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}}
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|46}}
{{ED intro}}


== Theory ==
== Theory ==
In the opinion of some{{Who}}, 46edo is the first equal division to deal adequately with the [[13-limit]], though others award that distinction to [[41edo]]. 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. (Their sum, [[87edo]], does better than both in the 13-limit at the expense of a high note count.) Like 41, 46 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]]. 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.
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]].


[[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.  
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|prec=2}}
{{Harmonics in equal|46|columns=9}}
{{Harmonics in equal|46|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Line 26: Line 29:
! &#35;
! &#35;
! Cents
! Cents
! Approximate Ratios<ref name="interval ratios" group="note">Based on treating 46edo as a 2.3.5.7.11.13.17.23 subgroup, without ratios of 15 (except the superparticulars). 46edo has intervals involving the 15th harmonic poorly approximated, except for 15/8 and 16/15 themselves, 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>
! 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]]
! colspan="3" | [[SKULO interval names| SKULO notation]] (K or S = 1, U = 2)
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>5</sup>A1 and ^^d2)
! colspan="2" | [[Solfege| Solfeges]]
! 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
Line 44: Line 48:
|-
|-
| 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,  
| comma-wide unison, <br>super unison
super unison
| K1, S1
| K1, S1
| KD, SD
| KD, SD
Line 57: Line 60:
|-
|-
| 2
| 2
| 52.174
| 52.2
| [[28/27]], [[36/35]], [[33/32]]
| [[28/27]], [[36/35]], [[33/32]]
| downminor 2nd
| downminor 2nd
Line 69: Line 72:
|-
|-
| 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,  
| minor 2nd, <br>classic augmented unison
classic augmented unison
| m2, kkA1
| m2, kkA1
| Eb, kkD#
| Eb, kkD#
Line 82: Line 84:
|-
|-
| 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,  
| classic minor 2nd, <br>comma-narrow aug unison
comma-narrow aug unison
| Km2, kA1
| Km2, kA1
| KEb, kD#
| KEb, kD#
Line 95: Line 96:
|-
|-
| 5
| 5
| 130.435
| 130.4
| [[13/12]], [[14/13]], [[15/14]]
| [[13/12]], [[14/13]], [[15/14]]
| dupminor 2nd
| dupminor 2nd
Line 107: Line 108:
|-
|-
| 6
| 6
| 156.522
| 156.5
| [[12/11]], [[11/10]], [[23/21]]
| [[12/11]], [[11/10]], [[23/21]]
| dudmajor 2nd
| dudmajor 2nd
| vvM2
| vvM2
| vvE
| vvE
| greater neutral second,  
| greater neutral second, <br>super aug unison
super aug unison
| N2, sA1
| N2, sA1
| uE, sD#
| uE, sD#
Line 120: Line 120:
|-
|-
| 7
| 7
| 182.609
| 182.6
| [[10/9]]
| [[10/9]]
| downmajor 2nd
| downmajor 2nd
Line 132: Line 132:
|-
|-
| 8
| 8
| 208.696
| 208.7
| [[9/8]]
| [[9/8]]
| major 2nd
| major 2nd
Line 144: Line 144:
|-
|-
| 9
| 9
| 234.783
| 234.8
| [[8/7]], [[23/20]]
| [[8/7]], [[23/20]]
| upmajor 2nd
| upmajor 2nd
Line 156: Line 156:
|-
|-
| 10
| 10
| 260.870
| 260.9
| [[7/6]]
| [[7/6]]
| downminor 3rd
| downminor 3rd
Line 168: Line 168:
|-
|-
| 11
| 11
| 286.957
| 287.0
| [[13/11]], [[20/17]]
| [[13/11]], [[20/17]]
| minor 3rd
| minor 3rd
Line 180: Line 180:
|-
|-
| 12
| 12
| 313.043
| 313.0
| [[6/5]]
| [[6/5]]
| upminor 3rd
| upminor 3rd
Line 192: Line 192:
|-
|-
| 13
| 13
| 339.130
| 339.1
| [[11/9]], [[17/14]], [[28/23]]
| [[11/9]], [[17/14]], [[28/23]]
| dupminor 3rd
| dupminor 3rd
Line 204: Line 204:
|-
|-
| 14
| 14
| 365.217
| 365.2
| [[16/13]], [[26/21]], [[21/17]]
| [[16/13]], [[26/21]], [[21/17]]
| dudmajor 3rd
| dudmajor 3rd
Line 216: Line 216:
|-
|-
| 15
| 15
| 391.304
| 391.3
| [[5/4]]
| [[5/4]]
| downmajor 3rd
| downmajor 3rd
Line 228: Line 228:
|-
|-
| 16
| 16
| 417.391
| 417.4
| [[14/11]], [[23/18]]
| [[14/11]], [[23/18]]
| major 3rd
| major 3rd
Line 240: Line 240:
|-
|-
| 17
| 17
| 443.478
| 443.5
| [[9/7]], [[13/10]], [[22/17]]
| [[9/7]], [[13/10]], [[22/17]]
| upmajor 3rd
| upmajor 3rd
Line 252: Line 252:
|-
|-
| 18
| 18
| 469.565
| 469.6
| [[21/16]], [[17/13]]
| [[21/16]], [[17/13]]
| down 4th
| down 4th
Line 264: Line 264:
|-
|-
| 19
| 19
| 495.652
| 495.7
| [[4/3]]
| [[4/3]]
| perfect 4th
| perfect 4th
Line 276: Line 276:
|-
|-
| 20
| 20
| 521.739
| 521.7
| [[27/20]], [[23/17]]
| [[27/20]], [[23/17]]
| up 4th
| up 4th
Line 288: Line 288:
|-
|-
| 21
| 21
| 547.826
| 547.8
| [[11/8]]
| [[11/8]]
| dup 4th
| dup 4th
Line 300: Line 300:
|-
|-
| 22
| 22
| 573.913
| 573.9
| [[7/5]], [[18/13]], [[32/23]]
| [[7/5]], [[18/13]], [[32/23]]
| dudaug 4th,  
| dudaug 4th, <br>dim 5th
dim 5th
| vvA4, d5
| vvA4, d5
| vvG#, Ab
| vvG#, Ab
Line 313: Line 312:
|-
|-
| 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,  
| comma-narrow aug 4th, <br>comma-wide dim 5th
comma-wide dim 5th
| kA4, Kd5
| kA4, Kd5
| kG#, KAb
| kG#, KAb
Line 326: Line 324:
|-
|-
| 24
| 24
| 626.087
| 626.1
| [[10/7]], [[13/9]], [[23/16]]
| [[10/7]], [[13/9]], [[23/16]]
| aug 4th, dupdim 5th
| aug 4th, dupdim 5th
| A4, ^^d5
| A4, ^^d5
| G#, ^^Ab
| G#, ^^Ab
| augmented 4th,  
| augmented 4th, <br>classic diminished 5th
classic diminished 5th
| A4, KKd5
| A4, KKd5
| G#, KKAb
| G#, KKAb
Line 339: Line 336:
|-
|-
| 25
| 25
| 652.174
| 652.2
| [[16/11]]
| [[16/11]]
| dud 5th
| dud 5th
| vv5
| vv5
| vvA
| vvA
| super augmented 4th,  
| super augmented 4th, <br>unter 5th
unter 5th
| SA4, u5
| SA4, u5
| SG#, uA
| SG#, uA
Line 352: Line 348:
|-
|-
| 26
| 26
| 678.261
| 678.3
| [[40/27]], [[34/23]]
| [[40/27]], [[34/23]]
| down 5th
| down 5th
Line 364: Line 360:
|-
|-
| 27
| 27
| 704.348
| 704.3
| [[3/2]]
| [[3/2]]
| perfect 5th
| perfect 5th
Line 376: Line 372:
|-
|-
| 28
| 28
| 730.435
| 730.4
| [[32/21]], [[26/17]]
| [[32/21]], [[26/17]]
| up 5th
| up 5th
Line 388: Line 384:
|-
|-
| 29
| 29
| 756.522
| 756.5
| [[14/9]], [[20/13]], [[17/11]]
| [[14/9]], [[20/13]], [[17/11]]
| downminor 6th
| downminor 6th
Line 400: Line 396:
|-
|-
| 30
| 30
| 782.609
| 782.6
| [[11/7]]
| [[11/7]]
| minor 6th
| minor 6th
Line 412: Line 408:
|-
|-
| 31
| 31
| 808.696
| 808.7
| [[8/5]]
| [[8/5]]
| upminor 6th
| upminor 6th
Line 424: Line 420:
|-
|-
| 32
| 32
| 834.783
| 834.8
| [[13/8]], [[21/13]], [[34/21]]
| [[13/8]], [[21/13]], [[34/21]]
| dupminor 6th
| dupminor 6th
Line 436: Line 432:
|-
|-
| 33
| 33
| 860.870
| 860.9
| [[18/11]], [[28/17]], [[23/14]]
| [[18/11]], [[28/17]], [[23/14]]
| dudmajor 6th
| dudmajor 6th
Line 448: Line 444:
|-
|-
| 34
| 34
| 886.957
| 887.0
| [[5/3]]
| [[5/3]]
| downmajor 6th
| downmajor 6th
Line 460: Line 456:
|-
|-
| 35
| 35
| 913.043
| 913.0
| [[22/13]], [[17/10]]
| [[22/13]], [[17/10]]
| major 6th
| major 6th
Line 472: Line 468:
|-
|-
| 36
| 36
| 939.130
| 939.1
| [[12/7]]
| [[12/7]]
| upmajor 6th
| upmajor 6th
Line 484: Line 480:
|-
|-
| 37
| 37
| 965.217
| 965.2
| [[7/4]], [[40/23]]
| [[7/4]], [[40/23]]
| downminor 7th
| downminor 7th
Line 496: Line 492:
|-
|-
| 38
| 38
| 991.304
| 991.3
| [[16/9]], [[23/13]]
| [[16/9]], [[23/13]]
| minor 7th
| minor 7th
Line 508: Line 504:
|-
|-
| 39
| 39
| 1017.391
| 1017.4
| [[9/5]]
| [[9/5]]
| upminor 7th
| upminor 7th
Line 520: Line 516:
|-
|-
| 40
| 40
| 1043.478
| 1043.5
| [[11/6]], [[20/11]], [[42/23]]
| [[11/6]], [[20/11]], [[42/23]]
| dupminor 7th
| dupminor 7th
Line 532: Line 528:
|-
|-
| 41
| 41
| 1069.565
| 1069.6
| [[24/13]], [[13/7]], [[28/15]]
| [[24/13]], [[13/7]], [[28/15]]
| dudmajor 7th
| dudmajor 7th
Line 545: Line 541:
|-
|-
| 42
| 42
| 1095.652
| 1095.7
| [[15/8]], [[32/17]], [[17/9]]
| [[15/8]], [[32/17]], [[17/9]]
| downmajor 7th
| downmajor 7th
Line 558: Line 554:
|-
|-
| 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,  
| major 7th, <br>classic diminished 8ve
classic diminished 8ve
| M7, KKd8
| M7, KKd8
| C#, KKDb
| C#, KKDb
Line 571: Line 566:
|-
|-
| 44
| 44
| 1147.826
| 1147.8
| [[27/14]], [[35/18]], [[64/33]]
| [[27/14]], [[35/18]], [[64/33]]
| upmajor 7th
| upmajor 7th
Line 583: Line 578:
|-
|-
| 45
| 45
| 1173.913
| 1173.9
| [[160/81]], [[63/32]], [[96/49]]
| [[160/81]], [[63/32]], [[96/49]]
| down 8ve
| down 8ve
Line 595: Line 590:
|-
|-
| 46
| 46
| 1200.000
| 1200.0
| [[2/1]]
| [[2/1]]
| perfect 8ve
| perfect 8ve
Line 619: Line 614:
| 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
|-
|-
| dupminor
| dupminor
| ilo
| ilo
| {a, b, 0, 0, 1}
| [a, b, 0, 0, 1>
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| dudmajor
| 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
|}
|}
Line 670: Line 665:
| zo
| zo
| 6:7:9
| 6:7:9
| 0-10-27
| 0–10–27
| C vEb G
| C vEb G
| Cvm
| Cvm
Line 677: 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 684: Line 679:
| ilo
| ilo
| 18:22:27
| 18:22:27
| 0-13-27
| 0–13–27
| C ^^Eb G
| C ^^Eb G
| C^^m
| C^^m
Line 691: Line 686:
| lu
| lu
| 22:27:33
| 22:27:33
| 0-14-27
| 0–14–27
| C vvE G
| C vvE G
| Cvv
| Cvv
Line 698: Line 693:
| yo
| yo
| 4:5:6
| 4:5:6
| 0-15-27
| 0–15–27
| C vE G
| C vE G
| Cv
| Cv
Line 705: 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]].


{| class="wikitable center-all"
==== Evo flavor ====
! Steps
<imagemap>
| 0
File:46-EDO_Evo_Sagittal.svg
| 1
desc none
| 2
rect 80 0 300 50 [[Sagittal_notation]]
| 3
rect 300 0 511 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| 4
rect 20 80 120 106 [[81/80]]
| 5
rect 120 80 240 106 [[33/32]]
|-
default [[File:46-EDO_Evo_Sagittal.svg]]
! Symbol
</imagemap>
| [[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]]
|}


=== Ups and downs notation ===
==== Revo flavor ====
Using [[Helmholtz-Ellis notation|Helmholtz&ndash;Ellis]] accidentals, 46edo can also be notated using [[ups and downs notation]]:
<imagemap>
{{Sharpness-sharp5}}
File:46-EDO_Revo_Sagittal.svg
Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with three arrows.
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 ==
== Approximation to JI ==
[[File:46ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 15-limit intervals approximated in 46edo]]
[[File:46ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 15-limit intervals approximated in 46edo]]


=== Interval mappings ===
=== 17-odd-limit interval mappings ===
{{Q-odd-limit intervals|46|17}}
{{Q-odd-limit intervals|46|17}}


Line 756: Line 756:
| [[10/9]] || 36.1% || Normal || [[Mitonic]] || [[Unidec]], [[hendec]]
| [[10/9]] || 36.1% || Normal || [[Mitonic]] || [[Unidec]], [[hendec]]
|-
|-
| [[31/24]] || 70.2% || Weak || ? || ?
| [[31/24]] || 70.2% || Weak || [[Sensible]], [[sensi]] add-31 || [[Bison]] add-31, [[bisensi]] add-31
|}
|}


Line 777: Line 777:
|}
|}


''For the 23rd-octave temperament that 46edo supports which combines all above 23-note circles, see:'' [[Icositritonic]].
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 792: 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
Line 799: 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
Line 806: 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
Line 813: 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
Line 820: 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 827: 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| 46 73 107 129 159 170 188 }}]
| {{mapping| 46 73 107 129 159 170 188 }}
| +0.047
| +0.047
| 1.31
| 1.31
Line 834: 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| 46 73 107 129 159 170 188 208 }}]
| {{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.


=== Commas ===
=== Commas ===
This is a partial list of the [[commas]] that 46edo [[tempers out]] with its patent [[val]], {{val| 24 38 56 67 83 89 }}.  
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"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
|-
! [[Harmonic limit|Prime<br />Limit]]
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
Line 976: Line 978:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 998: Line 1,000:
| Thothotrigu
| 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
| 17
Line 1,013: Line 1,029:
| Semitonisma
| Semitonisma
|}
|}
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.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2 center-3 center-4"
{| 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
|-
|-
Line 1,036: Line 1,050:
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 11\46
| 11\46
| 286.957
| 286.957
| [[Amity_family#Gamity|Gamity]]
| [[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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| rowspan="11" | 2
| rowspan="11" | 2
Line 1,103: Line 1,117:
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
|-
|-
| 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
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== Scales ==
== Scales ==
{{Main| List of MOS scales in 46edo }}
* [[Plum]]
* [[Plum]]
* [[Compdye]]
* [[Smi2s]]


;Sensi
; Sensi
* [[Sensi5]]
* [[Sensi5]]
* [[Sensi8]]
* [[Sensi8]]
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* [[Sensi19]]
* [[Sensi19]]


;Elfleapday
; Elfleapday
* [[Elfleapday7]]
* [[Elfleapday7]]
* [[Elfleapday8d]]
* [[Elfleapday8d]]
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* [[Elfleapday12f]]
* [[Elfleapday12f]]


;Elfsensus
; Elfsensus
* [[Elfsensus7]]
* [[Elfsensus7]]
* [[Elfsensus8d]]
* [[Elfsensus8d]]
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* [[Elfsensus12]]
* [[Elfsensus12]]
* [[Elfsensus12f]]
* [[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. This corresponds to scale steps of 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&nbsp;&cent;) stands in for frequency ratio [[9/8]] (203.910¢).
* 8\46 (208.696{{c}}) stands in for frequency ratio [[9/8]] (203.910{{c}}).
* 7\46 (182.609&nbsp;&cent;) stands in for [[10/9]] (182.404¢).
* 7\46 (182.609{{c}}) stands in for [[10/9]] (182.404{{c}}).
* 6\46 (156.522&nbsp;&cent;) stands in for [[11/10]] (165.004¢) and [[12/11]] (150.637¢).
* 6\46 (156.522{{c}}) stands in for [[11/10]] (165.004{{c}}) and [[12/11]] (150.637{{c}}).
* 5\46 (130.435&nbsp;&cent;) stands in for [[13/12]] (138.573¢), [[14/13]] (128.298¢) and [[15/14]] (119.443¢).
* 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&nbsp;&cent;) stands in for [[16/15]] (111.731¢).
* 4\46 (104.348{{c}}) stands in for [[16/15]] (111.731{{c}}).
 
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
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|-
|-
! 7
! 7
| style="font-size: 16px;" | B{{flatdown|40}}
| style="font-size: 16px;" | G&#x1D12A;, B{{flatdown|40}}
|-
|-
! 9
! 9
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|-
|-
! 11
! 11
| style="font-size: 16px;" | F{{naturalup2|40}}
| style="font-size: 16px;" | E&#x266F;, F{{naturalup2|40}}
|-
|-
! 13
! 13
| style="font-size: 16px;" | A{{flatup2|40}}
| style="font-size: 16px;" | G&#x266F;, A{{flatup2|40}}
|-
|-
! 15
! 15
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== Instruments ==
== Instruments ==
* [[Skip fretting system 46 2 11]]: A skip-fretting system for playing 46-edo on a 23-edo stringed instrument.
=== Lumatone ===
* [[Lumatone mapping for 46edo]]
* [[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
2/1: string 3 fret 5
3/2: not easily accessible
5/4: string 5 fret 4


== Music ==
== Music ==
=== Modern renditions ===
=== Modern renditions ===
; {{W|Johann Sebastian Bach}}
; {{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)
* [https://www.youtube.com/watch?v=4yetEubmIk0 "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)


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=== 21st century ===
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=SCpbbov1fSQ ''music box 46edo''] (2025)
; [[Jake Freivald]] ([https://soundcloud.com/jdfreivald site])
; [[Jake Freivald]] ([https://soundcloud.com/jdfreivald site])
* [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://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}}  
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* [https://www.youtube.com/watch?v=JwjoT6ceU20 ''Arietta with 5 Variations'', for Organ]
* [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)
* [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]]
; [[Joseph Monzo]]
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* [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/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.
* [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.
; [[Tristan Bay]]
* [https://www.youtube.com/watch?v=s61YY80E_IA ''Catalyst''] (2025)
; vivi mouse
* [https://soundcloud.com/vivi-mouse-emoji/valentines-day ''valentine's day''] (2025)


== Notes ==
== Notes ==
Retrieved from "https://en.xen.wiki/w/46edo"