowiki> | -5 -10 9 </nowiki>&gt;
| style="text-align:center;" | C vvE G
| style="text-align:center;" |  
| style="text-align:center;" | C~
| | 57.27
| style="text-align:center;" | C mid
|-
| style="text-align:center;" | yo
| style="text-align:center;" | 4:5:6
| style="text-align:center;" | 0-13-24
| style="text-align:center;" | C vE G
| style="text-align:center;" | Cv
| style="text-align:center;" | C downmajor or C down
|-
| style="text-align:center;" | ru
| style="text-align:center;" | 14:18:27
| style="text-align:center;" | 0-15-24
| style="text-align:center;" | C ^E G
| style="text-align:center;" | C^
| style="text-align:center;" | C upmajor or C up
|}
|}
0-10-20 = D F Ab = Ddim = D dim


=Temperaments=
0-10-21 = D F ^Ab = Ddim(^5) = D dim up-five
[[List_of_edo-distinct_41et_rank_two_temperaments|List of edo-distinct 41et rank two temperaments]]
 
0-10-22 = D F vvA = Dm(~5) = D minor mid-five


=Intervals=
0-10-23 = D F vA = Dm(v5) = D minor down-five


{| class="wikitable"
0-10-24 = D F A = Dm = D minor
|-
 
! |
0-14-24 = D F# A = D = D or D major
! | Cents Value
 
! | Approximate Ratios
0-14-25 = D F# ^A = D(^5) = D up-five
in the [[11-limit|11-limit]]
 
! colspan="2" | [[Ups_and_Downs_Notation|Ups and Downs Notation]]
0-14-26 = D F# ^^A = D(^^5) = D double-up-five, or possibly Daug(vv5)
! | Andrew's  
 
Solfege Syllables
0-14-27 = D F# vA# = Daug(v5) = D aug down-five
! | Generator
 
! | Some MOS and MODMOS Scales Available
0-14-28 = D F# A# = Daug = D aug
|-
 
| style="text-align:center;" | 0
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
| style="text-align:center;" | 0.00
 
| | [[1/1|1/1]]
==Red-Blue Notation==
| style="text-align:center;" | P1
 
| style="text-align:center;" | D
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:
| | do
 
| |  
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.
| |  
 
Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.
 
The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.
 
If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups_and_Downs_Notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone.
 
=Selected just intervals by error=
The following table shows how [[Just-24|some prominent just intervals]] are represented in 41edo (ordered by absolute error).
 
{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
|-
| style="text-align:center;" | 1
| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
| style="text-align:center;" | 29.27
| style="text-align:center;" | '''0.484'''
| | [[81/80|81/80]], [[64/63]]
|-
| style="text-align:center;" | ^1
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | ^D
| style="text-align:center;" | 0.968
| | di
| |  
| |  
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 58.54
| style="text-align:center;" | 2.370
| | [[25/24|25/24]], [[28/27|28/27]], [[33/32|33/32]]
| style="text-align:center;" | ^^1, vm2
| style="text-align:center;" | ^^D, vEb
| | ro
| | [[Hemimiracle|Hemimiracle]]
| |
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 87.805
| style="text-align:center;" | 2.854
| | [[21/20|21/20]], [[22/21|22/21]]
| style="text-align:center;" | vA1, m2
| style="text-align:center;" | vD#, Eb
| | rih
| | 88cET (approx),
 
[[Octacot|octacot]]
| |
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| style="text-align:center;" | 117.07
| style="text-align:center;" | '''2.972'''
| | [[16/15|16/15]], [[15/14|15/14]]
| style="text-align:center;" | A1, m2
| style="text-align:center;" | D#, ^Eb
| | ra
| | [[Miracle|Miracle]]
| |
|-
|-
| style="text-align:center;" | 5
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 146.34
| style="text-align:center;" | 3.456
| | [[12/11|12/11]]
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D#, vvE
| | ru
| | [[Bohlen-Pierce|Bohlen-Pierce]]/[[bohpier|bohpier]]
| |
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 175.61
| style="text-align:center;" | 3.473
| | [[10/9|10/9]], [[11/10|11/10]]
|-
| style="text-align:center;" | vM2
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | vE
| style="text-align:center;" | 3.812
| | reh
| | [[Tetracot|Tetracot]]/[[bunya|bunya]]/[[Monkey|monkey]]
| | 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 204.88
| style="text-align:center;" | 3.940
| | [[9/8|9/8]]
| style="text-align:center;" | M2
| style="text-align:center;" | E
| | re
| | [[Baldy|Baldy]]
| | 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 234.15
| style="text-align:center;" | 4.296
| | [[8/7|8/7]]
| style="text-align:center;" | ^M2
| style="text-align:center;" | ^E
| | ri
| | [[Rodan|Rodan]]/[[guiron|guiron]]
| | 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
| style="text-align:center;" | 263.415
| style="text-align:center;" | '''4.780'''
| | [[7/6|7/6]]
| style="text-align:center;" | vm3
| style="text-align:center;" | vF
| | ma
| | [[Septimin|Septimin]]
| | 9-tone MOS: 5 4 5 5 4 5 4 5 4
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 292.68
| style="text-align:center;" | 5.342
| | [[32/27|32/27]], [[13/11]]
| style="text-align:center;" | m3
| style="text-align:center;" | F
| | meh
| | [[Quasitemp|Quasitemp]]
| |
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
| style="text-align:center;" | 321.95
| style="text-align:center;" | '''5.826'''
| | [[6/5|6/5]]
|-
| style="text-align:center;" | ^m3
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | ^F
| style="text-align:center;" | 6.310
| | me
| | [[Superkleismic|Superkleismic]]
| | 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 351.22
| style="text-align:center;" | 6.794
| | [[11/9|11/9]], [[27/22|27/22]]
| style="text-align:center;" | ~3
| style="text-align:center;" | ^^F, vGb
| | mu
| | [[Hemififths|Hemififths]]/[[karadeniz|karadeniz]]
| | 10-tone MOS: 5 2 5 5 2 5 5 5 2 5
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 380.49
| style="text-align:center;" | 7.285
| | [[5/4|5/4]]
| style="text-align:center;" | vM3
| style="text-align:center;" | vF#, Gb
| | mi
| | [[Magic|Magic]]/[[witchcraft|witchcraft]]
| | 10-tone MOS: 2 9 2 2 9 2 2 9 2 2
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 409.76
| style="text-align:center;" | 7.752
| | [[14/11|14/11]], [[81/64|81/64]]
| style="text-align:center;" | M3
| style="text-align:center;" | F#, ^Gb
| | maa
| | [[Hocus|Hocus]]
| |
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 439.02
| style="text-align:center;" | 7.769
| | [[9/7|9/7,]] [[32/25|32/25]]
| style="text-align:center;" | ^M3
| style="text-align:center;" | ^F#, vvG
| | mo
| |
| | 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| style="text-align:center;" | 468.29
| style="text-align:center;" | '''8.253'''
| | [[21/16|21/16]], [[13/10]]
|-
| style="text-align:center;" | v4
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | vG
| style="text-align:center;" | 10.122
| | fe
| | [[Barbad|Barbad]]
| |  
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 497.56
| style="text-align:center;" | 10.606
| | [[4/3|4/3]]
| style="text-align:center;" | P4
| style="text-align:center;" | G
| | fa
| | [[Schismatic|Schismatic]] ([[Helmholtz|helmholtz]], [[Garibaldi_temperament|garibaldi]], [[cassandra|cassandra]])
| |
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 526.83
| style="text-align:center;" | 11.225
| | [[15/11|15/11]], [[27/20|27/20]]
| style="text-align:center;" | ^4
| style="text-align:center;" | ^G
| | fih
| | [[Trismegistus|Trismegistus]]
| | 9-tone MOS: 5 5 3 5 5 5 5 3 5
|-
|-
| style="text-align:center;" | 19
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 556.1
| style="text-align:center;" | 13.595
| | [[11/8|11/8]]
| style="text-align:center;" | ~4
| style="text-align:center;" | ^^G, vAb
| | fu
| |
| |
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 585.37
| style="text-align:center;" | 14.079
| | [[7/5|7/5]]
|}
| style="text-align:center;" | vA4, d5
 
| style="text-align:center;" | vG#, Ab
=Relationship to 12-edo=
| | fi
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.
| | [[Pluto|Pluto]]
 
| |  
[[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"
|-
|-
| style="text-align:center;" | 21
![[Prime limit|Prime]]
| style="text-align:center;" | 614.63
[[Prime limit|Limit]]
| | [[10/7|10/7]]
! | [[Ratio]]
| style="text-align:center;" | A4, ^d5
! | [[Cents]]
| style="text-align:center;" | G#, ^Ab
! | [[Monzo]]
| | se
! colspan="2" |[[Color notation/Temperament Names|color name]]
| |
! | Name
| |
|-
|-
| style="text-align:center;" | 22
|3
| style="text-align:center;" | 643.90
| style="text-align:center;" |  
| | [[16/11|16/11]], [[13/9]]
| | 19.84
| style="text-align:center;" | ~5
| |<nowiki> | 65 -41 </nowiki>&gt;
| style="text-align:center;" | vvA
|Wa-41
| | su
|41-edo
| |
| | '41-tone' comma
| |  
|-
|-
| style="text-align:center;" | 23
|5
| style="text-align:center;" | 673.17
| style="text-align:center;" |  
| | [[22/15|22/15]], [[40/27|40/27]]
| | 57.27
| style="text-align:center;" | v5
| |<nowiki> | -5 -10 9 </nowiki>&gt;
| style="text-align:center;" | vA
|Tritriyo
| | sih
|y<sup>9</sup>
| |
| | shibboleth
| |  
|-
|-
| style="text-align:center;" | 24
|"
| style="text-align:center;" | 702.44
|
| | [[3/2|3/2]]
|31.57
| style="text-align:center;" | P5
|<nowiki> | -25 7 6 </nowiki>&gt;
| style="text-align:center;" | A
|Lala-tribiyo
| | sol
|LLy<sup>3</sup>
| |
|Ampersand's comma
| |
|-
|-
| style="text-align:center;" | 25
|"
| style="text-align:center;" | 731.71
| style="text-align:center;" | 3125/3072
| | [[32/21|32/21]], [[20/13]]
| | 29.61
| style="text-align:center;" | ^5
| |<nowiki> | -10 -1 5 </nowiki>&gt;
| style="text-align:center;" | ^A
|Laquinyo
| | si
|Ly<sup>5</sup>
| |
| | small diesis, magic comma
| |  
|-
|-
| style="text-align:center;" | 26
|"
| style="text-align:center;" | 760.98
| style="text-align:center;" | 20000/19683
| | [[14/9|14/9]], [[25/16|25/16]]
| | 27.66
| style="text-align:center;" | vm6
| |<nowiki> | 5 -9 4 </nowiki>&gt;
| style="text-align:center;" | ^^A, vBb
|Saquadyo
| | lo
|sy<sup>4</sup>
| |
| | minimal diesis, tetracot comma
| |  
|-
|-
| style="text-align:center;" | 27
|"
| style="text-align:center;" | 790.24
| style="text-align:center;" |  
| | [[11/7|11/7]], [[128/81|128/81]]
| | 25.71
| style="text-align:center;" | m6
| |<nowiki> | 20 -17 3 </nowiki>&gt;
| style="text-align:center;" | vA#, Bb
|Sasa-triyo
| | leh
|ssy<sup>3</sup>
| |
| | roda
| |  
|-
|-
| style="text-align:center;" | 28
|"
| style="text-align:center;" | 819.51
|32805/32768
| | [[8/5|8/5]]
|1.95
| style="text-align:center;" | ^m6
|<nowiki> | -15 8 1 </nowiki>&gt;
| style="text-align:center;" | A#, ^Bb
|Layo
| | le
|Ly
| |
|schisma
| |
|-
|-
| style="text-align:center;" | 29
|7
| style="text-align:center;" | 848.78
| style="text-align:center;" | 15625/15309
| | [[18/11|18/11]], [[44/27|44/27]]
| | 35.37
| style="text-align:center;" | ~6
| |<nowiki> | 0 -7 6 -1 </nowiki>&gt;
| style="text-align:center;" | ^A#, vvB
|Rutribiyo
| | lu
|ry<sup>6</sup>
| |
| | great BP diesis
| |  
|-
|-
| style="text-align:center;" | 30
|"
| style="text-align:center;" | 878.05
|
| | [[5/3|5/3]]
|22.41
| style="text-align:center;" | vM6
|<nowiki> | -10 7 8 -7 </nowiki>&gt;
| style="text-align:center;" | vB
|Lasepru-aquadbiyo
| | la
|Lr<sup>7</sup>y<sup>8</sup>
| |
|blackjackisma
| |
|-
|-
| style="text-align:center;" | 31
|"
| style="text-align:center;" | 907.32
| style="text-align:center;" | 875/864
| | [[27/16|27/16]]
| | 21.90
| style="text-align:center;" | M6
| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| style="text-align:center;" | B
|Zotriyo
| | laa
|zy<sup>3</sup>
| |
| | keema
| |  
|-
|-
| style="text-align:center;" | 32
|"
| style="text-align:center;" | 936.59
|3125/3087
| | [[12/7|12/7]]
|21.18
| style="text-align:center;" | ^M6
|<nowiki> | 0 -2 5 -3 </nowiki>&gt;
| style="text-align:center;" | ^B
|Triru-aquinyo
| | li
|r<sup>3</sup>y<sup>5</sup>
| |
|major BP diesis, gariboh
| |
|-
|-
| style="text-align:center;" | 33
|"
| style="text-align:center;" | 965.85
|
| | [[7/4|7/4]]
|19.95
| style="text-align:center;" | vm7
|<nowiki> | 10 -11 2 1 </nowiki>&gt;
| style="text-align:center;" | vC
|Sazoyoyo
| | ta
|szyy
| |
|tolerma
| |
|-
|-
| style="text-align:center;" | 34
|"
| style="text-align:center;" | 995.12
|33075/32768
| | [[16/9|16/9]]
|16.14
| style="text-align:center;" | m7
|<nowiki> | -15 3 2 2 </nowiki>&gt;
| style="text-align:center;" | C
|Labizoyo
| | teh
|Lzzyy
| |
|mirwomo comma
| |
|-
|-
| style="text-align:center;" | 35
|"
| style="text-align:center;" | 1024.39
|245/243
| | [[9/5|9/5]], [[20/11|20/11]]
|14.19
| style="text-align:center;" | ^m7
|<nowiki> | 0 -5 1 2 </nowiki>&gt;
| style="text-align:center;" | ^C
|Zozoyo
| | te
|zzy
| |  
|minor BP diesis, sensamagic
| |
|-
|"
|4000/3969
|13.47
|<nowiki> | 5 -4 3 -2 </nowiki>&gt;
|Rurutriyo
|rry<sup>3</sup>
|septimal semicomma, octagar
|-
|-
| style="text-align:center;" | 36
|"
| style="text-align:center;" | 1053.66
|
| | [[11/6|11/6]]
|9.15
| style="text-align:center;" | ~7
|<nowiki> | -15 0 -2 7 </nowiki>&gt;
| style="text-align:center;" | ^^C, vDb
|Lasepzo-agugu
| | tu
|Lz<sup>7</sup>gg
| |
|quince
| |
|-
|-
| style="text-align:center;" | 37
|"
| style="text-align:center;" | 1082.93
|1029/1024
| | [[15/8|15/8]]
|8.43
| style="text-align:center;" | vM7
|<nowiki> | -10 1 0 3 </nowiki>&gt;
| style="text-align:center;" | vC#, Db
|Latrizo
| | ti
|Lz<sup>3</sup>
| |
|gamelan residue, gamelisma
| |
|-
|"
|225/224
|7.71
|<nowiki> | -5 2 2 -1 </nowiki>&gt;
|Ruyoyo
|ryy
|septimal kleisma, marvel comma
|-
|-
| style="text-align:center;" | 38
|"
| style="text-align:center;" | 1112.195
|16875/16807
| | [[40/21|40/21]], [[21/11|21/11]]
|6.99
| style="text-align:center;" | M7
|<nowiki> | 0 3 4 -5 </nowiki>&gt;
| style="text-align:center;" | C#, ^Db
|Quinru-aquadyo
| | taa
|r<sup>5</sup>y<sup>4</sup>
| |
|small BP diesis, mirkwai
| |
|-
|-
| style="text-align:center;" | 39
|"
| style="text-align:center;" | 1141.46
|10976/10935
| | [[48/25|48/25]], [[27/14|27/14]], [[64/33|64/33]]
|6.48
| style="text-align:center;" | ^M7
|<nowiki> | 5 -7 -1 3 </nowiki>&gt;
| style="text-align:center;" | C#^, vvD
|Satrizo-agu
| | to
|sz<sup>3</sup>g
| |  
|hemimage
| |  
|-
|"
|5120/5103
|5.76
|<nowiki> | 10 -6 1 -1 </nowiki>&gt;
|Saruyo
|sry
|Beta 5, Garibaldi comma, hemifamity
|-
|"
|
|3.80
|<nowiki> | 25 -14 0 -1 </nowiki>&gt;
|Sasaru
|ssr
|Beta 2, septimal schisma, garischisma
|-
|-
| style="text-align:center;" | 40
|"
| style="text-align:center;" | 1170.73
|2401/2400
| | [[160/81|160/81]], [[63/32]]
|0.72
| style="text-align:center;" | v8
|<nowiki> | -5 -1 -2 4 </nowiki>&gt;
| style="text-align:center;" | vD
|Bizozogu
| | da
|z<sup>4</sup>gg
| |
|Breedsma
| |
|-
|-
| style="text-align:center;" | 41
|11
| style="text-align:center;" | 1200
|
| | 2/1
|29.72
| style="text-align:center;" | P8
|<nowiki> | 15 0 1 0 -5 </nowiki>&gt;
| style="text-align:center;" | D
|Saquinlu-ayo
| | do
|s1u<sup>5</sup>y
| |
|thuja comma
| |  
|-
|}
|"
 
|245/242
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
|21.33
 
|<nowiki> | -1 0 1 2 -2 </nowiki>&gt;
{| class="wikitable"
|Luluzozoyo
|1uuzzy
|cassacot
|-
|-
! | quality
|"
! | [[Kite's color notation|color]]
|100/99
! | monzo format
|17.40
! | examples
|<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
|Luyoyo
|1uyy
|Ptolemy's comma, ptolemisma
|-
|-
| style="text-align:center;" | downminor
|"
| style="text-align:center;" | zo
|1344/1331
| style="text-align:center;" | (a, b, 0, 1)
|16.83
| style="text-align:center;" | 7/6, 7/4
|<nowiki> | 6 1 0 1 -3 </nowiki>&gt;
|Trilu-azo
|1u<sup>3</sup>z
|hemimin
|-
|-
| style="text-align:center;" | minor
|"
| style="text-align:center;" | fourthward wa
|896/891
| style="text-align:center;" | (a, b) with b &lt; -1
|9.69
| style="text-align:center;" | 32/27, 16/9
|<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
|Saluzo
|s1uz
|undecimal semicomma, pentacircle (minthma * gentle)
|-
|-
| style="text-align:center;" | upminor
|"
| style="text-align:center;" | gu
|65536/65219
| style="text-align:center;" | (a, b, -1)
|8.39
| style="text-align:center;" | 6/5, 9/5
|<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
|Satrilu-aruru
|s1u<sup>3</sup>rr
|orgonisma
|-
|-
| style="text-align:center;" | mid
|"
| style="text-align:center;" | ilo
|243/242
| style="text-align:center;" | (a, b, 0, 0, 1)
|7.14
| style="text-align:center;" | 11/9, 11/6
|<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
|Lulu
|1uu
|neutral third comma, rastma
|-
|-
| style="text-align:center;" | "
|"
| style="text-align:center;" | lu
|385/384
| style="text-align:center;" | (a, b, 0, 0, -1)
|4.50
| style="text-align:center;" | 12/11, 18/11
|<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
|Lozoyo
|1ozg
|undecimal kleisma, keenanisma
|-
|-
| style="text-align:center;" | downmajor
|"
| style="text-align:center;" | yo
| style="text-align:center;" | 441/440
| style="text-align:center;" | (a, b, 1)
| | 3.93
| style="text-align:center;" | 5/4, 5/3
| |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
|Luzozogu
|1uzzg
| | Werckmeister's undecimal septenarian schisma, werckisma
|-
|-
| style="text-align:center;" | major
|"
| style="text-align:center;" | fifthward wa
| style="text-align:center;" | 1375/1372
| style="text-align:center;" | (a, b) with b &gt; 1
| | 3.78
| style="text-align:center;" | 9/8, 27/16
| |<nowiki> | -2 0 3 -3 1 </nowiki>&gt;
|Lotriruyo
|1or<sup>3</sup>y
| | moctdel
|-
|-
| style="text-align:center;" | upmajor
|"
| style="text-align:center;" | ru
| style="text-align:center;" | 540/539
| style="text-align:center;" | (a, b, 0, -1)
| | 3.21
| style="text-align:center;" | 9/7, 12/7
| |<nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
|}
|Lururuyo
All 41edo chords can be named using ups and downs. Here are the zo, gu, ilo, yo and ru triads:
|1urry
 
| | Swets' comma, swetisma
{| class="wikitable"
|-
|-
! | [[Kite's color notation|color of the 3rd]]
|"
! | JI chord
| style="text-align:center;" | 3025/3024
! | notes as edosteps
| | 0.57
! | notes of C chord
| |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
! | written name
|Loloruyoyo
! | spoken name
|1ooryy
| | Lehmerisma
|-
|-
| style="text-align:center;" | zo
|"
| style="text-align:center;" | 6:7:9
| style="text-align:center;" |  
| style="text-align:center;" | 0-9-24
| | 0.15
| style="text-align:center;" | C vEb G
| |<nowiki> | -1 2 -4 5 -2 </nowiki>&gt;
| style="text-align:center;" | Cvm
|Luluquinzo-aquadgu
| style="text-align:center;" | C downminor
|1uuz<sup>5</sup>g<sup>4</sup>
| | odiheim
|-
|-
| style="text-align:center;" | gu
|13
| style="text-align:center;" | 10:12:15
| style="text-align:center;" | 105/104
| style="text-align:center;" | 0-11-24
| | 16.57
| style="text-align:center;" | C ^Eb G
| |<nowiki> | -3 1 1 1 0 -1 </nowiki>&gt;
| style="text-align:center;" | C^m
|Thuzoyo
| style="text-align:center;" | C upminor
|3uzy
| | small tridecimal comma, animist
|-
|-
| style="text-align:center;" | ilo
|"
| style="text-align:center;" | 18:22:27
| style="text-align:center;" | 28672/28431
| style="text-align:center;" | 0-12-24
| | 14.61
| style="text-align:center;" | C vvE G
| |<nowiki> | 12 -7 0 1 0 -1 </nowiki>&gt;
| style="text-align:center;" | C~
|Sathuzo
| style="text-align:center;" | C mid
|s3uz
| | secorian
|-
|-
| style="text-align:center;" | yo
|"
| style="text-align:center;" | 4:5:6
| style="text-align:center;" | 275/273
| style="text-align:center;" | 0-13-24
| | 12.64
| style="text-align:center;" | C vE G
| |<nowiki> | 0 -1 2 -1 1 -1 </nowiki>&gt;
| style="text-align:center;" | Cv
|Thuloruyoyo
| style="text-align:center;" | C downmajor or C down
|3u1oryy
| | gassorma
|-
|"
| style="text-align:center;" | 144/143
| | 12.06
| |<nowiki> | 4 2 0 0 -1 -1 </nowiki>&gt;
|Thulu
|3u1u
| | grossma
|-
|-
| style="text-align:center;" | ru
|"
| style="text-align:center;" | 14:18:27
| style="text-align:center;" | 196/195
| style="text-align:center;" | 0-15-24
| | 8.86
| style="text-align:center;" | C ^E G
| |<nowiki> | 2 -1 -1 2 0 -1 </nowiki>&gt;
| style="text-align:center;" | C^
|Thuzozogu
| style="text-align:center;" | C upmajor or C up
|3uzzg
|}
| | mynucuma
0-10-20 = D F Ab = Ddim = D dim
|-
 
|"
0-10-21 = D F ^Ab = Ddim(^5) = D dim up-five
| style="text-align:center;" | 640/637
 
| | 8.13
0-10-22 = D F vvA = Dm(~5) = D minor mid-five
| |<nowiki> | 7 0 1 -2 0 -1 </nowiki>&gt;
 
|Thururuyo
0-10-23 = D F vA = Dm(v5) = D minor down-five
|3urry
 
| | huntma
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 double-up-five, or possibly Daug(vv5)
 
0-14-27 = D F# vA# = Daug(v5) = D aug down-five
 
0-14-28 = D F# A# is Daug = D aug
 
etc.
 
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].
 
This chart shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with xen. 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.
 
[[File:41-edo spiral.png|673x673px]]
 
The same spiral, but with notes not intervals:
 
[[File:41-edo spiral with notes.png|730x730px]]
 
==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 41edo (ordered by absolute error).
 
{| class="wikitable"
|-
|-
| | '''Interval, complement'''
|"
| | '''Error (abs., in [[cent|cents]])'''
| style="text-align:center;" | 1188/1183
|-
| | 7.30
| style="text-align:center;" | '''[[4/3|4/3]], [[3/2|3/2]]'''
| |<nowiki> | 2 3 0 -1 1 -2 </nowiki>&gt;
| style="text-align:center;" | '''0.484'''
|Thuthuloru
|3uu1or
| | kestrel comma
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
|"
| style="text-align:center;" | 0.968
| style="text-align:center;" | 325/324
| | 5.34
| |<nowiki> | -2 -4 2 0 0 1 </nowiki>&gt;
|Thoyoyo
|3oyy
| | marveltwin
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
|"
| style="text-align:center;" | 2.370
| style="text-align:center;" | 352/351
| | 4.93
| |<nowiki> | 5 -3 0 0 1 -1 </nowiki>&gt;
|Thulo
|3u1o
| | minthma
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
|"
| style="text-align:center;" | 2.854
| style="text-align:center;" | 364/363
|-
| | 4.76
| style="text-align:center;" | '''[[8/7|8/7]], [[7/4|7/4]]'''
| |<nowiki> | 2 -1 0 1 -2 1 </nowiki>&gt;
| style="text-align:center;" | '''2.972'''
|Tholuluzo
|3o1uuz
| | gentle comma
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
|"
| style="text-align:center;" | 3.456
| style="text-align:center;" | 847/845
| | 4.09
| |<nowiki> | 0 0 -1 1 2 -2 </nowiki>&gt;
|Thuthulolozogu
|3uu1oozg
| | cuthbert
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
|"
| style="text-align:center;" | 3.473
| style="text-align:center;" | 729/728
| | 2.38
| |<nowiki> | -3 6 0 -1 0 -1 </nowiki>&gt;
|Lathuru
|L3ur
| | squbema
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
|"
| style="text-align:center;" | 3.812
| style="text-align:center;" | 4096/4095
| | 0.42
| |<nowiki> | 12 -2 -1 -1 0 -1 </nowiki>&gt;
|Sathurugu
|s3urg
| | tridecimal schisma, Sagittal schismina
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
|"
| style="text-align:center;" | 3.940
| style="text-align:center;" | 10648/10647
| | 0.16
| |<nowiki> | 3 -2 0 -1 3 -2 </nowiki>&gt;
|Thuthutrilo-aru
|3uu1o<sup>3</sup>r
| | harmonisma
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
|17
| style="text-align:center;" | 4.296
| style="text-align:center;" | 2187/2176
| | 8.73
| |<nowiki> | -7 7 0 0 0 0 -1 </nowiki>&gt;
|Lasu
|L17u
| | septendecimal comma
|-
|-
| style="text-align:center;" | '''[[11/8|11/8]], [[16/11|16/11]]'''
|"
| style="text-align:center;" | '''4.780'''
|256/255
|6.78
|<nowiki> | 8 -1 -1 0 0 0 -1 </nowiki>&gt;
|Sugu
|17ug
|septendecimal kleisma
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
|"
| style="text-align:center;" | 5.342
|715/714
|2.42
|<nowiki> | -1 -1 1 -1 1 1 -1 </nowiki>&gt;
|Sutholoruyo
|17u3o1ory
|septendecimal bridge comma
|-
|-
| style="text-align:center;" | '''[[5/4|5/4]], [[8/5|8/5]]'''
|19
| style="text-align:center;" | '''5.826'''
| style="text-align:center;" | 210/209
| | 8.26
| |<nowiki> | 1 1 1 1 -1 0 0 -1 </nowiki>&gt;
|Nuluzoyo
|19u1uzy
| | spleen comma
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
|"
| style="text-align:center;" | 6.310
|1216/1215
|1.42
|<nowiki> | 6 -5 -1 0 0 0 0 1 </nowiki>&gt;
|Sanogu
|s19og
|Eratosthenes' comma
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
|"
| style="text-align:center;" | 6.794
| style="text-align:center;" | 513/512
| | 3.38
| |<nowiki> | -9 3 0 0 0 0 0 1 </nowiki>&gt;
|Lano
|L19o
| | undevicesimal comma, Boethius' comma
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
|23
| style="text-align:center;" | 7.285
|736/729
|16.54
|<nowiki> | 5 -6 0 0 0 0 0 0 1 </nowiki>&gt;
|Sa-twenty-tho
|s23o
|vicesimotertial comma
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
|29
| style="text-align:center;" | 7.752
|145/144
|11.98
|<nowiki> | -4 -2 1 0 0 0 0 0 0 1 </nowiki>&gt;
|Twenty-noyo
|29oy
|29th-partial chroma
|}
 
=Temperaments=
[[List_of_edo-distinct_41et_rank_two_temperaments|List of edo-distinct 41et rank two temperaments]]
 
Table of Temperaments by generator
 
{| class="wikitable"
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
! |Degree
| style="text-align:center;" | 7.769
! | Cents
! | Generator
! | Some MOS and MODMOS Scales Available
|-
|-
| style="text-align:center;" | '''[[16/13|16/13]], [[13/8|13/8]]'''
| style="text-align:center;" | 0
| style="text-align:center;" | '''8.253'''
| style="text-align:center;" | 0.00
| |
| |
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 1
| style="text-align:center;" | 10.122
| style="text-align:center;" | 29.27
| |
| |
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 2
| style="text-align:center;" | 10.606
| style="text-align:center;" | 58.54
| | [[Hemimiracle|Hemimiracle]]
| |
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 3
| style="text-align:center;" | 11.225
| style="text-align:center;" | 87.805
| | 88cET (approx),
 
[[Octacot|octacot]]
| |
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 4
| style="text-align:center;" | 13.595
| style="text-align:center;" | 117.07
| | [[Miracle|Miracle]]
| |
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 5
| style="text-align:center;" | 14.079
| style="text-align:center;" | 146.34
|}
| | [[Bohlen-Pierce|Bohlen-Pierce]]/[[bohpier|bohpier]]
 
| |  
=Instruments=
[[File:41-EDD_elektrische_gitaar.jpg|alt=41-EDD elektrische gitaar.jpg|560x745px|41-EDD elektrische gitaar.jpg]]
 
''41-EDO Electric guitar, by Gregory Sanchez.''
 
[[File:Ron_Sword_with_a_41ET_Guitar.jpg|alt=Ron_Sword_with_a_41ET_Guitar.jpg|Ron_Sword_with_a_41ET_Guitar.jpg]]
 
''41-EDO Classical guitar, by Ron Sword.''
 
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.
 
[[File:Caleb's Kite guitar.jpg|480x640px]]
 
A possible system to tune keyboards in 41EDO is discussed in http://launch.groups.yahoo.com/group/tuning/message/74155.
 
=Scales and modes=
 
A list of [[41edo_modes|41edo modes]] (MOS and others).
 
===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).
 
{| class="wikitable"
|-
|-
| | Overtones in "Mode 8":
| style="text-align:center;" | 6
| | 8
| style="text-align:center;" | 175.61
| | 9
| | [[Tetracot|Tetracot]]/[[bunya|bunya]]/[[Monkey|monkey]]
| | 10
| | 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5
| | 11
|-
| | 12
| style="text-align:center;" | 7
| | 13
| style="text-align:center;" | 204.88
| | 14
| | [[Baldy|Baldy]]
| | 15
| | 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1
| | 16
|-
|-
| style="text-align:center;" | 8
| | ...as JI Ratio from 1/1:
| style="text-align:center;" | 234.15
| | 1/1
| | [[Rodan|Rodan]]/[[guiron|guiron]]
| | 9/8
| | 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1
| | 5/4
|-
| | 11/8
| style="text-align:center;" | 9
| | 3/2
| style="text-align:center;" | 263.415
| | 13/8
| | [[Septimin|Septimin]]
| | 7/4
| | 9-tone MOS: 5 4 5 5 4 5 4 5 4
| | 15/8
|-
| | 2/1
| style="text-align:center;" | 10
|-
| style="text-align:center;" | 292.68
| | ...in cents:
| | [[Quasitemp|Quasitemp]]
| | 0
| |
| | 203.9
|-
| | 386.3
| style="text-align:center;" | 11
| | 551.3
| style="text-align:center;" | 321.95
| | 702.0
| | [[Superkleismic|Superkleismic]]
| | 840.5
| | 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 351.22
| | [[Hemififths|Hemififths]]/[[karadeniz|karadeniz]]
| | 10-tone MOS: 5 2 5 5 2 5 5 5 2 5
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 380.49
| | [[Magic|Magic]]/[[witchcraft|witchcraft]]
| | 10-tone MOS: 2 9 2 2 9 2 2 9 2 2
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 409.76
| | [[Hocus|Hocus]]
| |
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 439.02
| |
| | 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 468.29
| | [[Barbad|Barbad]]
| |
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 497.56
| | [[Schismatic|Schismatic]] ([[Helmholtz|helmholtz]], [[Garibaldi_temperament|garibaldi]], [[cassandra|cassandra]])
| |
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 526.83
| | [[Trismegistus|Trismegistus]]
| | 9-tone MOS: 5 5 3 5 5 5 5 3 5
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 556.1
| |
| |
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 585.37
| | [[Pluto|Pluto]]
| |
|}
 
=Scales and modes=
A list of [[41edo_modes|41edo modes]] (MOS and others).
 
===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).
 
{| class="wikitable"
|-
| | Overtones in "Mode 8":
| | 8
| | 9
| | 10
| | 11
| | 12
| | 13
| | 14
| | 15
| | 16
|-
| | ...as JI Ratio from 1/1:
| | 1/1
| | 9/8
| | 5/4
| | 11/8
| | 3/2
| | 13/8
| | 7/4
| | 15/8
| | 2/1
|-
| | ...in cents:
| | 0
| | 203.9
| | 386.3
| | 551.3
| | 702.0
| | 840.5
| | 968.8
| | 968.8
| | 1088.3
| | 1088.3
Line 1,607: Line 1,736:
|}
|}


==Notation==
==Instruments==
[[File:41-EDD_elektrische_gitaar.jpg|alt=41-EDD elektrische gitaar.jpg|560x745px|41-EDD elektrische gitaar.jpg]]


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:
''41-EDO Electric guitar, by Gregory Sanchez.''
 
[[File:Ron_Sword_with_a_41ET_Guitar.jpg|alt=Ron_Sword_with_a_41ET_Guitar.jpg|Ron_Sword_with_a_41ET_Guitar.jpg]]


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.
''41-EDO Classical guitar, by Ron Sword.''


Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.
The [[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.


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:Caleb's Kite guitar.jpg|480x640px]]


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 possible system to tune keyboards in 41EDO is discussed in http://launch.groups.yahoo.com/group/tuning/message/74155.


=Music=
=Music=
Retrieved from "https://en.xen.wiki/w/41edo"