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{{interwiki
| de = 46-EDO
| en = 46edo
| es =
| ja =
}}
{{Infobox ET}}
{{Infobox ET}}
The '''46 equal divisions of the octave''' ('''46edo'''), or the '''46(-tone) equal temperament''' ('''46tet''', '''46et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 46 [[equal]]ly-sized steps. Each step has a size of about 26.1 [[cent]]s, an interval close in size to [[66/65]], the interval from [[13/11]] to [[6/5]].
{{ED intro}}


== Theory ==
== Theory ==
46et tempers out [[507/500]], [[91/90]], [[686/675]], [[2048/2025]], [[121/120]], [[245/243]], [[126/125]], [[169/168]], [[176/175]], [[896/891]], [[196/195]], [[1029/1024]], [[5120/5103]], [[385/384]], and [[441/440]] among other intervals, with various consequences. [[Rank-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]] and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves. In the opinion of some, 46et is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo|41et]]. In fact, while 41 is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and zeta integral edo]] but not a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap edo]], 46 is zeta gap but not zeta peak or zeta integral.  
In the opinion of some, 46edo is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]] or [[53edo]]. 46edo's fifth is slightly sharp of just, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. Many say that sharp fifths give a characteristic bright sound to 5-limit triads, and consider the sound of meantone triads to be more mellow in comparison. 46edo is not a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], but it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta gap edo]]. It is distinctly [[consistent]] in the [[9-odd-limit]], and it is consistent to the [[13-odd-limit]] or the no-15 no-19 [[23-odd-limit]].


The fifth of 46 equal is 2.39 cents sharp, which some people (e.g. [[Margo Schulter]]) prefer, sometimes strongly, over both the [[3/2|just fifth]] and fifths of temperaments with flat fifths, such as meantone. It gives a characteristic bright sound to triads, distinct from the mellowness of a meantone triad.
46edo is also notable for being the smallest equal division to approximate harmonics 3, 5, 7, 11, and 13 with less than 25% [[relative interval error]].
 
[[Rank-2 temperament]]s it [[support]]s include [[sensi]], [[valentine]], [[shrutar]], [[rodan]], [[leapday]], and [[unidec]]. The [[11-odd-limit]] [[minimax tuning]] for valentine, (11/7)<sup>1/10</sup>, is only 0.01 cents flat of 3\46 octaves.  


[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.
[[Magic22 as srutis #Shrutar.5B22.5D_as_srutis|Shrutar22 as srutis]] describes a possible use of 46edo for [[Indian]] music.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|46}}
{{Harmonics in equal|46|columns=9}}
{{Harmonics in equal|46|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}


=== Divisors ===
=== Subsets and supersets ===
46edo can be treated as two [[23edo]]'s separated by an interval of 26.087 cents.
46edo can be treated as two circles of [[23edo]] separated by an interval of 26.087 cents.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4"
{| class="wikitable center-all right-2 left-3 left-4 left-7"
|-
|-
! #
! &#35;
! Cents
! Cents
! Approximate Ratios<nowiki>*</nowiki>
! Approximate ratios<ref name="interval ratios" group="note">{{sg|limit=2.3.5.7.11.13.17.23&nbsp;subgroup}} However, ratios of 15 are not included here, as except for 15/8 and 16/15 themselves 46edo has intervals involving the 15th harmonic poorly approximated in general. This is because, while the 3rd and 5th harmonics are sharp and their deviations from just intonation add up, 7, 11, and 13 are all tuned flat, making the difference even larger. This prevents it from being [[consistent]] in the [[15-odd-limit]], as there is a discrepancy approximating [[15/13]] and [[26/15]]&mdash;9\46 is closer to 15/13 by a hair, but 10\46 represents the difference between 46edo's 15/8 and 13/8 and is more likely to appear in chords actually functioning as 15/13.</ref>
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
! Solfege
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>5</sup>A1 and ^^d2)
! colspan="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)
! colspan="2" | [[Solfege]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| perfect unison
| perfect unison
| P1
| P1
| D
| D
| Perfect unison
| P1
| D
| da
| do
| do
|-
|-
| 1
| 1
| 26.087
| 26.1
| [[81/80]], [[64/63]], [[49/48]]
| [[81/80]], [[64/63]], [[49/48]]
| up unison
| up unison
| ^1
| ^1
| ^D
| ^D
| comma-wide unison, <br>super unison
| K1, S1
| KD, SD
| du
| di
| di
|-
|-
| 2
| 2
| 52.174
| 52.2
| [[28/27]], [[36/35]], [[33/32]]
| [[28/27]], [[36/35]], [[33/32]]
| downminor 2nd
| downminor 2nd
| vm2
| vm2
| vEb
| vEb
| subminor 2nd, uber unison
| sm2, U1
| sEb, UD
| fro
| ro
| ro
|-
|-
| 3
| 3
| 78.261
| 78.3
| [[25/24]], [[21/20]], [[22/21]], [[24/23]], [[23/22]]
| [[25/24]], [[21/20]], [[22/21]], [[24/23]], [[23/22]]
| minor 2nd
| minor 2nd
| m2
| m2
| Eb
| Eb
| minor 2nd, <br>classic augmented unison
| m2, kkA1
| Eb, kkD#
| fra
| rih
| rih
|-
|-
| 4
| 4
| 104.348
| 104.3
| [[16/15]], [[17/16]], [[18/17]]
| [[16/15]], [[17/16]], [[18/17]]
| upminor 2nd
| upminor 2nd
| ^m2
| ^m2
| ^Eb
| ^Eb
| classic minor 2nd, <br>comma-narrow aug unison
| Km2, kA1
| KEb, kD#
| fru
| ra
| ra
|-
|-
| 5
| 5
| 130.435
| 130.4
| [[13/12]], [[14/13]], [[15/14]]
| [[13/12]], [[14/13]], [[15/14]]
| downmid 2nd
| dupminor 2nd
| v~2
| ^^m2
| ^^Eb
| ^^Eb
| ru (as in supraminor)
| lesser neutral second, augmented unison
| n2, A1
| UEb, D#
| fri
| ru<ref name="intervals2" group="note">/u/ as in s'''u'''praminor</ref>
|-
|-
| 6
| 6
| 156.522
| 156.5
| [[12/11]], [[11/10]], [[23/21]]
| [[12/11]], [[11/10]], [[23/21]]
| upmid 2nd
| dudmajor 2nd
| ^~2
| vvM2
| vvE
| vvE
| ruh (as in submajor)
| greater neutral second, <br>super aug unison
| N2, sA1
| uE, sD#
| ri
| ruh<ref name="intervals3" group="note">/ʌ/ as in s'''u'''bmajor</ref>
|-
|-
| 7
| 7
| 182.609
| 182.6
| [[10/9]]
| [[10/9]]
| downmajor 2nd
| downmajor 2nd
| vM2
| vM2
| vE
| vE
| classic/comma-narrow major 2nd
| kM2
| kE
| ro
| reh
| reh
|-
|-
| 8
| 8
| 208.696
| 208.7
| [[9/8]]
| [[9/8]]
| major 2nd
| major 2nd
| M2
| M2
| E
| E
| major 2nd
| M2
| E
| ra
| re
| re
|-
|-
| 9
| 9
| 234.783
| 234.8
| [[8/7]], [[23/20]]
| [[8/7]], [[23/20]]
| upmajor 2nd
| upmajor 2nd
| ^M2
| ^M2
| ^E
| ^E
| supermajor 2nd
| SM2
| SE
| ru
| ri
| ri
|-
|-
| 10
| 10
| 260.870
| 260.9
| [[7/6]]
| [[7/6]]
| downminor 3rd
| downminor 3rd
| vm3
| vm3
| vF
| vF
| subminor 3rd
| sm3
| sF
| no
| ma
| ma
|-
|-
| 11
| 11
| 286.957
| 287.0
| [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
| minor 3rd
| minor 3rd
| m3
| m3
| F
| F
| minor 3rd
| m3
| F
| na
| meh
| meh
|-
|-
| 12
| 12
| 313.043
| 313.0
| [[6/5]]
| [[6/5]]
| upminor 3rd
| upminor 3rd
| ^m3
| ^m3
| ^F
| ^F
| classic minor 3rd
| Km3
| KF
| nu
| me
| me
|-
|-
| 13
| 13
| 339.130
| 339.1
| [[11/9]], [[17/14]], [[28/23]]
| [[11/9]], [[17/14]], [[28/23]]
| downmid 3rd
| dupminor 3rd
| v~3
| ^^m3
| ^^F
| ^^F
| mu
| lesser neutral 3rd
| n3
| UF
| ni
| mu<ref name="intervals2" group="note" />
|-
|-
| 14
| 14
| 365.217
| 365.2
| [[16/13]], [[26/21]], [[21/17]]
| [[16/13]], [[26/21]], [[21/17]]
| upmid 3rd
| dudmajor 3rd
| ^~3
| vvM3
| vvF#
| vvF#
| muh
| greater neutral 3rd
| N3
| uF#
| mi
| muh<ref name="intervals3" group="note" />
|-
|-
| 15
| 15
| 391.304
| 391.3
| [[5/4]]
| [[5/4]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
| vF#
| vF#
| classic major 3rd
| kM3
| kF#
| mo
| mi
| mi
|-
|-
| 16
| 16
| 417.391
| 417.4
| [[14/11]], [[23/18]]
| [[14/11]], [[23/18]]
| major 3rd
| major 3rd
| M3
| M3
| F#
| F#
| major 3rd
| M3
| F#
| ma
| maa
| maa
|-
|-
| 17
| 17
| 443.478
| 443.5
| [[9/7]], [[13/10]], [[22/17]]
| [[9/7]], [[13/10]], [[22/17]]
| upmajor 3rd
| upmajor 3rd
| ^M3
| ^M3
| ^F#
| ^F#
| supermajor 3rd
| SM3
| SF#
| mu
| mo
| mo
|-
|-
| 18
| 18
| 469.565
| 469.6
| [[21/16]], [[17/13]]
| [[21/16]], [[17/13]]
| down 4th
| down 4th
| v4
| v4
| vG
| vG
| sub 4th
| s4
| sG
| fo
| fe
| fe
|-
|-
| 19
| 19
| 495.652
| 495.7
| [[4/3]]
| [[4/3]]
| perfect 4th
| P4
| G
| perfect 4th
| perfect 4th
| P4
| P4
| G
| G
| fa
| fa
| fa
|-
|-
| 20
| 20
| 521.739
| 521.7
| [[27/20]], [[23/17]]
| [[27/20]], [[23/17]]
| up 4th
| up 4th
| ^4
| ^4
| ^G
| ^G
| comma-wide 4th
| K4
| KG
| fu
| fih
| fih
|-
|-
| 21
| 21
| 547.826
| 547.8
| [[11/8]]
| [[11/8]]
| downmid 4th
| dup 4th
| v~4
| ^^4
| ^^G
| ^^G
| uber 4th, sub diminished 5th
| U4, sd5
| UG, sAb
| fi/sho
| fu
| fu
|-
|-
| 22
| 22
| 573.913
| 573.9
| [[7/5]], [[18/13]], [[32/23]]
| [[7/5]], [[18/13]], [[32/23]]
| upmid 4th, dim 5th
| dudaug 4th, <br>dim 5th
| ^~4, d5
| vvA4, d5
| vvG#, Ab
| vvG#, Ab
| classic augmented 4th, diminished 5th
| kkA4, d5
| kkG#, Ab
| pi/sha
| fi
| fi
|-
|-
| 23
| 23
| 600.000
| 600.0
| [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
| downaug 4th, updim 5th
| downaug 4th, updim 5th
| vA4, ^d5
| vA4, ^d5
| vG#, ^Ab
| vG#, ^Ab
| comma-narrow aug 4th, <br>comma-wide dim 5th
| kA4, Kd5
| kG#, KAb
| po/shu
| seh
| seh
|-
|-
| 24
| 24
| 626.087
| 626.1
| [[10/7]], [[13/9]], [[23/16]]
| [[10/7]], [[13/9]], [[23/16]]
| aug 4th, downmid 5th
| aug 4th, dupdim 5th
| A4, v~5
| A4, ^^d5
| G#, ^^Ab
| G#, ^^Ab
| augmented 4th, <br>classic diminished 5th
| A4, KKd5
| G#, KKAb
| pa/shi
| se
| se
|-
|-
| 25
| 25
| 652.174
| 652.2
| [[16/11]]
| [[16/11]]
| upmid 5th
| dud 5th
| ^~5
| vv5
| vvA
| vvA
| super augmented 4th, <br>unter 5th
| SA4, u5
| SG#, uA
| pu/si
| su
| su
|-
|-
| 26
| 26
| 678.261
| 678.3
| [[40/27]], [[34/23]]
| [[40/27]], [[34/23]]
| down 5th
| down 5th
| v5
| v5
| vA
| vA
| comma-narrow 5th
| k5
| kA
| so
| sih
| sih
|-
|-
| 27
| 27
| 704.348
| 704.3
| [[3/2]]
| [[3/2]]
| perfect 5th
| perfect 5th
| P5
| P5
| A
| A
| perfect 5th
| P5
| A
| sa
| sol
| sol
|-
|-
| 28
| 28
| 730.435
| 730.4
| [[32/21]], [[26/17]]
| [[32/21]], [[26/17]]
| up 5th
| up 5th
| ^5
| ^5
| ^A
| ^A
| super 5th
| S5
| SA
| su
| si
| si
|-
|-
| 29
| 29
| 756.522
| 756.5
| [[14/9]], [[20/13]], [[17/11]]
| [[14/9]], [[20/13]], [[17/11]]
| downminor 6th
| downminor 6th
| vm6
| vm6
| vBb
| vBb
| subminor 6th
| sm6
| sBb
| flo
| lo
| lo
|-
|-
| 30
| 30
| 782.609
| 782.6
| [[11/7]]
| [[11/7]]
| minor 6th
| minor 6th
| m6
| m6
| Bb
| Bb
| minor 6th
| m6
| Bb
| fla
| leh
| leh
|-
|-
| 31
| 31
| 808.696
| 808.7
| [[8/5]]
| [[8/5]]
| upminor 6th
| upminor 6th
| ^m6
| ^m6
| ^Bb
| ^Bb
| classic minor 6th
| Km6
| KBb
| flu
| le
| le
|-
|-
| 32
| 32
| 834.783
| 834.8
| [[13/8]], [[21/13]], [[34/21]]
| [[13/8]], [[21/13]], [[34/21]]
| downmid 6th
| dupminor 6th
| v~6
| ^^m6
| ^^Bb
| ^^Bb
| lu
| lesser neutral 6th
| n6
| UBb
| fli
| lu<ref name="intervals2" group="note" />
|-
|-
| 33
| 33
| 860.870
| 860.9
| [[18/11]], [[28/17]], [[23/14]]
| [[18/11]], [[28/17]], [[23/14]]
| upmid 6th
| dudmajor 6th
| ^~6
| vvM6
| vvB
| vvB
| luh
| greater neutral 6th
| N6
| uB
| li
| luh<ref name="intervals3" group="note" />
|-
|-
| 34
| 34
| 886.957
| 887.0
| [[5/3]]
| [[5/3]]
| downmajor 6th
| downmajor 6th
| vM6
| vM6
| vB
| vB
| classic major 6th
| kM6
| kB
| lo
| la
| la
|-
|-
| 35
| 35
| 913.043
| 913.0
| [[22/13]], [[17/10]]
| [[22/13]], [[17/10]]
| major 6th
| major 6th
| M6
| M6
| B
| B
| major 6th
| M6
| B
| la
| laa
| laa
|-
|-
| 36
| 36
| 939.130
| 939.1
| [[12/7]]
| [[12/7]]
| upmajor 6th
| upmajor 6th
| ^M6
| ^M6
| ^B
| ^B
| supermajor 6th
| SM6
| SB
| lu
| li
| li
|-
|-
| 37
| 37
| 965.217
| 965.2
| [[7/4]], [[40/23]]
| [[7/4]], [[40/23]]
| downminor 7th
| downminor 7th
| vm7
| vm7
| vC
| vC
| subminor 7th
| sm7
| sC
| tho
| ta
| ta
|-
|-
| 38
| 38
| 991.304
| 991.3
| [[16/9]], [[23/13]]
| [[16/9]], [[23/13]]
| minor 7th
| minor 7th
| m7
| m7
| C
| C
| minor 7th
| m7
| C
| tha
| teh
| teh
|-
|-
| 39
| 39
| 1017.391
| 1017.4
| [[9/5]]
| [[9/5]]
| upminor 7th
| upminor 7th
| ^m7
| ^m7
| ^C
| ^C
| classic/comma-wide minor 7th
| Km7
| KC
| thu
| te
| te
|-
|-
| 40
| 40
| 1043.478
| 1043.5
| [[11/6]], [[20/11]], [[42/23]]
| [[11/6]], [[20/11]], [[42/23]]
| downmid 7th
| dupminor 7th
| v~7
| ^^m7
| ^^C
| ^^C
| tu
| lesser neutral 7th, sub diminished 8ve
| n7, sd8
| UC, sDb
| thi
| tu<ref name="intervals2" group="note" />
|-
|-
| 41
| 41
| 1069.565
| 1069.6
| [[24/13]], [[13/7]], [[28/15]]
| [[24/13]], [[13/7]], [[28/15]]
| upmid 7th
| dudmajor 7th
| ^~7
| vvM7
| vvC#
| vvC#
| tuh
| greater neutral 7th,
diminished 8ve
| N7, d8
| uC#, Db
| ti
| tuh<ref name="intervals3" group="note" />
|-
|-
| 42
| 42
| 1095.652
| 1095.7
| [[15/8]], [[32/17]], [[17/9]]
| [[15/8]], [[32/17]], [[17/9]]
| downmajor 7th
| downmajor 7th
| vM7
| vM7
| vC#
| vC#
| classic major 7th,
comma-wide dim 8ve
| kM7, Kd8
| kC#, KDb
| to
| ti
| ti
|-
|-
| 43
| 43
| 1121.739
| 1121.7
| [[48/25]], [[40/21]], [[21/11]], [[23/12]], [[44/23]]
| [[48/25]], [[40/21]], [[21/11]], [[23/12]], [[44/23]]
| major 7th
| major 7th
| M7
| M7
| C#
| C#
| major 7th, <br>classic diminished 8ve
| M7, KKd8
| C#, KKDb
| ta
| taa
| taa
|-
|-
| 44
| 44
| 1147.826
| 1147.8
| [[27/14]], [[35/18]], [[64/33]]
| [[27/14]], [[35/18]], [[64/33]]
| upmajor 7th
| upmajor 7th
| ^M7
| ^M7
| ^C#
| ^C#
| supermajor 7th, unter 8ve
| SM7, u8
| SC#, uD
| tu
| to
| to
|-
|-
| 45
| 45
| 1173.913
| 1173.9
| [[160/81]], [[63/32]], [[96/49]]
| [[160/81]], [[63/32]], [[96/49]]
| down 8ve
| down 8ve
| v8
| v8
| vD
| vD
| comma-narrow 8ve, sub 8ve
| k8/s8
| kD, sD
| do
| da
| da
|-
|-
| 46
| 46
| 1200.000
| 1200.0
| [[2/1]]
| [[2/1]]
| perfect 8ve
| perfect 8ve
| P8
| P8
| D
| D
| perfect 8ve
| P8
| D
| da
| do
| do
|}
|}
<nowiki>*</nowiki> Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46edo has the 15th harmonic poorly approximated in general, because, while both the 3rd and 5th harmonics are sharp by a fair amount and they add up, all the other primes are flat, making the difference even larger, to the extent that it is not [[consistent]] in the [[15-odd-limit]]. This can be demonstrated with the discrepancy approximating [[15/13]] (and its inversion [[26/15]]). 9\46edo is closer to 15/13 by a hair; 10\46edo represents the difference between, for instance, 46edo's 15/8 and 13/8, and is more likely to appear in chords actually functioning as 15/13.


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 center-all"
{| class="wikitable center-all"
|-
|-
! quality
! Quality
! color
! Color
! monzo format
! Monzo Format
! examples
! Examples
|-
|-
| downminor
| downminor
| zo
| zo
| {a, b, 0, 1}
| [a, b, 0, 1>
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| minor
| minor
| fourthward wa
| fourthward wa
| {a, b}, b &lt; -1
| [a, b>, b &lt; −1
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
| gu
| gu
| {a, b, -1}
| [a, b, −1>
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| downmid
| dupminor
| ilo
| ilo
| {a, b, 0, 0, 1}
| [a, b, 0, 0, 1>
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| upmid
| dudmajor
| lu
| lu
| {a, b, 0, 0, -1}
| [a, b, 0, 0, −1>
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| downmajor
| downmajor
| yo
| yo
| {a, b, 1}
| [a, b, 1>
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| major
| major
| fifthward wa
| fifthward wa
| {a, b}, b &gt; 1
| [a, b>, b &gt; 1
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| upmajor
| upmajor
| ru
| ru
| {a, b, 0, -1}
| [a, b, 0, −1>
| 9/7, 12/7
| 9/7, 12/7
|}
|}
All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. 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). Here are the zo, gu, ilo, lu, yo and ru triads:
All 46edo chords can be named using ups and downs. Alterations are always enclosed in parentheses, additions never are. Ups or downs immediately after the chord root affect the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Here are the zo, gu, ilo, lu, yo and ru triads:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! color of the 3rd
! 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
|-
|-
| zo
| zo
| 6:7:9
| 6:7:9
| 0-10-27
| 0–10–27
| C vEb G
| C vEb G
| Cvm
| Cvm
Line 471: Line 672:
| gu
| gu
| 10:12:15
| 10:12:15
| 0-12-27
| 0–12–27
| C ^Eb G
| C ^Eb G
| C^m
| C^m
Line 478: Line 679:
| ilo
| ilo
| 18:22:27
| 18:22:27
| 0-13-27
| 0–13–27
| C ^^Eb G
| C ^^Eb G
| Cv~
| C^^m
| C downmid
| C dupminor
|-
|-
| lu
| lu
| 22:27:33
| 22:27:33
| 0-14-27
| 0–14–27
| C vvE G
| C vvE G
| C^~
| Cvv
| C upmid
| C dudmajor or C dud
|-
|-
| yo
| yo
| 4:5:6
| 4:5:6
| 0-15-27
| 0–15–27
| C vE G
| C vE G
| Cv
| Cv
Line 499: Line 700:
| ru
| ru
| 14:18:21
| 14:18:21
| 0-17-27
| 0–17–27
| C ^E G
| C ^E G
| C^
| C^
| C upmajor or C up
| C upmajor or C up
|}
|}
For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].
For a more complete list, see [[Ups and downs notation #Chords and chord progressions]].


== Notation ==
== Notation ==
=== Sagittal ===
=== Ups and downs notation ===
The following table shows [[sagittal notation]] accidentals in one apotome for 46edo.
46edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Sharpness-sharp5a}}
Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp5}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[39edo #Sagittal notation|39edo]].
 
==== Evo flavor ====
<imagemap>
File:46-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 511 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:46-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:46-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 557 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:46-EDO_Revo_Sagittal.svg]]
</imagemap>
 
== Approximation to JI ==
[[File:46ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 15-limit intervals approximated in 46edo]]


{| class="wikitable center-all"
=== 17-odd-limit interval mappings ===
! Steps
{{Q-odd-limit intervals|46|17}}
| 0
| 1
| 2
| 3
| 4
| 5
|-
! Symbol
| [[File:Sagittal natural.png]]
| [[File:Sagittal pai.png]]
| [[File:Sagittal pakai.png]]
| [[File:Sagittal jatai.png]]
| [[File:Sagittal sharp pao.png]]
| [[File:Sagittal sharp.png]]
|}


== JI approximation ==
=== Consistent circles ===
=== 15-odd-limit interval mappings ===
46edo is home to a number of [[consistent circle]]s, both ones closing after generating all 46 notes and ones closing after generating 23edo.
The following table shows how [[15-odd-limit intervals]] are represented in 46edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
{| class="wikitable center-1 center-2 center-3"
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | 46-note circles by gen. with related temperaments organized by period
|+style=white-space:nowrap| 15-odd-limit intervals by direct mapping (even if inconsistent)
|-
|-
! Interval, complement
! [[Interval]] !! [[Closing error|Closing<br />Error]] !! [[Circle #Definitions|Consistency]] !! 1\1 !! 1\2
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
|-
| [[11/7]], [[14/11]]
| [[68/65]] || 25.9% || Normal || [[Valentine]] || [[Semivalentine]]
| 0.117
| 0.4
|-
|-
| [[9/5]], [[10/9]]
| [[10/9]] || 36.1% || Normal || [[Mitonic]] || [[Unidec]], [[hendec]]
| 0.205
| 0.8
|-
|-
| [[13/7]], [[14/13]]
| [[31/24]] || 70.2% || Weak || [[Sensible]], [[sensi]] add-31 || [[Bison]] add-31, [[bisensi]] add-31
| 2.137
|}
| 8.2
 
|-
{| class="wikitable center-all left-4"
| [[13/11]], [[22/13]]
|+ style="font-size: 105%;" | 23-note circles by gen. with related half-octave temperaments
| 2.253
| 8.6
|-
| '''[[3/2]], [[4/3]]'''
| '''2.393'''
| '''9.2'''
|-
| [[5/3]], [[6/5]]
| 2.598
| 10.0
|-
| '''[[11/8]], [[16/11]]'''
| '''3.492'''
| '''13.4'''
|-
| '''[[7/4]], [[8/7]]'''
| '''3.609'''
| '''13.8'''
|-
| [[9/8]], [[16/9]]
| 4.786
| 18.3
|-
| '''[[5/4]], [[8/5]]'''
| '''4.991'''
| '''19.1'''
|-
| '''[[13/8]], [[16/13]]'''
| '''5.745'''
| '''22.0'''
|-
| [[11/6]], [[12/11]]
| 5.885
| 22.6
|-
| [[7/6]], [[12/7]]
| 6.001
| 23.0
|-
| [[15/8]], [[16/15]]
| 7.383
| 28.3
|-
| [[13/12]], [[24/13]]
| 8.138
| 31.2
|-
| [[11/9]], [[18/11]]
| 8.278
| 31.7
|-
| [[9/7]], [[14/9]]
| 8.394
| 32.2
|-
|-
| [[11/10]], [[20/11]]
! [[Interval]] !! [[Closing error|Closing<br />Error]] !! [[Circle #Definitions|Consistency]] !! Temperaments
| 8.482
| 32.5
|-
|-
| [[7/5]], [[10/7]]
| [[17/16]] || 53.5% || Weak || [[Diaschismic]]
| 8.599
| 33.0
|-
|-
| [[18/13]], [[13/9]]
| [[23/21]] || 85.7% || Weak || [[Bison]]
| 10.531
| 40.4
|-
|-
| [[13/10]], [[20/13]]
| [[44/39]] || 12.3% || Super-strong || [[Abigail]]  
| 10.736
| 41.2
|-
|-
| [[15/11]], [[22/15]]
| [[21/17]] || 53.6% || Weak || ?
| 10.875
| 41.7
|-
|-
| [[15/14]], [[28/15]]
| [[14/11]] || 10.2% || Super-strong || ?
| 10.992
| 42.1
|-
|-
| ''[[15/13]], [[26/15]]''
| [[21/16]] || 107% || Sub-weak || ?
| ''12.958''
| ''49.7''
|}
|}
{{15-odd-limit|46}}
 
For the 23rd-octave temperament that 46edo supports which combines all above 23-note circles, see [[icositritonic]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 649: Line 793:
| 2.3
| 2.3
| {{monzo| 73 -46 }}
| {{monzo| 73 -46 }}
| [{{val| 46 73 }}]
| {{mapping| 46 73 }}
| -0.755
| −0.755
| 0.75
| 0.75
| 2.89
| 2.89
Line 656: Line 800:
| 2.3.5
| 2.3.5
| 2048/2025, 78732/78125
| 2048/2025, 78732/78125
| [{{val| 46 73 107 }}]
| {{mapping| 46 73 107 }}
| -1.219
| −1.219
| 0.90
| 0.90
| 3.45
| 3.45
Line 663: Line 807:
| 2.3.5.7
| 2.3.5.7
| 126/125, 245/243, 1029/1024
| 126/125, 245/243, 1029/1024
| [{{val| 46 73 107 129 }}]
| {{mapping| 46 73 107 129 }}
| -0.595
| −0.595
| 1.34
| 1.34
| 5.12
| 5.12
Line 670: Line 814:
| 2.3.5.7.11
| 2.3.5.7.11
| 121/120, 126/125, 176/175, 245/243
| 121/120, 126/125, 176/175, 245/243
| [{{val| 46 73 107 129 159 }}]
| {{mapping| 46 73 107 129 159 }}
| -0.274
| −0.274
| 1.36
| 1.36
| 5.20
| 5.20
Line 677: Line 821:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 91/90, 121/120, 169/168, 176/175, 245/243
| 91/90, 121/120, 169/168, 176/175, 245/243
| [{{val| 46 73 107 129 159 170 }}]
| {{mapping| 46 73 107 129 159 170 }}
| +0.030
| +0.030
| 1.41
| 1.41
Line 684: Line 828:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 91/90, 121/120, 154/153, 169/168, 176/175, 245/243
| 91/90, 121/120, 154/153, 169/168, 176/175, 245/243
| [{{val| 41 65 95 115 142 152 174 188 }}]
| {{mapping| 46 73 107 129 159 170 188 }}
| +0.047
| +0.047
| 1.31
| 1.31
Line 691: Line 835:
| 2.3.5.7.11.13.17.23
| 2.3.5.7.11.13.17.23
| 91/90, 121/120, 154/153, 169/168, 176/175, 208/207, 231/230
| 91/90, 121/120, 154/153, 169/168, 176/175, 208/207, 231/230
| [{{val| 41 65 95 115 142 152 174 188 208 }}]
| {{mapping| 46 73 107 129 159 170 188 208 }}
| +0.101
| +0.101
| 1.23
| 1.23
| 4.72
| 4.72
|}
|}
* 46et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-limit, and others. The next equal temperaments doing better in the aforementioned subgroups are 72, 72, 94, respectively. 46et is even more prominent in the no-19 23-limit, and the next equal temperament doing better in this subgroup is 140.


46et is lower in relative error than any previous equal temperaments in the 17-, 19-, 23-limit, and others. The next equal temperaments doing better in the aforementioned subgroups are 72, 72, 94, respectively. 46et is even more prominent in the no-19 23-limit, and the next equal temperament doing better in this subgroup is 140.  
=== Commas ===
This is a partial list of the [[commas]] that 46et [[tempering out|tempers out]] with its [[patent val]], {{val| 24 38 56 67 83 89 }}.
 
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| <abbr title="50331648/48828125">(16 digits)</abbr>
| {{monzo| 24 1 -11 }}
| 52.50
| Salegu
| [[Magus comma]]
|-
| 5
| <abbr title="1990656/1953125">(14 digits)</abbr>
| {{monzo| 13 5 -9 }}
| 32.95
| Satritrigu
| [[Valentine comma]]
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| <abbr title="1600000/1594323">(14 digits)</abbr>
| {{monzo| 9 -13 5 }}
| 6.15
| Saquinyo
| [[Amity comma]]
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.80
| Zotrigu
| Starling comma
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma, aberschisma
|-
| 7
| <abbr title="2147483648/2144153025">(20 digits)</abbr>
| {{monzo| 31 -6 -2 -6 }}
| 2.69
| Sasa-tribiru-agugu
| [[Pessoalisma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[121/120]]
| {{monzo|-3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[385/384]]
| {{monzo|-7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[169/168]]
| {{monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Thothoru
| Buzurgisma, dhanvantarisma
|-
| 13
| [[196/195]]
| {{monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[507/500]]
| {{monzo| -2 1 -3 0 0 2 }}
| 24.07
| Thothotrigu
|
|-
| 13
| [[351/350]]
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Thorugugu
| Ratwolfsma
|-
| 13
| [[352/351]]
| {{monzo| 5 -3 0 0 1 -1 }}
| 4.93
| Thulo
| Minor minthma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2 center-3"
{| class="wikitable center-1 center-2 center-3 center-4"
|-
|-
! Periods <br>per 8ve
! Periods<br>per 8ve
! Generator
! Generator
! Cents
! Cents
! Temperaments
! Temperaments
! MOS Scales
! MOS scales
! L:s
! L:s
|-
|-
| 1
| rowspan="11" | 1
| 1\46
| 1\46
| 26.087
| 26.087
Line 716: Line 1,047:
|  
|  
|-
|-
| 1
| 3\46
| 3\46
| 78.261
| 78.261
| [[Valentine]]
| [[Valentine]]
| 1L 14s (15-tone)<br>15L 1s (16-tone)<br>16L 15s (31-tone)
| 1L&nbsp;14s (15-tone)<br>15L&nbsp;1s (16-tone)<br>16L&nbsp;15s (31-tone)
| 4:3 ~ [[Maximal evenness|quasi-equal]]<br>3:1<br>2:1 ~ QE
| 4:3 ~ [[Maximal evenness|quasi-equal]]<br>3:1<br>2:1 ~ QE
|-
|-
| 1
| 5\46
| 5\46
| 130.435
| 130.435
| [[Twothirdtonic]]
| [[Twothirdtonic]]
| [[1L 8s]] (9-tone)<br>[[9L 1s]] (10-tone)<br>9L 10s (19-tone)<br>9L 19s (28-tone)<br>9L 28s (37-tone)
| [[1L&nbsp;8s]] (9-tone)<br>[[9L&nbsp;1s]] (10-tone)<br>9L&nbsp;10s (19-tone)<br>9L&nbsp;19s (28-tone)<br>9L&nbsp;28s (37-tone)
| 6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
| 6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 1
| 7\46
| 7\46
| 182.609
| 182.609
| [[Minortone]] / [[mitonic]]
| [[Minortone]] / [[mitonic]]
| [[1L 5s]] (6-tone)<br>[[6L 1s]] (7-tone)<br>7L 6s (13-tone)<br>13L 7s (20-tone)<br>13L 20s (33-tone)
| [[1L&nbsp;5s]] (6-tone)<br>[[6L&nbsp;1s]] (7-tone)<br>7L&nbsp;6s (13-tone)<br>13L&nbsp;7s (20-tone)<br>13L&nbsp;20s (33-tone)
| 11:7<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
| 11:7<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
|-
|-
| 1
| 9\46
| 9\46
| 234.783
| 234.783
| [[Rodan]]
| [[Rodan]]
| [[1L 4s]] (5-tone)<br>[[1L 5s]] (6-tone)<br>[[5L 6s]] (11-tone)<br>5L 11s (16-tone)<br>5L 16s (21-tone)<br>5L 21s (26-tone)<br>5L 26s (31-tone)<br>5L 31s (36-tone)<br>5L 36s (41-tone)
| [[1L&nbsp;4s]] (5-tone)<br>[[1L&nbsp;5s]] (6-tone)<br>[[5L&nbsp;6s]] (11-tone)<br>5L&nbsp;11s (16-tone)<br>5L&nbsp;16s (21-tone)<br>5L&nbsp;21s (26-tone)<br>5L&nbsp;26s (31-tone)<br>5L&nbsp;31s (36-tone)<br>5L&nbsp;36s (41-tone)
| 10:9 ~QE<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE, Pathological
| 10:9 ~QE<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 1
| 11\46
| 11\46
| 286.957
| 286.957
|  
| [[Amity_family#Gamity|Gamity]]
| [[4L 1s]] (5-tone)<br>[[4L 5s]] (9-tone)<br>4L 9s (13-tone)<br>4L 13s (17-tone)<br>4L 17s (21-tone)<br>21L 4s (25-tone)
| [[4L&nbsp;1s]] (5-tone)<br>[[4L&nbsp;5s]] (9-tone)<br>4L&nbsp;9s (13-tone)<br>4L&nbsp;13s (17-tone)<br>4L&nbsp;17s (21-tone)<br>21L&nbsp;4s (25-tone)
| 11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
|-
|-
| 1
| 13\46
| 13\46
| 339.130
| 339.130
| [[Amity]] / [[hitchcock]]
| [[Amity]] / [[hitchcock]]
| [[4L 3s]] (7-tone)<br>[[7L 4s]] (11-tone)<br>7L 11s (18-tone)<br>7L 18s (25-tone)<br>7L 25s (32-tone)<br>7L 32s (39-tone)
| [[4L&nbsp;3s]] (7-tone)<br>[[7L&nbsp;4s]] (11-tone)<br>7L&nbsp;11s (18-tone)<br>7L&nbsp;18s (25-tone)<br>7L&nbsp;25s (32-tone)<br>7L&nbsp;32s (39-tone)
| 7:6 ~ QE<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE! Pathological
| 7:6 ~ QE<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 1
| 15\46
| 15\46
| 391.304
| 391.304
| [[Magus]] / [[amigo]]
| [[Magus]] / [[amigo]]
| [[1L 2s]] (3-tone)<br>[[3L 1s]] (4-tone)<br>[[3L 4s]] (7-tone)<br>[[3L 7s]] (10-tone)<br>3L 10s (13-tone)<br>3L 13s (16-tone)<br>3L 16s (19-tone)<br>3L 19s (21-tone)<br>3L 21s (24-tone)<br>3L 24s (27-tone)<br>3L 27s (30-tone)<br>3L 30s (33-tone)<br>3L 33s (36-tone)<br>3L 36s (39-tone)<br>3L 39s (42-tone)
| [[1L&nbsp;2s]] (3-tone)<br>[[3L&nbsp;1s]] (4-tone)<br>[[3L&nbsp;4s]] (7-tone)<br>[[3L&nbsp;7s]] (10-tone)<br>3L&nbsp;10s (13-tone)<br>3L&nbsp;13s (16-tone)<br>3L&nbsp;16s (19-tone)<br>3L&nbsp;19s (21-tone)<br>3L&nbsp;21s (24-tone)<br>3L&nbsp;24s (27-tone)<br>3L&nbsp;27s (30-tone)<br>3L&nbsp;30s (33-tone)<br>3L&nbsp;33s (36-tone)<br>3L&nbsp;36s (39-tone)<br>3L&nbsp;39s (42-tone)
| 16:15 ~ QE<br>15:1<br>14:1<br>13:1<br>12:1<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
| 16:15 ~ QE<br>15:1<br>14:1<br>13:1<br>12:1<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 1
| 17\46
| 17\46
| 443.478
| 443.478
| [[Sensi]]
| [[Sensi]]
| [[3L 2s]] (5-tone)<br>[[3L 5s]] (8-tone)<br>[[8L 3s]] (11-tone)<br>8L 11s (19-tone)<br>19L 8s (27-tone)
| [[3L&nbsp;2s]] (5-tone)<br>[[3L&nbsp;5s]] (8-tone)<br>[[8L&nbsp;3s]] (11-tone)<br>8L&nbsp;11s (19-tone)<br>19L&nbsp;8s (27-tone)
| 12:5<br>7:5<br>5:2<br>3:2 ~ QE, Golden<br>2:1
| 12:5<br>7:5<br>5:2<br>3:2 ~ QE, Golden<br>2:1
|-
|-
| 1
| 19\46
| 19\46
| 495.652
| 495.652
| [[Leapday]]
| [[Leapday]]
| [[2L 3s]] (5-tone)<br>[[5L 2s]] (7-tone)<br>[[5L 7s]] (12-tone)<br>12L 5s (17-tone)<br>17L 12s (29-tone)
| [[2L&nbsp;3s]] (5-tone)<br>[[5L&nbsp;2s]] (7-tone)<br>[[5L&nbsp;7s]] (12-tone)<br>12L&nbsp;5s (17-tone)<br>17L&nbsp;12s (29-tone)
| 11:8<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 11:8<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
|-
|-
| 1
| 21\46
| 21\46
| 547.826
| 547.826
| [[Heinz]]
| [[Heinz]]
| [[2L 3s]] (5-tone)<br>[[2L 5s]] (7-tone)<br>[[2L 7s]] (9-tone)<br>[[2L 9s]] (11-tone)<br>11L 2s (13-tone)<br>11L 13s (24-tone)<br>11L 24s (35-tone)
| [[2L&nbsp;3s]] (5-tone)<br>[[2L&nbsp;5s]] (7-tone)<br>[[2L&nbsp;7s]] (9-tone)<br>[[2L&nbsp;9s]] (11-tone)<br>11L&nbsp;2s (13-tone)<br>11L&nbsp;13s (24-tone)<br>11L&nbsp;24s (35-tone)
| 17:4<br>13:4<br>9:4<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
| 17:4<br>13:4<br>9:4<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 2
| rowspan="11" | 2
| 1\46
| 1\46
| 26.087
| 26.087
Line 793: Line 1,114:
|  
|  
|-
|-
| 2
| 2\46
| 2\46
| 52.174
| 52.174
| [[Shrutar]]
| [[Shrutar]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[2L 10s]] (12-tone)<br>2L 12s (14-tone)<br>2L 14s (16-tone)<br>2L 16s (18-tone)<br>2L 18s (20-tone)<br>2L 20s (22-tone)<br>22L 2s (24-tone)
| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[2L&nbsp;8s]] (10-tone)<br>[[2L&nbsp;10s]] (12-tone)<br>2L&nbsp;12s (14-tone)<br>2L&nbsp;14s (16-tone)<br>2L&nbsp;16s (18-tone)<br>2L&nbsp;18s (20-tone)<br>2L&nbsp;20s (22-tone)<br>22L&nbsp;2s (24-tone)
| 21:2<br>19:2<br>17:2<br>15:2<br>13:2<br>11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 21:2<br>19:2<br>17:2<br>15:2<br>13:2<br>11:2<br>9:2<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
|-
|-
| 2
| 3\46
| 3\46
| 78.261
| 78.261
| [[Semivalentine]]
| [[Semivalentine]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[2L 10s]] (12-tone)<br>2L 12s (14-tone)<br>14L 2s (16-tone)<br>16L 14s (30-tone)
| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[2L&nbsp;8s]] (10-tone)<br>[[2L&nbsp;10s]] (12-tone)<br>2L&nbsp;12s (14-tone)<br>14L&nbsp;2s (16-tone)<br>16L&nbsp;14s (30-tone)
| 20:3<br>17:3<br>14:3<br>11:3<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 20:3<br>17:3<br>14:3<br>11:3<br>8:3<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
|-
|-
| 2
| 4\46
| 4\46
| 104.348
| 104.348
| [[Srutal]] / [[diaschismic]]
| [[Srutal]] / [[diaschismic]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[2L 8s]] (10-tone)<br>[[10L 2s]] (12-tone)<br>12L 10s (22-tone)<br>12L 22s (34-tone)
| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[2L&nbsp;8s]] (10-tone)<br>[[10L&nbsp;2s]] (12-tone)<br>12L&nbsp;10s (22-tone)<br>12L&nbsp;22s (34-tone)
| 19:4<br>15:4<br>11:4<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
| 19:4<br>15:4<br>11:4<br>7:4<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
|-
|-
| 2
| 5\46
| 5\46
| 130.435
| 130.435
|  
|  
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[2L 6s]] (8-tone)<br>[[8L 2s]] (10-tone)<br>8L 10s (18-tone)<br>18L 10s (28-tone)
| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[2L&nbsp;6s]] (8-tone)<br>[[8L&nbsp;2s]] (10-tone)<br>8L&nbsp;10s (18-tone)<br>18L&nbsp;10s (28-tone)
| 18:5<br>13:5<br>8:5 ~ Golden<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 18:5<br>13:5<br>8:5 ~ Golden<br>5:3 ~ Golden<br>3:2 ~ QE, Golden<br>2:1 ~ QE
|-
|-
| 2
| 6\46
| 6\46
| 156.522
| 156.522
| [[Bison]]
| [[Bison]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>8L 6s (14-tone)<br>8L 14s (22-tone)<br>8L 22s (30-tone)<br>8L 30s (38-tone
| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[6L&nbsp;2s]] (8-tone)<br>8L&nbsp;6s (14-tone)<br>8L&nbsp;14s (22-tone)<br>8L&nbsp;22s (30-tone)<br>8L&nbsp;30s (38-tone
| 17:6<br>11:6<br>6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE, Pathological
| 17:6<br>11:6<br>6:5 ~ QE<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 2
| 7\46
| 7\46
| 182.609
| 182.609
| [[Unidec]] / [[hendec]]
| [[Unidec]] / [[hendec]]
| 2L 2s (4-tone)<br>[[2L 4s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>6L 8s (14-tone)<br>6L 14s (20-tone)<br>20L 6s (26-tone)
| 2L&nbsp;2s (4-tone)<br>[[2L&nbsp;4s]] (6-tone)<br>[[6L&nbsp;2s]] (8-tone)<br>6L&nbsp;8s (14-tone)<br>6L&nbsp;14s (20-tone)<br>20L&nbsp;6s (26-tone)
| 16:7<br>9:7<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
| 16:7<br>9:7<br>7:2<br>5:2<br>3:2 ~ QE, Golden<br>2:1 ~ QE
|-
|-
| 2
| 8\46
| 8\46
| 208.696
| 208.696
| [[Abigail]]
| [[Abigail]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[6L 2s]] (8-tone)<br>6L 8s (14-tone)<br>6L 14s (20-tone)<br>6L 20s (26-tone)<br>6L 26s (32-tone)<br>6L 32s (38-tone)<br>6L 38s (44-tone)
| 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[6L&nbsp;2s]] (8-tone)<br>6L&nbsp;8s (14-tone)<br>6L&nbsp;14s (20-tone)<br>6L&nbsp;20s (26-tone)<br>6L&nbsp;26s (32-tone)<br>6L&nbsp;32s (38-tone)<br>6L&nbsp;38s (44-tone)
| 15:8<br>8:7 ~ QE<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
| 15:8<br>8:7 ~ QE<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 2
| 9\46
| 9\46
| 234.783
| 234.783
| [[Echidnic]]
| [[Echidnic]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[6L 4s]] (10-tone)<br>10L 6s (16-tone)<br>10L 16s (26-tone)<br>10L 26s (36-tone)
| 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[6L&nbsp;4s]] (10-tone)<br>10L&nbsp;6s (16-tone)<br>10L&nbsp;16s (26-tone)<br>10L&nbsp;26s (36-tone)
| 14:9<br>9:5<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
| 14:9<br>9:5<br>5:4 ~ QE<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 2
| 10\46
| 10\46
| 260.87
| 260.87
| [[Bamity]]
| [[Bamity]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[4L 6s]] (10-tone)<br>4L 10s (14-tone)<br>14L 4s (18-tone)<br>14L 18s (32-tone)
| 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[4L&nbsp;6s]] (10-tone)<br>4L&nbsp;10s (14-tone)<br>14L&nbsp;4s (18-tone)<br>14L&nbsp;18s (32-tone)
| 13:10<br>10:3<br>7:3<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
| 13:10<br>10:3<br>7:3<br>4:3 ~ QE<br>3:1<br>2:1 ~ QE
|-
|-
| 2
| 11\46
| 11\46
| 286.957
| 286.957
| [[Vines]]
| [[Vines]]
| 2L 2s (4-tone)<br>[[4L 2s]] (6-tone)<br>[[4L 6s]] (10-tone)<br>4L 10s (14-tone)<br>4L 14s (18-tone)<br>4L 18s (22-tone)<br>4L 22s (26-tone)<br>4L 26s (30-tone)<br>4L 30s (34-tone)<br>4L 34s (38-tone)<br>4L 38s (42-tone)
| 2L&nbsp;2s (4-tone)<br>[[4L&nbsp;2s]] (6-tone)<br>[[4L&nbsp;6s]] (10-tone)<br>4L&nbsp;10s (14-tone)<br>4L&nbsp;14s (18-tone)<br>4L&nbsp;18s (22-tone)<br>4L&nbsp;22s (26-tone)<br>4L&nbsp;26s (30-tone)<br>4L&nbsp;30s (34-tone)<br>4L&nbsp;34s (38-tone)<br>4L&nbsp;38s (42-tone)
| 12:11 ~ QE<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1 ~ Pathological<br>2:1 ~ QE, Pathological
| 12:11 ~ QE<br>11:1<br>10:1<br>9:1<br>8:1<br>7:1<br>6:1<br>5:1<br>4:1<br>3:1<br>2:1 ~ QE
|-
|-
| 23
| 23
Line 872: Line 1,183:


== Scales ==
== Scales ==
{{Main| List of MOS scales in 46edo }}
* [[Plum]]
* [[Plum]]
* [[Compdye]]
* [[Smi2s]]
; Sensi
* [[Sensi5]]
* [[Sensi5]]
* [[Sensi8]]
* [[Sensi8]]
* [[Sensi11]]
* [[Sensi11]]
* [[Sensi19]]
* [[Sensi19]]
; Elfleapday
* [[Elfleapday7]]
* [[Elfleapday8d]]
* [[Elfleapday9]]
* [[Elfleapday10]]
* [[Elfleapday12]]
* [[Elfleapday12f]]
; Elfsensus
* [[Elfsensus7]]
* [[Elfsensus8d]]
* [[Elfsensus9]]
* [[Elfsensus10]]
* [[Elfsensus12]]
* [[Elfsensus12f]]


=== Harmonic scales ===
=== Harmonic scales ===
46edo represents [[harmonic series|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. In steps-in-between, that's 8, 7, 6, 6, 5, 5, 5, 4.
46edo represents [[harmonic series|overtones]] 8 through 16 (written as [[JI]] ratios 8:9:10:11:12:13:14:15:16) with degrees 0, 8, 15, 21, 27, 32, 37, 42, 46. This corresponds to scale steps of 8, 7, 6, 6, 5, 5, 5, 4.
* 8\46 (208.696{{c}}) stands in for frequency ratio [[9/8]] (203.910{{c}}).
* 7\46 (182.609{{c}}) stands in for [[10/9]] (182.404{{c}}).
* 6\46 (156.522{{c}}) stands in for [[11/10]] (165.004{{c}}) and [[12/11]] (150.637{{c}}).
* 5\46 (130.435{{c}}) stands in for [[13/12]] (138.573{{c}}), [[14/13]] (128.298{{c}}) and [[15/14]] (119.443{{c}}).
* 4\46 (104.348{{c}}) stands in for [[16/15]] (111.731{{c}}).
 
{| class="wikitable center-all"
|-
! Harmonic
! Note (starting from C)
|-
! 1
| style="font-size: 16px;" | C
|-
! 3
| style="font-size: 16px;" | G
|-
! 5
| style="font-size: 16px;" | E{{naturaldown|40}}
|-
! 7
| style="font-size: 16px;" | G&#x1D12A;, B{{flatdown|40}}
|-
! 9
| style="font-size: 16px;" | D
|-
! 11
| style="font-size: 16px;" | E&#x266F;, F{{naturalup2|40}}
|-
! 13
| style="font-size: 16px;" | G&#x266F;, A{{flatup2|40}}
|-
! 15
| style="font-size: 16px;" | B{{naturaldown|40}}
|}
 
== Instruments ==
=== Lumatone ===
* [[Lumatone mapping for 46edo]]
 
=== Skip fretting ===
'''[[Skip fretting system 46 2 11]]''' is a [[skip fretting]] system for playing 46-edo on a 23-edo stringed instrument.
 
 
'''Skip fretting system 46 7 11''' is another skip fretting system for 46edo. The examples on this page are for 7-string [[guitar]].
 
; Harmonics
1/1: string 2 open


* 8\46 (208.696¢) stands in for frequency ratio [[9/8]] (203.910¢).
2/1: string 3 fret 5
* 7\46 (182.609¢) stands in for [[10/9]] (182.404¢).
 
* 6\46 (156.522¢) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).
3/2: not easily accessible
* 5\46 (130.435¢) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).
 
* 4\46 (104.348¢) stands in for [[16/15]] (111.731¢).
5/4: string 5 fret 4


== Music ==
== Music ==
; [[Aaron Krister Johnson]]
=== Modern renditions ===
* [https://aaronkristerjohnson.bandcamp.com/track/satiesque ''Satiesque''] (2014)
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=wMEdFl2puL0 "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)
* [https://www.youtube.com/watch?v=4yetEubmIk0 "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=_YiCc81ktBI ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
 
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=GXIoWsvFMzI ''Maple Leaf Rag''] (1899) – with syntonic comma adjustment, arranged for harpsichord and rendered by Claudi Meneghin (2024)
 
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=SCpbbov1fSQ ''music box 46edo''] (2025)  


; [[Gene Ward Smith]]
; [[Jake Freivald]] ([https://soundcloud.com/jdfreivald site])
* [https://www.archive.org/details/Chromosounds ''Chromosounds''] [http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3 play]
* [https://soundcloud.com/jdfreivald/a-seed-planted-yet-another ''A Seed Planted - (Yet another version: 46 EDO)''] [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/a_seed_planted-starling-46-EDO.mp3 play]{{dead link}}
* [https://www.archive.org/details/MusicForYourEars ''Music For Your Ears''] [https://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] – The central portion is in [[27edo]]; the rest is in 46edo.


; [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
Line 901: Line 1,293:
* [https://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-2 ''Hypnocloudsmack 2''] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2009%20Hypnocloudsmack%202.mp3 play]{{dead link}} (2012)
* [https://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-2 ''Hypnocloudsmack 2''] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2009%20Hypnocloudsmack%202.mp3 play]{{dead link}} (2012)
* [https://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-3 ''Hypnocloudsmack 3''] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2013%20Hypnocloudsmack%203.mp3 play]{{dead link}} (2012)
* [https://andrewheathwaite.bandcamp.com/track/hypnocloudsmack-3 ''Hypnocloudsmack 3''] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2013%20Hypnocloudsmack%203.mp3 play]{{dead link}} (2012)
; [[Aaron Krister Johnson]]
* [https://aaronkristerjohnson.bandcamp.com/track/satiesque ''Satiesque''] (2014)


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Bach_BWV_1029_E46-Alto-Sax-+-Harpsichord.mp3 Bach BWV 1029 in 46 equal] Claudi Meneghin version
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Bach_BWV_1029_E46-Alto-Sax-+-Harpsichord.mp3 Bach BWV 1029 in 46 equal] Claudi Meneghin version
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Bach_Contrapunctus_4-Jeux14-E46.mp3 Bach Contrapunctus 4] Claudi Meneghin version
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Bach_Contrapunctus_4-Jeux14-E46.mp3 Bach Contrapunctus 4] Claudi Meneghin version
* [https://www.youtube.com/watch?v=-6ORgyqom5M ''Chaconne et Fugue à 5 "Les Regrets"'']
* [https://www.youtube.com/watch?v=1r55c2nppe8 ''El Rossinyol'']
* [https://www.youtube.com/watch?v=JwjoT6ceU20 ''Arietta with 5 Variations'', for Organ]
* [https://www.youtube.com/watch?v=rmgWC_jruSg ''Fugue in 46edo (as Sensi)'', for two organs] (2024)
; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/light-at-the-end Light at the End]'' (2020)
; [[Joseph Monzo]]
* [https://www.youtube.com/watch?v=gL90m2ri5kw ''I Love the Blackness (unfinished)''] (2008)
; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=QDFLKyTXZgs ''Fugue in 46EDO Bohlen-Pierce-Stearns 9 (Sensi Extension) LssLsLsLs "Lambda"'']
; [[Gene Ward Smith]]
* [https://www.archive.org/details/Chromosounds ''Chromosounds''] [http://clones.soonlabel.com/public/micro/gene_ward_smith/chromosounds/GWS-GPO-Jazz-chromosounds.mp3 play]
* [https://www.archive.org/details/MusicForYourEars ''Music For Your Ears''] [https://www.archive.org/download/MusicForYourEars/musicfor.mp3 play] – The central portion is in [[27edo]]; the rest is in 46edo.


; [[Jake Freivald]] ([https://soundcloud.com/jdfreivald site])
; [[Tristan Bay]]
* [https://soundcloud.com/jdfreivald/a-seed-planted-yet-another ''A Seed Planted - (Yet another version: 46 EDO)''] [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/a_seed_planted-starling-46-EDO.mp3 play]{{dead link}}
* [https://www.youtube.com/watch?v=s61YY80E_IA ''Catalyst''] (2025)
 
; vivi mouse
* [https://soundcloud.com/vivi-mouse-emoji/valentines-day ''valentine's day''] (2025)


; [[Inthar lus Lăneaf]]
== Notes ==
* [[:File:Locrian Suite Gavotte.mp3|''Locrian Suite Gavotte'']] (2020)
<references group="note" />


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[[Category:Diaschismic]]
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