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== Theory ==
== Theory ==
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]].  
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|columns=9}}
{{Harmonics in equal|46|columns=11}}
{{Harmonics in equal|46|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 46edo (continued)}}
{{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">{{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>
! 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]]
|-
| 0
| 0.0
| [[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="3" | [[SKULO interval names|SKULO notation]] (K or S = 1, U = 2)
! colspan="2" | [[Solfege]]
! colspan="2" | [[Solfège]]s
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| perfect unison
| perfect unison
| P1
| P1
Line 46: Line 293:
| 1
| 1
| 26.1
| 26.1
| [[81/80]], [[64/63]], [[49/48]]
| up unison
| up unison
| ^1
| ^1
Line 58: Line 304:
| 2
| 2
| 52.2
| 52.2
| [[28/27]], [[36/35]], [[33/32]]
| downminor 2nd
| downminor 2nd
| vm2
| vm2
Line 70: Line 315:
| 3
| 3
| 78.3
| 78.3
| [[25/24]], [[21/20]], [[22/21]], [[24/23]], [[23/22]]
| minor 2nd
| minor 2nd
| m2
| m2
Line 82: Line 326:
| 4
| 4
| 104.3
| 104.3
| [[16/15]], [[17/16]], [[18/17]]
| upminor 2nd
| upminor 2nd
| ^m2
| ^m2
Line 94: Line 337:
| 5
| 5
| 130.4
| 130.4
| [[13/12]], [[14/13]], [[15/14]]
| dupminor 2nd
| dupminor 2nd
| ^^m2
| ^^m2
Line 106: Line 348:
| 6
| 6
| 156.5
| 156.5
| [[12/11]], [[11/10]], [[23/21]]
| dudmajor 2nd
| dudmajor 2nd
| vvM2
| vvM2
Line 118: Line 359:
| 7
| 7
| 182.6
| 182.6
| [[10/9]]
| downmajor 2nd
| downmajor 2nd
| vM2
| vM2
Line 130: Line 370:
| 8
| 8
| 208.7
| 208.7
| [[9/8]]
| major 2nd
| major 2nd
| M2
| M2
Line 142: Line 381:
| 9
| 9
| 234.8
| 234.8
| [[8/7]], [[23/20]]
| upmajor 2nd
| upmajor 2nd
| ^M2
| ^M2
Line 154: Line 392:
| 10
| 10
| 260.9
| 260.9
| [[7/6]]
| downminor 3rd
| downminor 3rd
| vm3
| vm3
Line 166: Line 403:
| 11
| 11
| 287.0
| 287.0
| [[13/11]], [[20/17]]
| minor 3rd
| minor 3rd
| m3
| m3
Line 178: Line 414:
| 12
| 12
| 313.0
| 313.0
| [[6/5]]
| upminor 3rd
| upminor 3rd
| ^m3
| ^m3
Line 190: Line 425:
| 13
| 13
| 339.1
| 339.1
| [[11/9]], [[17/14]], [[28/23]]
| dupminor 3rd
| dupminor 3rd
| ^^m3
| ^^m3
Line 202: Line 436:
| 14
| 14
| 365.2
| 365.2
| [[16/13]], [[26/21]], [[21/17]]
| dudmajor 3rd
| dudmajor 3rd
| vvM3
| vvM3
Line 214: Line 447:
| 15
| 15
| 391.3
| 391.3
| [[5/4]]
| downmajor 3rd
| downmajor 3rd
| vM3
| vM3
Line 226: Line 458:
| 16
| 16
| 417.4
| 417.4
| [[14/11]], [[23/18]]
| major 3rd
| major 3rd
| M3
| M3
Line 238: Line 469:
| 17
| 17
| 443.5
| 443.5
| [[9/7]], [[13/10]], [[22/17]]
| upmajor 3rd
| upmajor 3rd
| ^M3
| ^M3
Line 250: Line 480:
| 18
| 18
| 469.6
| 469.6
| [[21/16]], [[17/13]]
| down 4th
| down 4th
| v4
| v4
Line 262: Line 491:
| 19
| 19
| 495.7
| 495.7
| [[4/3]]
| perfect 4th
| perfect 4th
| P4
| P4
Line 274: Line 502:
| 20
| 20
| 521.7
| 521.7
| [[27/20]], [[23/17]]
| up 4th
| up 4th
| ^4
| ^4
Line 286: Line 513:
| 21
| 21
| 547.8
| 547.8
| [[11/8]]
| dup 4th
| dup 4th
| ^^4
| ^^4
Line 298: Line 524:
| 22
| 22
| 573.9
| 573.9
| [[7/5]], [[18/13]], [[32/23]]
| dudaug 4th, <br>dim 5th
| dudaug 4th, <br>dim 5th
| vvA4, d5
| vvA4, d5
Line 310: Line 535:
| 23
| 23
| 600.0
| 600.0
| [[17/12]], [[24/17]]
| downaug 4th, updim 5th
| downaug 4th, updim 5th
| vA4, ^d5
| vA4, ^d5
Line 322: Line 546:
| 24
| 24
| 626.1
| 626.1
| [[10/7]], [[13/9]], [[23/16]]
| aug 4th, dupdim 5th
| aug 4th, dupdim 5th
| A4, ^^d5
| A4, ^^d5
Line 334: Line 557:
| 25
| 25
| 652.2
| 652.2
| [[16/11]]
| dud 5th
| dud 5th
| vv5
| vv5
Line 346: Line 568:
| 26
| 26
| 678.3
| 678.3
| [[40/27]], [[34/23]]
| down 5th
| down 5th
| v5
| v5
Line 358: Line 579:
| 27
| 27
| 704.3
| 704.3
| [[3/2]]
| perfect 5th
| perfect 5th
| P5
| P5
Line 370: Line 590:
| 28
| 28
| 730.4
| 730.4
| [[32/21]], [[26/17]]
| up 5th
| up 5th
| ^5
| ^5
Line 382: Line 601:
| 29
| 29
| 756.5
| 756.5
| [[14/9]], [[20/13]], [[17/11]]
| downminor 6th
| downminor 6th
| vm6
| vm6
Line 394: Line 612:
| 30
| 30
| 782.6
| 782.6
| [[11/7]]
| minor 6th
| minor 6th
| m6
| m6
Line 406: Line 623:
| 31
| 31
| 808.7
| 808.7
| [[8/5]]
| upminor 6th
| upminor 6th
| ^m6
| ^m6
Line 418: Line 634:
| 32
| 32
| 834.8
| 834.8
| [[13/8]], [[21/13]], [[34/21]]
| dupminor 6th
| dupminor 6th
| ^^m6
| ^^m6
Line 430: Line 645:
| 33
| 33
| 860.9
| 860.9
| [[18/11]], [[28/17]], [[23/14]]
| dudmajor 6th
| dudmajor 6th
| vvM6
| vvM6
Line 442: Line 656:
| 34
| 34
| 887.0
| 887.0
| [[5/3]]
| downmajor 6th
| downmajor 6th
| vM6
| vM6
Line 454: Line 667:
| 35
| 35
| 913.0
| 913.0
| [[22/13]], [[17/10]]
| major 6th
| major 6th
| M6
| M6
Line 466: Line 678:
| 36
| 36
| 939.1
| 939.1
| [[12/7]]
| upmajor 6th
| upmajor 6th
| ^M6
| ^M6
Line 478: Line 689:
| 37
| 37
| 965.2
| 965.2
| [[7/4]], [[40/23]]
| downminor 7th
| downminor 7th
| vm7
| vm7
Line 490: Line 700:
| 38
| 38
| 991.3
| 991.3
| [[16/9]], [[23/13]]
| minor 7th
| minor 7th
| m7
| m7
Line 502: Line 711:
| 39
| 39
| 1017.4
| 1017.4
| [[9/5]]
| upminor 7th
| upminor 7th
| ^m7
| ^m7
Line 514: Line 722:
| 40
| 40
| 1043.5
| 1043.5
| [[11/6]], [[20/11]], [[42/23]]
| dupminor 7th
| dupminor 7th
| ^^m7
| ^^m7
Line 526: Line 733:
| 41
| 41
| 1069.6
| 1069.6
| [[24/13]], [[13/7]], [[28/15]]
| dudmajor 7th
| dudmajor 7th
| vvM7
| vvM7
Line 539: Line 745:
| 42
| 42
| 1095.7
| 1095.7
| [[15/8]], [[32/17]], [[17/9]]
| downmajor 7th
| downmajor 7th
| vM7
| vM7
Line 552: Line 757:
| 43
| 43
| 1121.7
| 1121.7
| [[48/25]], [[40/21]], [[21/11]], [[23/12]], [[44/23]]
| major 7th
| major 7th
| M7
| M7
Line 564: Line 768:
| 44
| 44
| 1147.8
| 1147.8
| [[27/14]], [[35/18]], [[64/33]]
| upmajor 7th
| upmajor 7th
| ^M7
| ^M7
Line 576: Line 779:
| 45
| 45
| 1173.9
| 1173.9
| [[160/81]], [[63/32]], [[96/49]]
| down 8ve
| down 8ve
| v8
| v8
Line 588: Line 790:
| 46
| 46
| 1200.0
| 1200.0
| [[2/1]]
| perfect 8ve
| perfect 8ve
| P8
| P8
Line 606: 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 < -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 > 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 654: 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 669: 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 676: 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 683: 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 690: 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 697: Line 898:
| ru
| ru
| 14:18:21
| 14:18:21
| 0-17-27
| 0–17–27
| C ^E G
| C ^E G
| C^
| C^
Line 705: Line 906:


== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
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).
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5a}}
{{Sharpness-sharp5-szg}}
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}}
=== Kite's ups and downs notation ===
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).
{{Ups and downs sharpness}}


=== Sagittal notation ===
=== Sagittal notation ===
Line 715: Line 918:


==== Evo flavor ====
==== Evo flavor ====
<imagemap>
{{Sagittal chart|Evo}}
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 ====
==== Revo flavor ====
<imagemap>
{{Sagittal chart}}
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 ==
== Approximation to JI ==
Line 775: Line 962:


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]].
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[214zpi]]
| 46.0089748051542
| 26.0818678330031
| 7.495674
| 1.356067
| 17.747832
| 46edo
| 1199.76592031814
| 14
| 11
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 1,028: 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 1,043: Line 1,213:
| Semitonisma
| Semitonisma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 1,083: Line 1,254:
| [[Rodan]]
| [[Rodan]]
| [[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)
| [[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
Line 1,093: Line 1,264:
| 13\46
| 13\46
| 339.130
| 339.130
| [[Amity]] / [[hitchcock]]
| [[Amity]] / [[Amity_family#Hitchcock|Hitchcock]]
| [[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)
| [[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
Line 1,101: Line 1,272:
| [[Magus]] / [[amigo]]
| [[Magus]] / [[amigo]]
| [[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)
| [[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
Line 1,156: Line 1,327:
| [[Bison]]
| [[Bison]]
| 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
| 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
Line 1,168: Line 1,339:
| [[Abigail]]
| [[Abigail]]
| 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)
| 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
Line 1,186: Line 1,357:
| [[Vines]]
| [[Vines]]
| 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)
| 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 1,264: Line 1,435:


== Instruments ==
== Instruments ==
* [[Skip fretting system 46 2 11]]: A skip-fretting system for playing 46-edo on a 23-edo stringed instrument.
=== Lumatone ===
* [[Skip fretting system 46 7 11]]
* [[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]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=SCpbbov1fSQ ''music box 46edo''] (2025)  
* [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 1,297: 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]]
; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/light-at-the-end Light at the End]'' (2020)
* [https://soundcloud.com/morphosyntax-1/light-at-the-end ''Light at the End''] (2020)


; [[Joseph Monzo]]
; [[Joseph Monzo]]
Line 1,320: Line 1,527:
* [https://www.youtube.com/watch?v=s61YY80E_IA ''Catalyst''] (2025)
* [https://www.youtube.com/watch?v=s61YY80E_IA ''Catalyst''] (2025)


== Notes ==
; [[vivi mouse]]
<references group="note" />
* ''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"