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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 [[zeta peak edo]], but it is a [[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 edo to approximate odd harmonics 1–13 with less than 25% [[relative interval error|relative error]].


[[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=11}}
{{Harmonics in equal|46|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}
 
=== As a tuning of other temperaments ===
46edo is a superb tuning for [[sensi]] and [[leapday]]. It also [[support]]s but tunes less optimally [[valentine]], [[shrutar]], [[rodan]], 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, but [[77edo]] has a lower average error overall.


=== Subsets and supersets ===
=== Subsets and supersets ===
46edo can be treated as two circles of [[23edo]] 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. Supersets [[414edo]] and [[460edo]] are also notable for their high consistency.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 left-7"
{| class="wikitable center-all right-2 left-3 left-4 left-7"
|-
|-
! &#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">{{safesubst:Subgroup note|limit=2.3.5.7.11.13.17.23-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]]
! [[Kite's ups and downs notation|Ups and downs notation]]
! colspan="3" | [[SKULO interval names| SKULO notation]] (K or S = 1, U = 2)
! colspan="2" | [[Solfege| Solfeges]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 26.1
| [[49/48]], [[64/63]], [[81/80]]
| {{UDnote|step=1}}
|-
| 2
| 52.2
| [[28/27]], [[33/32]], [[36/35]]
| {{UDnote|step=2}}
|-
| 3
| 78.3
| [[21/20]], [[22/21]], [[24/23]], [[23/22]], [[25/24]]
| {{UDnote|step=3}}
|-
| 4
| 104.3
| [[16/15]], [[17/16]], [[18/17]]
| {{UDnote|step=4}}
|-
| 5
| 130.4
| [[13/12]], [[14/13]], [[15/14]]
| {{UDnote|step=5}}
|-
| 6
| 156.5
| [[11/10]], [[12/11]], [[23/21]]
| {{UDnote|step=6}}
|-
| 7
| 182.6
| [[10/9]]
| {{UDnote|step=7}}
|-
| 8
| 208.7
| [[9/8]]
| {{UDnote|step=8}}
|-
| 9
| 234.8
| [[8/7]], [[23/20]]
| {{UDnote|step=9}}
|-
| 10
| 260.9
| [[7/6]]
| {{UDnote|step=10}}
|-
| 11
| 287.0
| [[13/11]], [[20/17]]
| {{UDnote|step=11}}
|-
| 12
| 313.0
| [[6/5]]
| {{UDnote|step=12}}
|-
| 13
| 339.1
| [[11/9]], [[17/14]], [[28/23]]
| {{UDnote|step=13}}
|-
| 14
| 365.2
| [[16/13]], [[21/17]], [[26/21]]
| {{UDnote|step=14}}
|-
| 15
| 391.3
| [[5/4]]
| {{UDnote|step=15}}
|-
| 16
| 417.4
| [[14/11]], [[23/18]]
| {{UDnote|step=16}}
|-
| 17
| 443.5
| [[9/7]], [[13/10]], [[22/17]]
| {{UDnote|step=17}}
|-
| 18
| 469.6
| [[17/13]], [[21/16]]
| {{UDnote|step=18}}
|-
| 19
| 495.7
| [[4/3]]
| {{UDnote|step=19}}
|-
| 20
| 521.7
| [[23/17]], [[27/20]]
| {{UDnote|step=20}}
|-
| 21
| 547.8
| [[11/8]]
| {{UDnote|step=21}}
|-
| 22
| 573.9
| [[7/5]], [[18/13]], [[32/23]]
| {{UDnote|step=22}}
|-
| 23
| 600.0
| [[17/12]], [[24/17]]
| {{UDnote|step=23}}
|-
| 24
| 626.1
| [[10/7]], [[13/9]], [[23/16]]
| {{UDnote|step=24}}
|-
| 25
| 652.2
| [[16/11]]
| {{UDnote|step=25}}
|-
| 26
| 678.3
| [[34/23]], [[40/27]]
| {{UDnote|step=26}}
|-
| 27
| 704.3
| [[3/2]]
| {{UDnote|step=27}}
|-
| 28
| 730.4
| [[26/17]], [[32/21]]
| {{UDnote|step=28}}
|-
| 29
| 756.5
| [[14/9]], [[17/11]], [[20/13]]
| {{UDnote|step=29}}
|-
| 30
| 782.6
| [[11/7]]
| {{UDnote|step=30}}
|-
| 31
| 808.7
| [[8/5]]
| {{UDnote|step=31}}
|-
| 32
| 834.8
| [[13/8]], [[21/13]], [[34/21]]
| {{UDnote|step=32}}
|-
| 33
| 860.9
| [[18/11]], [[23/14]], [[28/17]]
| {{UDnote|step=33}}
|-
| 34
| 887.0
| [[5/3]]
| {{UDnote|step=34}}
|-
| 35
| 913.0
| [[17/10]], [[22/13]]
| {{UDnote|step=35}}
|-
| 36
| 939.1
| [[12/7]]
| {{UDnote|step=36}}
|-
| 37
| 965.2
| [[7/4]], [[40/23]]
| {{UDnote|step=37}}
|-
| 38
| 991.3
| [[16/9]], [[23/13]]
| {{UDnote|step=38}}
|-
| 39
| 1017.4
| [[9/5]]
| {{UDnote|step=39}}
|-
| 40
| 1043.5
| [[11/6]], [[20/11]], [[42/23]]
| {{UDnote|step=40}}
|-
| 41
| 1069.6
| [[13/7]], [[24/13]], [[28/15]]
| {{UDnote|step=41}}
|-
| 42
| 1095.7
| [[15/8]], [[17/9]], [[32/17]]
| {{UDnote|step=42}}
|-
| 43
| 1121.7
| [[21/11]], [[23/12]], [[40/21]], [[44/23]], [[48/25]]
| {{UDnote|step=43}}
|-
| 44
| 1147.8
| [[27/14]], [[35/18]], [[64/33]]
| {{UDnote|step=44}}
|-
| 45
| 1173.9
| [[63/32]], [[96/49]], [[160/81]]
| {{UDnote|step=45}}
|-
| 46
| 1200.0
| [[2/1]]
| {{UDnote|step=46}}
|}
<references group="note" />
=== Proposed interval names and solfèges ===
{| class="wikitable center-all right-2 left-2 left-3 left-6 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! colspan="3" | [[Kite's ups and downs notation|Ups and downs notation]]<br>([[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" | [[Solfège]]s
|-
| 0
| 0.0
| perfect unison
| perfect unison
| P1
| P1
Line 44: Line 292:
|-
|-
| 1
| 1
| 26.087
| 26.1
| [[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 303:
|-
|-
| 2
| 2
| 52.174
| 52.2
| [[28/27]], [[36/35]], [[33/32]]
| downminor 2nd
| downminor 2nd
| vm2
| vm2
Line 69: Line 314:
|-
|-
| 3
| 3
| 78.261
| 78.3
| [[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 325:
|-
|-
| 4
| 4
| 104.348
| 104.3
| [[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 336:
|-
|-
| 5
| 5
| 130.435
| 130.4
| [[13/12]], [[14/13]], [[15/14]]
| dupminor 2nd
| dupminor 2nd
| ^^m2
| ^^m2
Line 107: Line 347:
|-
|-
| 6
| 6
| 156.522
| 156.5
| [[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 358:
|-
|-
| 7
| 7
| 182.609
| 182.6
| [[10/9]]
| downmajor 2nd
| downmajor 2nd
| vM2
| vM2
Line 132: Line 369:
|-
|-
| 8
| 8
| 208.696
| 208.7
| [[9/8]]
| major 2nd
| major 2nd
| M2
| M2
Line 144: Line 380:
|-
|-
| 9
| 9
| 234.783
| 234.8
| [[8/7]], [[23/20]]
| upmajor 2nd
| upmajor 2nd
| ^M2
| ^M2
Line 156: Line 391:
|-
|-
| 10
| 10
| 260.870
| 260.9
| [[7/6]]
| downminor 3rd
| downminor 3rd
| vm3
| vm3
Line 168: Line 402:
|-
|-
| 11
| 11
| 286.957
| 287.0
| [[13/11]], [[20/17]]
| minor 3rd
| minor 3rd
| m3
| m3
Line 180: Line 413:
|-
|-
| 12
| 12
| 313.043
| 313.0
| [[6/5]]
| upminor 3rd
| upminor 3rd
| ^m3
| ^m3
Line 192: Line 424:
|-
|-
| 13
| 13
| 339.130
| 339.1
| [[11/9]], [[17/14]], [[28/23]]
| dupminor 3rd
| dupminor 3rd
| ^^m3
| ^^m3
Line 204: Line 435:
|-
|-
| 14
| 14
| 365.217
| 365.2
| [[16/13]], [[26/21]], [[21/17]]
| dudmajor 3rd
| dudmajor 3rd
| vvM3
| vvM3
Line 216: Line 446:
|-
|-
| 15
| 15
| 391.304
| 391.3
| [[5/4]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 228: Line 457:
|-
|-
| 16
| 16
| 417.391
| 417.4
| [[14/11]], [[23/18]]
| major 3rd
| major 3rd
| M3
| M3
Line 240: Line 468:
|-
|-
| 17
| 17
| 443.478
| 443.5
| [[9/7]], [[13/10]], [[22/17]]
| upmajor 3rd
| upmajor 3rd
| ^M3
| ^M3
Line 252: Line 479:
|-
|-
| 18
| 18
| 469.565
| 469.6
| [[21/16]], [[17/13]]
| down 4th
| down 4th
| v4
| v4
Line 264: Line 490:
|-
|-
| 19
| 19
| 495.652
| 495.7
| [[4/3]]
| perfect 4th
| perfect 4th
| P4
| P4
Line 276: Line 501:
|-
|-
| 20
| 20
| 521.739
| 521.7
| [[27/20]], [[23/17]]
| up 4th
| up 4th
| ^4
| ^4
Line 288: Line 512:
|-
|-
| 21
| 21
| 547.826
| 547.8
| [[11/8]]
| dup 4th
| dup 4th
| ^^4
| ^^4
Line 300: Line 523:
|-
|-
| 22
| 22
| 573.913
| 573.9
| [[7/5]], [[18/13]], [[32/23]]
| dudaug 4th, <br>dim 5th
| dudaug 4th,  
dim 5th
| vvA4, d5
| vvA4, d5
| vvG#, Ab
| vvG#, Ab
Line 313: Line 534:
|-
|-
| 23
| 23
| 600.000
| 600.0
| [[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 545:
|-
|-
| 24
| 24
| 626.087
| 626.1
| [[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 556:
|-
|-
| 25
| 25
| 652.174
| 652.2
| [[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 567:
|-
|-
| 26
| 26
| 678.261
| 678.3
| [[40/27]], [[34/23]]
| down 5th
| down 5th
| v5
| v5
Line 364: Line 578:
|-
|-
| 27
| 27
| 704.348
| 704.3
| [[3/2]]
| perfect 5th
| perfect 5th
| P5
| P5
Line 376: Line 589:
|-
|-
| 28
| 28
| 730.435
| 730.4
| [[32/21]], [[26/17]]
| up 5th
| up 5th
| ^5
| ^5
Line 388: Line 600:
|-
|-
| 29
| 29
| 756.522
| 756.5
| [[14/9]], [[20/13]], [[17/11]]
| downminor 6th
| downminor 6th
| vm6
| vm6
Line 400: Line 611:
|-
|-
| 30
| 30
| 782.609
| 782.6
| [[11/7]]
| minor 6th
| minor 6th
| m6
| m6
Line 412: Line 622:
|-
|-
| 31
| 31
| 808.696
| 808.7
| [[8/5]]
| upminor 6th
| upminor 6th
| ^m6
| ^m6
Line 424: Line 633:
|-
|-
| 32
| 32
| 834.783
| 834.8
| [[13/8]], [[21/13]], [[34/21]]
| dupminor 6th
| dupminor 6th
| ^^m6
| ^^m6
Line 436: Line 644:
|-
|-
| 33
| 33
| 860.870
| 860.9
| [[18/11]], [[28/17]], [[23/14]]
| dudmajor 6th
| dudmajor 6th
| vvM6
| vvM6
Line 448: Line 655:
|-
|-
| 34
| 34
| 886.957
| 887.0
| [[5/3]]
| downmajor 6th
| downmajor 6th
| vM6
| vM6
Line 460: Line 666:
|-
|-
| 35
| 35
| 913.043
| 913.0
| [[22/13]], [[17/10]]
| major 6th
| major 6th
| M6
| M6
Line 472: Line 677:
|-
|-
| 36
| 36
| 939.130
| 939.1
| [[12/7]]
| upmajor 6th
| upmajor 6th
| ^M6
| ^M6
Line 484: Line 688:
|-
|-
| 37
| 37
| 965.217
| 965.2
| [[7/4]], [[40/23]]
| downminor 7th
| downminor 7th
| vm7
| vm7
Line 496: Line 699:
|-
|-
| 38
| 38
| 991.304
| 991.3
| [[16/9]], [[23/13]]
| minor 7th
| minor 7th
| m7
| m7
Line 508: Line 710:
|-
|-
| 39
| 39
| 1017.391
| 1017.4
| [[9/5]]
| upminor 7th
| upminor 7th
| ^m7
| ^m7
Line 520: Line 721:
|-
|-
| 40
| 40
| 1043.478
| 1043.5
| [[11/6]], [[20/11]], [[42/23]]
| dupminor 7th
| dupminor 7th
| ^^m7
| ^^m7
Line 532: Line 732:
|-
|-
| 41
| 41
| 1069.565
| 1069.6
| [[24/13]], [[13/7]], [[28/15]]
| dudmajor 7th
| dudmajor 7th
| vvM7
| vvM7
Line 545: Line 744:
|-
|-
| 42
| 42
| 1095.652
| 1095.7
| [[15/8]], [[32/17]], [[17/9]]
| downmajor 7th
| downmajor 7th
| vM7
| vM7
Line 558: Line 756:
|-
|-
| 43
| 43
| 1121.739
| 1121.7
| [[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 767:
|-
|-
| 44
| 44
| 1147.826
| 1147.8
| [[27/14]], [[35/18]], [[64/33]]
| upmajor 7th
| upmajor 7th
| ^M7
| ^M7
Line 583: Line 778:
|-
|-
| 45
| 45
| 1173.913
| 1173.9
| [[160/81]], [[63/32]], [[96/49]]
| down 8ve
| down 8ve
| v8
| v8
Line 595: Line 789:
|-
|-
| 46
| 46
| 1200.000
| 1200.0
| [[2/1]]
| perfect 8ve
| perfect 8ve
| P8
| P8
Line 614: Line 807:
! Quality
! Quality
! Color
! Color
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
| downminor
| downminor
| zo
| zo
| {a, b, 0, 1}
| {{Monzo| ''a'' ''b'' 0 1 }}
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| minor
| minor
| fourthward wa
| fourthward wa
| {a, b}, b &lt; -1
| {{Monzo| ''a'' ''b'' }}, ''b'' < −1
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| upminor
| upminor
| gu
| gu
| {a, b, -1}
| {{Monzo| ''a'' ''b'' −1 }}
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| dupminor
| dupminor
| ilo
| ilo
| {a, b, 0, 0, 1}
| {{Monzo| ''a'' ''b'' 0 0 1 }}
| 11/9, 11/6
| 11/9, 11/6
|-
|-
| dudmajor
| dudmajor
| lu
| lu
| {a, b, 0, 0, -1}
| {{Monzo| ''a'' ''b'' 0 0 −1 }}
| 12/11, 18/11
| 12/11, 18/11
|-
|-
| downmajor
| downmajor
| yo
| yo
| {a, b, 1}
| {{Monzo| ''a'' ''b'' 1 }}
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| major
| major
| fifthward wa
| fifthward wa
| {a, b}, b &gt; 1
| {{Monzo| ''a'' ''b'' }}, ''b'' > 1
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| upmajor
| upmajor
| ru
| ru
| {a, b, 0, -1}
| {{Monzo| ''a'' ''b'' 0 −1 }}
| 9/7, 12/7
| 9/7, 12/7
|}
|}
Line 662: Line 855:
|-
|-
! 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 677: Line 870:
| 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 877:
| 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 884:
| 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 891:
| 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 898:
| 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 ===
=== Stein–Zimmermann–Gould notation ===
The following table shows [[sagittal notation]] accidentals in one apotome for 46edo.
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}


{| class="wikitable center-all"
=== Kite's ups and downs notation ===
! Steps
46edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp, etc., and likewise down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
| 0
{{Ups and downs sharpness}}
| 1
 
| 2
=== Sagittal notation ===
| 3
This notation uses the same sagittal sequence as [[39edo #Sagittal notation|39edo]].
| 4
 
| 5
==== Evo flavor ====
|-
{{Sagittal chart|Evo}}
! 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]]
|}


=== Ups and downs notation ===
==== Revo flavor ====
Using [[Helmholtz-Ellis notation|Helmholtz&ndash;Ellis]] accidentals, 46edo can also be notated using [[ups and downs notation]]:
{{Sagittal chart}}
{{Sharpness-sharp5}}
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.


== 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}}
{{Q-odd-limit intervals|46|17}}


=== Consistent circles ===
=== Consistent circles ===
Line 756: Line 940:
| [[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 961:
|}
|}


''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 977:
| 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 984:
| 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 991:
| 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 998:
| 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 1,005:
| 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 1,012:
| 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 1,019:
| 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 1,162:
| 19.13
| 19.13
| Thozogu
| Thozogu
| Superleap
| Superleap comma, biome comma
|-
|-
| 13
| 13
Line 998: Line 1,184:
| 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 1,213:
| Semitonisma
| Semitonisma
|}
|}
 
<references group="note" />
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 1,235:
| 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]] / [[Amity_family#Hitchcock|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 1,302:
| 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
Line 1,169: Line 1,368:


== Scales ==
== Scales ==
{{Main| List of MOS scales in 46edo }}
* [[Plum]]
* [[Plum]]
* [[Compdye]]
* [[Smi2s]]


;Sensi
; Sensi
* [[Sensi5]]
* [[Sensi5]]
* [[Sensi8]]
* [[Sensi8]]
Line 1,177: Line 1,380:
* [[Sensi19]]
* [[Sensi19]]


;Elfleapday
; Elfleapday
* [[Elfleapday7]]
* [[Elfleapday7]]
* [[Elfleapday8d]]
* [[Elfleapday8d]]
Line 1,185: Line 1,388:
* [[Elfleapday12f]]
* [[Elfleapday12f]]


;Elfsensus
; Elfsensus
* [[Elfsensus7]]
* [[Elfsensus7]]
* [[Elfsensus8d]]
* [[Elfsensus8d]]
Line 1,192: Line 1,395:
* [[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
Line 1,222: Line 1,425:
|-
|-
! 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=4yetEubmIk0 "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Bach_BWV_1029_E46-Alto-Sax-+-Harpsichord.mp3 Sonata No. 3 in G minor, BWV 1029]{{dead link}} (late 1730s – early 1740s) – arranged and rendered by Claudi Meneghin
* [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)
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2014/02/Bach_Contrapunctus_4-Jeux14-E46.mp3 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080]{{dead link}} (1742–1749) – arranged and rendered by Claudi Meneghin
* "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – arranged and rendered by Claudi Meneghin
** [https://www.youtube.com/watch?v=4yetEubmIk0 organ rendering] (2024) · [https://www.youtube.com/watch?v=fwGBD-oGNxI harpsichord rendering] (2025)


; {{W|Nicolaus Bruhns}}
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=_YiCc81ktBI ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=_YiCc81ktBI ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
; {{W|Frederick Chopin}}
* [https://www.youtube.com/shorts/nCKtRb7fD2w ''Prelude op. 28 no. 7 in A major''] (1836) – rendered by Claudi Meneghin (2025)
* ''"Waterfall" Étude from 12 Études, op. 10'' (1829–1832) – with syntonic comma adjustment, rendered with fortepiano tone by Claudi Meneghin (2025)
** [https://www.youtube.com/shorts/AkKCmAtAQCg ''fortepiano rendering'']
** [https://www.youtube.com/shorts/QCN0H2Gnmk4 ''sine wave rendering'']


; {{W|Scott Joplin}}
; {{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)
* ''Maple Leaf Rag'' (1899) – arranged for harpsichord and rendered by Claudi Meneghin
** with syntonic comma adjustment ([https://www.youtube.com/watch?v=GXIoWsvFMzI 2024], [https://www.youtube.com/watch?v=55iWVqnaox8&lc=Ugz472kn24J1Bw_vHtV4AaABAg 2025])
** without syntonic comma adjustment (super-Pythagorean) ([https://www.youtube.com/shorts/2OOXUO2H9qA 2026])
 
; Claude Larson ({{W|Klaus Netzle|German Wikipedia|s=de}})
* [https://www.youtube.com/shorts/fxEWnqvZtjY "Memory Image" from ''Soundscapes Vol.2''] (1986) – covered by [[Bryan Deister]] (2025)
 
; {{W|Akira Yamaoka}}
* [https://www.youtube.com/shorts/WyY-7Im0zkw ''Lakeview Hotel Music Box''] via ''{{W|Silent Hill 2}}'' (2001) – microtonal cover in 46edo by [[Bryan Deister]] (2026)


=== 21st century ===
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=SCpbbov1fSQ ''music box 46edo''] (2025)
* [https://www.youtube.com/shorts/K-Gu_YkjYS4 ''46edo improv''] (2025)
* [https://www.youtube.com/shorts/IWYJPyWE_Ao ''46edo waltz''] (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''] [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/a_seed_planted-starling-46-EDO.mp3 play]{{dead link}}  
 
; [[groundfault]]
* "Glimmer Extrication", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/glimmer-extrication-46edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=969 YouTube (16:09–21:11)]


; [[Andrew Heathwaite]]
; [[Andrew Heathwaite]]
Line 1,261: Line 1,505:


; [[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
* [https://www.youtube.com/watch?v=-6ORgyqom5M ''Les Regrets''] (2019) – chaconne & fugue a 5, for recorder, cornett, viola da braccio, bassoon, gamba, and continuo.
* [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=1r55c2nppe8 ''El Rossinyol''] (2020) – for organ
* [https://www.youtube.com/watch?v=-6ORgyqom5M ''Chaconne et Fugue à 5 "Les Regrets"'']
* [https://www.youtube.com/watch?v=JwjoT6ceU20 ''Arietta with 5 Variations''] (2020) – for organ
* [https://www.youtube.com/watch?v=1r55c2nppe8 ''El Rossinyol'']
* [https://www.youtube.com/watch?v=rmgWC_jruSg ''Sensi Fugue''] (2024) – fugue for two organs, in sensi, 46edo tuning
* [https://www.youtube.com/watch?v=JwjoT6ceU20 ''Arietta with 5 Variations'', for Organ]
* [https://www.youtube.com/shorts/iZQElKqq6Hw "Closing Jingle"], from [https://www.youtube.com/playlist?list=PLC6ZSKWKnVz14UQd32K1sXc2zvstkPN6V ''Mikrokosmos''] (2026)
* [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]]
Line 1,278: Line 1,524:
* [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.


== Notes ==
; [[Tristan Bay]]
<references group="note" />
* [https://www.youtube.com/watch?v=s61YY80E_IA ''Catalyst''] (2025)
 
; [[vivi mouse]]
* ''valentine's day'', from ''2025'' (2025) – [https://soundcloud.com/vivi-mouse-emoji/valentines-day SoundCloud] | [https://vivimouse.bandcamp.com/track/valentines-day Bandcamp] – in valentine, 46edo tuning


[[Category:Chromosounds]]
[[Category:Chromosounds]]
Retrieved from "https://en.xen.wiki/w/46edo"