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{{Interwiki
| en = 43edo
| de = 43-EDO
}}
{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|43}} One step of 43edo was named ''[[méride]]'' by [[Joseph Sauveur]] (1653-1716) in 1696. The méride and eptaméride were the first logarithmic interval measures proposed. Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it. <ref>[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens&#45;Fokker&#58; Logarithmic Interval Measures]</ref>
{{ED intro}}
 
== History ==
The French Baroque acoustician {{w|Joseph Sauveur}}, who was ironically hearing and speech impaired, based his tuning system on 43 equal tones to the octave, calling one step a '''méride'''. Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it.<ref>[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens&#45;Fokker&#58; Logarithmic Interval Measures]</ref>
 
The composer [[Juhan Puhm]] uses 43edo in some of his fortepiano suites and prefers it to [[31edo]].


== Theory ==
== Theory ==
43edo is strongly associated with [[meantone]], particularly [[1/5-comma meantone]], being a good tuning system in the 5, 7, 11, and 13-limit. In the 7-limit, it supports septimal meantone, as it tempers out 3136/3125, along with 126/125 and 225/224. The version of 11-limit meantone is the one tempering out [[99/98]], [[176/175]] and [[441/440]], sometimes called Huygens. 43-equal has the first good 13-limit meantone available as an equal division of the octave. The baroque, French, ironically hearing and speech impaired acoustician [[wikipedia: Joseph Sauveur|Joseph Sauveur]] based his system on 43 equal tones to the octave, calling them "merides". Further information: [http://tonalsoft.com/enc/m/meride.aspx Tonalsoft encyclopedia entry of meride].
43edo is strongly associated with [[meantone]]. Specifically, it is for all practical purposes equivalent to [[1/5-comma meantone]], as it tunes the perfect fifth flat of [[3/2]] and major third sharp of [[5/4]] by slightly more than four cents on both of them. Its approximations to [[7/4]] and [[11/8]] are noticeably sharp, whereas the [[13/8]] is a little flat. Except for 9/7, 11/9, 14/9, and 18/11, all [[15-odd-limit]] intervals have [[consistent]] approximations in 43edo, making it an excellent tuning in the [[7-limit|7-]], [[11-limit|11-]], and [[13-limit]].  


The composer [[Juhan Puhm]] uses 43edo in some of his meantone suites for fortepiano and prefers it to [[31edo]].
=== Prime harmonics ===
{{Harmonics in equal|43|columns=11}}
{{Harmonics in equal|43|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}


In the 13-limit, we get two versions of meantone equivalent in 43et, one, [[Meantone family #Meridetone|meridetone]], tempering out [[78/77]], the other, [[Meantone family #Grosstone|grosstone]], [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, and grosstone {{val| 0 1 4 10 18 -16 }}; 43 supplies the [[optimal patent val]] for meridetone.
=== As a tuning for other temperaments ===
Besides the syntonic comma, 43et also tempers out the [[hypovishnuzma]] and the [[escapade comma]], so that six chromatic semitones make a perfect fourth and eight minor seconds make a major sixth. In the 7-limit, it supports septimal meantone, as it tempers out [[126/125]], [[225/224]], and [[3136/3125]]. The version of 11-limit meantone is the one tempering out [[99/98]], [[176/175]], and [[441/440]], sometimes called [[huygens]]. In the 13-limit it supports [[meridetone]], which tempers out [[78/77]], and [[grosstone]], which tempers out [[144/143]]. Meridetone has generator map {{val| 0 1 4 10 18 27 }}, for which 43 supplies the [[optimal patent val]] for, and grosstone {{val| 0 1 4 10 18 -16 }}.


The 43 patent val {{val| 43 68 100 121 149 159 }} maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the [[jerome]] temperament, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7-, 11-, 13-, 17-, 19- and 23-limit. It also provides the optimal patent val for the 11- and 13-limit [[amavil]] temperament, which is not a meantone temperament. The [[thuja]] temperament is also a possibility, in which five generators, (~11/8)<sup>5</sup> = ~5/1, with [[mos]] of 15 and 28.
43edo's patent val {{val| 43 68 100 121 149 159 }} maps 5 to 100 steps, allowing the divison of 5 into 20 equal parts, leading to the [[jerome]] temperament, an interesting higher-limit system for which 43 supplies the optimal patent val in the 7-, 11-, 13-, 17-, 19-, and even 23-limit. It also provides the optimal patent val for the 11- and 13-limit [[amavil]] temperament, which is not meantone. [[Thuja]] is also a possibility, whose 11-limit extension makes five 11/8's stack to a major third (i.e. {{nowrap|(11/8)<sup>5</sup> → 5/1}}), with [[mos scale]]s of 15 and 28.


=== Prime harmonics ===
=== Subsets and supersets ===
Although not [[consistent]], it performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'', with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. Mappings for composite harmonics and ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for an almost-complete version of the first 32 harmonics in the harmonic series, although the limited consistency will give some unusual results. Indeed, the step size of 43edo is very close to the [[64/63|septimal comma (64/63)]], while two steps is close to [[32/31]], and four steps to [[16/15]].
{{Harmonics in equal|43}}
 
=== Divisors ===
43edo is the 14th [[prime edo]], following [[41edo]] and coming before [[47edo]].
43edo is the 14th [[prime edo]], following [[41edo]] and coming before [[47edo]].


== Intervals ==
== Intervals ==
The distance from C to C♯ is 3 edosteps (or keys, frets). Thus one edostep equals one third of a sharp.  
The distance from C to C♯ is 3 edosteps (or keys, frets). Thus one edostep equals one third of a sharp.  
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! #
! #
! Cents
! Cents
! Approximate 17-limit Ratios
! Approximate ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and vd2)
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| [[1/1]]
| P1
| P1
| perfect unison
| perfect unison
Line 35: Line 44:
|-
|-
| 1
| 1
| 27.907
| 27.9
| ''36/35'', 50/49, 64/63, 65/64, 66/65
| ''[[36/35]]'', [[50/49]], [[64/63]], [[65/64]], [[66/65]]
| ^1, d2
| ^1, d2
| up unison, dim 2nd
| up unison, dim 2nd
Line 42: Line 51:
|-
|-
| 2
| 2
| 55.814
| 55.8
| ''49/48'', 33/32
| [[26/25]], [[27/26]], [[33/32]], [[40/39]], ''[[49/48]]''
| vA1, ^d2
| vA1, ^d2
| downaug unison, updim 2nd
| downaug unison, updim 2nd
Line 49: Line 58:
|-
|-
| 3
| 3
| 83.721
| 83.7
| 25/24, 21/20, ''28/27'', 22/21, ''18/17''
| ''[[18/17]]'', [[21/20]], [[22/21]], [[25/24]], ''[[28/27]]''
| vm2
| A1, vm2
| downminor 2nd
| aug 1sn, downminor 2nd
| vEb
| D#, vEb
|-
|-
| 4
| 4
| 111.628
| 111.6
| 16/15, 15/14, 17/16
| [[15/14]], [[16/15]], [[17/16]]
| m2
| m2
| minor 2nd
| minor 2nd
Line 63: Line 72:
|-
|-
| 5
| 5
| 139.535
| 139.5
| 12/11, 13/12, 14/13
| [[12/11]], [[13/12]], [[14/13]]
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
Line 70: Line 79:
|-
|-
| 6
| 6
| 167.442
| 167.4
| 11/10
| [[11/10]]
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 77: Line 86:
|-
|-
| 7
| 7
| 195.349
| 195.3
| 9/8, 10/9
| [[9/8]], [[10/9]]
| M2
| M2
| major 2nd
| major 2nd
Line 84: Line 93:
|-
|-
| 8
| 8
| 223.256
| 223.3
| 8/7
| [[8/7]]
| ^M2
| ^M2, d3
| upmajor 2nd
| upmajor 2nd, dim 3rd
| ^E
| ^E, Fb
|-
|-
| 9
| 9
| 251.163
| 251.2
| 15/13
| [[15/13]]
| vA2, ^d3
| vA2, ^d3
| downaug 2nd, updim 3rd
| downaug 2nd, updim 3rd
Line 98: Line 107:
|-
|-
| 10
| 10
| 279.070
| 279.1
| 7/6, 13/11
| [[7/6]], [[13/11]], [[20/17]]
| vm3
| A2, vm3
| downminor 3rd
| aug 2nd, downminor 3rd
| vF
| E#, vF
|-
|-
| 11
| 11
| 306.977
| 307.0
| 6/5
| [[6/5]]
| m3
| m3
| minor 3rd
| minor 3rd
Line 112: Line 121:
|-
|-
| 12
| 12
| 334.884
| 334.9
| 39/32, 17/14
| [[17/14]], ''[[27/22]]'', [[39/32]], [[40/33]]
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
Line 119: Line 128:
|-
|-
| 13
| 13
| 362.791
| 362.8
| 16/13, 21/17, ''11/9''
| ''[[11/9]]'', [[16/13]], [[21/17]], [[26/21]]
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
Line 126: Line 135:
|-
|-
| 14
| 14
| 390.698
| 390.7
| 5/4
| [[5/4]]
| M3
| M3
| major 3rd
| major 3rd
Line 133: Line 142:
|-
|-
| 15
| 15
| 418.605
| 418.6
| ''9/7'', 14/11
| ''[[9/7]]'', [[14/11]]
| ^M3
| ^M3, d4
| upmajor 3rd
| upmajor 3rd, dim 4th
| ^F#
| ^F#, Gb
|-
|-
| 16
| 16
| 446.512
| 446.5
| 13/10
| [[13/10]], [[22/17]]
| vA3, ^d4
| vA3, ^d4
| downaug 3rd, updim 4th
| downaug 3rd, updim 4th
Line 147: Line 156:
|-
|-
| 17
| 17
| 474.419
| 474.4
| 21/16
| [[21/16]]
| v4
| v4
| down 4th
| down 4th
Line 154: Line 163:
|-
|-
| 18
| 18
| 502.326
| 502.3
| 4/3
| [[4/3]]
| P4
| P4
| perfect 4th
| perfect 4th
Line 161: Line 170:
|-
|-
| 19
| 19
| 530.233
| 530.2
| 15/11
| [[15/11]]
| ^4
| ^4
| up 4th
| up 4th
Line 168: Line 177:
|-
|-
| 20
| 20
| 558.140
| 558.1
| 11/8, 18/13
| [[11/8]], [[18/13]]
| vA4
| vA4
| downaug 4th
| downaug 4th
Line 175: Line 184:
|-
|-
| 21
| 21
| 586.047
| 586.0
| 45/32, 7/5, 24/17
| [[7/5]], [[24/17]], [[45/32]]
| A4, vd5
| A4, vd5
| aug 4th, downdim 5th
| aug 4th, downdim 5th
Line 182: Line 191:
|-
|-
| 22
| 22
| 613.953
| 614.0
| 64/45, 10/7, 17/12
| [[10/7]], [[17/12]], [[64/45]]
| ^A4, d5
| ^A4, d5
| upaug 4th, dim 5th
| upaug 4th, dim 5th
Line 189: Line 198:
|-
|-
| 23
| 23
| 641.860
| 641.9
| 16/11, 13/9
| [[13/9]], [[16/11]]
| ^d5
| ^d5
| updim 5th
| updim 5th
Line 196: Line 205:
|-
|-
| 24
| 24
| 669.767
| 669.8
| 22/15
| [[22/15]]
| v5
| v5
| down 5th
| down 5th
Line 203: Line 212:
|-
|-
| 25
| 25
| 697.674
| 697.7
| 3/2
| [[3/2]]
| P5
| P5
| perfect 5th
| perfect 5th
Line 210: Line 219:
|-
|-
| 26
| 26
| 725.581
| 725.6
| 32/21
| [[32/21]]
| ^5
| ^5
| up 5th
| up 5th
Line 217: Line 226:
|-
|-
| 27
| 27
| 753.488
| 753.5
| 20/13
| [[17/11]], [[20/13]]
| vA5, ^d6
| vA5, ^d6
| downaug 5th, updim 6th
| downaug 5th, updim 6th
Line 224: Line 233:
|-
|-
| 28
| 28
| 781.395
| 781.4
| ''14/9'', 11/7
| [[11/7]], ''[[14/9]]''
| vm6
| A5, vm6
| downminor 6th
| aug 5th, downminor 6th
| vBb
| A#, vBb
|-
|-
| 29
| 29
| 809.302
| 809.3
| 8/5
| [[8/5]]
| m6
| m6
| minor 6th
| minor 6th
Line 238: Line 247:
|-
|-
| 30
| 30
| 837.209
| 837.2
| 13/8, 34/21, ''18/11''
| [[13/8]], ''[[18/11]]'', [[21/13]], [[34/21]]
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 245: Line 254:
|-
|-
| 31
| 31
| 865.116
| 865.1
| 64/39, 28/17
| [[28/17]], [[33/20]], ''[[44/27]]'', [[64/39]]
| vM6
| vM6
| downmajor 6th
| downmajor 6th
Line 252: Line 261:
|-
|-
| 32
| 32
| 893.023
| 893.0
| 5/3
| [[5/3]]
| M6
| M6
| major 6th
| major 6th
Line 259: Line 268:
|-
|-
| 33
| 33
| 920.930
| 920.9
| 12/7, 22/13
| [[12/7]], [[22/13]], [[17/10]]
| ^M6
| ^M6, d7
| upmajor 6th
| upmajor 6th, dim 7th
| ^B
| ^B, Cb
|-
|-
| 34
| 34
| 948.837
| 948.8
| 26/15
| [[26/15]]
| vA6, ^d7
| vA6, ^d7
| downaug 6th, updim 7th
| downaug 6th, updim 7th
Line 273: Line 282:
|-
|-
| 35
| 35
| 976.744
| 976.7
| 7/4
| [[7/4]]
| vm7
| A6, vm7
| downminor 7th
| aug 6th, downminor 7th
| vC
| B#, vC
|-
|-
| 36
| 36
| 1004.651
| 1004.7
| 16/9, 9/5
| [[9/5]], [[16/9]]
| m7
| m7
| minor 7th
| minor 7th
Line 287: Line 296:
|-
|-
| 37
| 37
| 1032.558
| 1032.6
| 20/11
| [[20/11]]
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 294: Line 303:
|-
|-
| 38
| 38
| 1060.465
| 1060.5
| 11/6, 24/13, 13/7
| [[11/6]], [[13/7]], [[24/13]]
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 301: Line 310:
|-
|-
| 39
| 39
| 1088.372
| 1088.4
| 15/8, 28/15, 32/17
| [[15/8]], [[28/15]], [[32/17]]
| M7
| M7
| major 7th
| major 7th
Line 308: Line 317:
|-
|-
| 40
| 40
| 1116.279
| 1116.3
| 48/25, 40/21, ''27/14'', 21/11, ''17/9''
| ''[[17/9]]'', [[21/11]], ''[[27/14]]'', [[40/21]], [[48/25]]
| ^M7
| ^M7, d8
| upmajor 7th
| upmajor 7th, dim 8ve
| ^C#
| ^C#, Db
|-
|-
| 41
| 41
| 1144.186
| 1144.2
| ''96/49'', 64/33
| [[25/13]], [[39/20]], [[52/27]], [[64/33]], ''[[96/49]]''
| vA7, ^d8
| vA7, ^d8
| downaug 7th, updim 8ve
| downaug 7th, updim 8ve
Line 322: Line 331:
|-
|-
| 42
| 42
| 1172.093
| 1172.1
| ''35/18'', 49/25, 63/32, 65/33, 128/65
| ''[[35/18]]'', [[49/25]], [[63/32]], [[65/33]], [[128/65]]
| A7, v8
| A7, v8
| aug 7th, down 8ve
| aug 7th, down 8ve
Line 329: Line 338:
|-
|-
| 43
| 43
| 1200.000
| 1200.0
| 2/1
| [[2/1]]
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 336: Line 345:
|}
|}


Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation #Chords and Chord Progressions]].
<nowiki>*</nowiki> As a 17-limit system


== JI approximation ==
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].
[[File:43ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 29-limit intervals approximated in 43edo]]
=== Selected just intervals ===
==== 15-odd-limit mappings ====
The following table shows how [[15-odd-limit intervals]] are represented in 43edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
 
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| Direct mapping (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| [[16/15]], [[15/8]]
| 0.103
| 0.4
|-
| [[13/12]], [[24/13]]
| 0.962
| 3.4
|-
| [[14/11]], [[11/7]]
| 1.097
| 3.9
|-
| [[11/10]], [[20/11]]
| 2.438
| 8.7
|-
| '''[[16/13]], [[13/8]]'''
| '''3.318'''
| '''11.9'''
|-
| [[15/13]], [[26/15]]
| 3.422
| 12.3
|-
| [[7/5]], [[10/7]]
| 3.534
| 12.7
|-
| '''[[4/3]], [[3/2]]'''
| '''4.281'''
| '''15.3'''
|-
| '''[[5/4]], [[8/5]]'''
| '''4.384'''
| '''15.7'''
|-
| [[18/13]], [[13/9]]
| 5.243
| 18.8
|-
| [[15/11]], [[22/15]]
| 6.718
| 24.1
|-
| '''[[11/8]], [[16/11]]'''
| '''6.822'''
| 24.4
|-
| [[13/10]], [[20/13]]
| 7.702
| 27.6
|-
| [[15/14]], [[28/15]]
| 7.815
| 28.0
|-
| '''[[8/7]], [[7/4]]'''
| '''7.918'''
| '''28.4'''
|-
| [[9/8]], [[16/9]]
| 8.561
| 30.7
|-
| [[6/5]], [[5/3]]
| 8.665
| 31.0
|-
| [[13/11]], [[22/13]]
| 10.140
| 36.3
|-
| [[12/11]], [[11/6]]
| 11.102
| 39.8
|-
| [[14/13]], [[13/7]]
| 11.237
| 40.3
|-
| ''[[9/7]], [[14/9]]''
| ''11.428''
| ''40.9''
|-
| [[7/6]], [[12/7]]
| 12.199
| 43.7
|-
| ''[[11/9]], [[18/11]]''
| ''12.524''
| ''44.9''
|-
| [[10/9]], [[9/5]]
| 12.945
| 46.4
|}
{{15-odd-limit|43}}


== Notation ==
== Notation ==
=== Red-Blue Notation ===
Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A♯ and B♭ are distinct and the distance between them is one meride. The whole tone is divided into seven merides so this means we can use "third-sharps", "two-thirds-sharps", "third-flats", and "two-thirds-flats" to reach the remaining notes between A and B; notes elsewhere on the scale can be notated similarly.
Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A♯ and B♭ are distinct and the distance between them is one meride. The whole tone is divided into seven merides so this means we can use "third-sharps", "two-thirds-sharps", "third-flats", and "two-thirds-flats" to reach the remaining notes between A and B; notes elsewhere on the scale can be notated similarly.


For people who aren't colorblind, a red-note/blue-note system (similar to that proposed for sixth-tones/[[36edo]]) can be used. (This is a different use of color than Kite's [[color notation]].) Now we have the following sequence of notes, each separated by one meride: A, red A, blue A♯, A♯, B♭, red B♭, blue B, B. (Note that there are red flats and blue sharps, but no red sharps or blue flats, because the latter are enharmonically equivalent to simpler notes: blue B♭ is actually just A♯, for instance).
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp3-szg}}
 
The notes between A and B can then be notated as A, A{{naturalup}}, A{{sharpdown}}, A♯, B♭, B{{flatup}}, B{{naturaldown}}, B. Note that A♯ is enharmonic to B{{flatdown}}, and B♭ is enharmonic to A{{sharpup}}.


The diatonic semitone is four steps, so for the region between B and C (or, E and F), we can use: B, C♭, red C♭/blue B♯ (''they are enharmonic equivalents''), B♯, and C. All of the notes in 43edo therefore have unambiguous names except for two: red C♭/blue B♯, and red F♭/blue E♯. It might also be possible to design special symbols for those two notes (resembling a cross between the letters B and C in the former case, and E and F in the latter).
The notes from B to C are B, C♭, B{{sharpdown}}&nbsp;/&nbsp;C{{flatup}}, B♯, and C. Similarily, the notes from E to F are E, F♭, E{{sharpdown}}&nbsp;/&nbsp;F{{flatup}}, E♯, and F. As with the red/blue note system described below, all notes in 43edo therefore have only one name, except for B{{sharpdown}}&nbsp;/&nbsp;C{{flatup}} and E{{sharpdown}}&nbsp;/&nbsp;F{{flatup}}.


If red C♭ and blue B♯ (and red F♭/blue E♯) are instead forced to be distinct, but the requirement that all notes be equally spaced is maintained, then we end up with a ''completely'' unambiguous red-note/blue-note notation for [[45edo]], which is another meantone (actually, a [[flattone]]) system.
Double or even triple arrows may arise if the arrows are taken to have their own layer of enharmonic spellings.


=== Ups and downs notation ===
=== Kite's ups and downs notation ===
The following shows how [[ups and downs notation]] can notate the third-sharps and third-flats through extended [[Helmholtz-Ellis notation|Helmholtz&ndash;Ellis]] symbols:
In [[Kite's ups and downs notation]], the "third-sharp" becomes an up and the "two-thirds-sharp" becomes a downsharp.
{{Sharpness-sharp3-extended}}
Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{Ups and downs sharpness}}


=== Sagittal ===
=== Sagittal notation ===
The following table shows [[sagittal notation]] accidentals in one apotome for 43do.  
This notation uses the same sagittal sequence as [[36edo #Sagittal notation|36edo]].


{| class="wikitable center-all"
==== Evo flavor ====
! Steps
<imagemap>
| 0
File:43-EDO_Evo_Sagittal.svg
| 1
desc none
| 2
rect 80 0 300 50 [[Sagittal_notation]]
| 3
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
|-
rect 20 80 120 106 [[64/63]]
! Symbol
default [[File:43-EDO_Evo_Sagittal.svg]]
| [[File:Sagittal natural.png]]
</imagemap>
| [[File:Sagittal tai.png]]
 
| [[File:Sagittal sharp tao.png]]
==== Revo flavor ====
| [[File:Sagittal sharp.png]]
<imagemap>
|}
File:43-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
default [[File:43-EDO_Revo_Sagittal.svg]]
</imagemap>
 
=== Red-Blue notation ===
For people who are not colorblind, a red-note/blue-note system (similar to that proposed for [[36edo]]) can also be used. Note that this is different from [[Kite's color notation]]. We have the following sequence of notes, each separated by one meride: {{colored note|A}}, {{colored note|red|A}}, {{colored note|blue|A♯}}, {{colored note|A♯}}, {{colored note|B♭}}, {{colored note|red|B♭}}, {{colored note|blue|B}}, {{colored note|B}}. (Note that red sharps or blue flats are enharmonically equivalent to simpler notes: {{colored note|red|A♯}} is enharmonic to B♭, and {{colored note|blue|B♭}} is actually just A♯).
 
The diatonic semitone is four steps, so for the region between B and C, we can use: {{colored note|B}}, {{colored note|C♭}}, {{colored note|blue|B♯}}&nbsp;/&nbsp;{{colored note|red|C♭}} (they are enharmonic equivalents), {{colored note|B♯}}, and {{colored note|C}}. All of the notes in 43edo therefore have only one name except for {{colored note|blue|B♯}}&nbsp;/&nbsp;{{colored note|red|C♭}}, and {{colored note|blue|E♯}}&nbsp;/&nbsp;{{colored note|red|F♭}}. It might also be possible to design special symbols for those two notes (resembling a cross between the letters B and C in the former case, and E and F in the latter).
 
If {{colored note|red|C♭}} and {{colored note|blue|B♯}} (and {{colored note|red|F♭}}&nbsp;/&nbsp;{{colored note|blue|E♯}}) are instead forced to be distinct, but the requirement that all notes be equally spaced is maintained, then we end up with a ''completely'' single-name red-note/blue-note notation for [[45edo]], which is another meantone (actually, a [[flattone]]) system.
 
== Approximation to JI ==
[[File:43ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 43edo]]
 
=== Interval mappings ===
{{Q-odd-limit intervals}}
 
=== Higher-limit JI ===
Although not [[consistent]], 43edo performs quite well in very high prime limits. It has unambiguous mappings for most prime harmonics up to ''113'', after which the demands on its pitch resolution finally become too great. The exceptions are 23, 41, 71, 89, and 103, which have more than 35% relative error (10 cents absolute error). This high-limit capability is useful for approaches based on the harmonic series, such as for creating [[#Ringer 43|Ringer scales]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for a complete set of approximations to the first 16 harmonics in the harmonic series and an almost-complete approximation of the first 32 harmonics, although the limited consistency will give some unusual results.
 
Within harmonics 1–63, 43edo approximates harmonics 15, 31, 37, 61, and 63 close to exactly – within less than a cent (less than 3% relative error). Indeed, one step of 43edo is very close to the [[64/63|septimal comma (64/63)]]; similarly, two steps is close to [[32/31]], and four steps tunes [[16/15]] almost perfectly. It approximates 3, 9, 13, 27, 39, 43, 53 and 61 flat. It approximates 5, 7, 11, 17, 19, 21, 25, 29, 33, 47, 49, 51, 57 and 59 sharp. Overall this gives 43edo a slightly sharp tendency/feeling.


== 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 492: Line 421:
|-
|-
| 2.3
| 2.3
| {{monzo| -68 43 }}
| {{Monzo| -68 43 }}
| [{{val| 43 68 }}]
| {{Mapping| 43 68 }}
| +1.35
| +1.35
| 1.35
| 1.35
Line 500: Line 429:
| 2.3.5
| 2.3.5
| 81/80, 50331648/48828125
| 81/80, 50331648/48828125
| [{{val| 43 68 100 }}]
| {{Mapping| 43 68 100 }}
| +0.27
| +0.27
| 1.88
| 1.88
Line 507: Line 436:
| 2.3.5.7
| 2.3.5.7
| 81/80, 126/125, 17280/16807
| 81/80, 126/125, 17280/16807
| [{{val| 43 68 100 121 }}]
| {{Mapping| 43 68 100 121 }}
| -0.51
| −0.51
| 2.11
| 2.11
| 7.56
| 7.56
Line 514: Line 443:
| 2.3.5.7.11
| 2.3.5.7.11
| 81/80, 99/98, 126/125, 864/847
| 81/80, 99/98, 126/125, 864/847
| [{{val| 43 68 100 121 149 }}]
| {{Mapping| 43 68 100 121 149 }}
| -0.80
| −0.80
| 1.98
| 1.98
| 7.08
| 7.08
Line 521: Line 450:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 78/77, 81/80, 99/98, 126/125, 144/143
| 78/77, 81/80, 99/98, 126/125, 144/143
| [{{val| 43 68 100 121 149 159 }}]
| {{Mapping| 43 68 100 121 149 159 }}
| -0.52
| −0.52
| 1.91
| 1.91
| 6.85
| 6.85
Line 528: Line 457:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 78/77, 81/80, 99/98, 120/119, 126/125, 144/143
| 78/77, 81/80, 99/98, 120/119, 126/125, 144/143
| [{{val| 43 68 100 121 149 159 176 }}]
| {{Mapping| 43 68 100 121 149 159 176 }}
| -0.52
| −0.52
| 1.81
| 1.81
| 6.49
| 6.49
|-
| 2.3.5.7.11.13.17.19
| 78/77, 81/80, 99/98, 120/119, 126/125, 135/133, 144/143
| {{Mapping| 43 68 100 121 149 159 176 183 }}
| −0.87
| 1.77
| 6.34
|}
|}


=== Commas ===
=== Commas ===
This is a partial list of the [[commas]] that 43edo [[tempers out]] with its patent [[val]], {{val| 43 68 100 121 149 159 176 195 209 213 }}.
This is a partial list of the 19-limit [[commas]] that 43et [[tempering out|tempers out]] with its patent [[val]], {{val| 43 68 100 121 149 159 176 183 }}.


{| 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>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cent]]s
! [[Cent]]s
Line 548: Line 484:
| 3
| 3
| <abbr title="328256967394537077627/295147905179352825856">(42 digits)</abbr>
| <abbr title="328256967394537077627/295147905179352825856">(42 digits)</abbr>
| {{monzo| -68 43 }}
| {{Monzo| -68 43 }}
| 184.07
| 184.07
| Tribilawa
| Tribilawa
Line 555: Line 491:
| 5
| 5
| <abbr title="254803968/244140625">(18 digits)</abbr>
| <abbr title="254803968/244140625">(18 digits)</abbr>
| {{monzo| 20 5 -12}}
| {{Monzo| 20 5 -12 }}
| 74.01
| 74.01
| Saquadtrigu
| Saquadtrigu
| [[Hypovishnuzma]]
| [[Hypovishnuzma]]
|-
| 5
| <abbr title="50331648/48828125">(16 digits)</abbr>
| {{Monzo| 24 1 -11 }}
| 52.50
| Salegu
| [[Magus comma]]
|-
|-
| 5
| 5
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Gu
| Syntonic comma, Didymus comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
| 5
| 5
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| <abbr title="4294967296/4271484375">(20 digits)</abbr>
| {{monzo| 32 -7 -9 }}
| {{Monzo| 32 -7 -9 }}
| 9.49
| 9.49
| Sasa-tritrigu
| Sasa-tritrigu
Line 576: Line 519:
| 5
| 5
| <abbr title="295578376007080078125/295147905179352825856">(42 digits)</abbr>
| <abbr title="295578376007080078125/295147905179352825856">(42 digits)</abbr>
| {{monzo| -68 18 17 }}
| {{Monzo| -68 18 17 }}
| 2.52
| 2.52
| Quinla-seyo
| Quinla-seyo
| [[Vavoom family|Vavoom comma]]
| [[Vavoom comma]]
|-
| 7
| [[59049/57344]]
| {{Monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
|-
| 7
| 7
| <abbr title="1119744/1071875">(14 digits)</abbr>
| [[3645/3584]]
| {{monzo| 9 7 -5 -3 }}
| {{Monzo| -9 6 1 -1 }}
| 75.64
| 29.22
| Triru-aquingu
| Laruyo
| [[Superpine|Superpine comma]]
| Schismean comma
|-
|-
| 7
| 7
| <abbr title="2500000/2470629">(14 digits)</abbr>
| <abbr title="2500000/2470629">(14 digits)</abbr>
| {{monzo| 5 -1 7 -7 }}
| {{Monzo| 5 -1 7 -7 }}
| 20.46
| 20.46
| Sepruyo
| Sepruyo
| [[Merman|Mermisma]]
| [[Mermisma]]
|-
|-
| 7
| 7
| [[126/125]]
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| {{Monzo| 1 2 -3 1 }}
| 13.80
| 13.80
| Zotrigu
| Zotrigu
Line 604: Line 554:
| 7
| 7
| <abbr title="2097152/2083725">(14 digits)</abbr>
| <abbr title="2097152/2083725">(14 digits)</abbr>
| {{monzo| 21 -5 -2 -3 }}
| {{Monzo| 21 -5 -2 -3 }}
| 11.12
| 11.12
| Satriru-agugu
| Satriru-agugu
| [[Bronzisma]]
| [[Bronzisma]]
|-
| 7
| <abbr title="257298363/256000000">(18 digits)</abbr>
| {{Monzo| -14 7 -6 6 }}
| 8.76
| Latribizogu
| [[Historisma]]
|-
|-
| 7
| 7
| [[225/224]]
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
Line 618: Line 575:
| 7
| 7
| [[3136/3125]]
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| {{Monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquingu
Line 624: Line 581:
|-
|-
| 7
| 7
| [[703125/702464|(12 digits)]]
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| {{Monzo| -11 2 7 -3 }}
| 1.63
| 1.63
| Latriru-asepyo
| Latriru-asepyo
| [[Meter comma]]
| [[Meter]]
|-
| 11
| [[1350/1331]]
| {{Monzo| 1 3 2 0 -3}}
| 24.54
| Trilu-ayoyo
| Large tetracot diesis
|-
|-
| 11
| 11
| [[99/98]]
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| 17.58
| Loruru
| Loruru
Line 639: Line 603:
| 11
| 11
| [[176/175]]
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| 9.86
| Lorugugu
| Lorugugu
Line 646: Line 610:
| 11
| 11
| [[441/440]]
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| 3.93
| Luzozogu
| Luzozogu
Line 653: Line 617:
| 11
| 11
| [[4000/3993]]
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3}}
| {{Monzo| 5 -1 3 0 -3}}
| 3.03
| 3.03
| Triluyo
| Triluyo
Line 659: Line 623:
|-
|-
| 11
| 11
| [[131072/130977]]
| <abbr title="131072/130977">(12 digits)</abbr>
| {{monzo| 17 -5 0 -2 -1 }}
| {{Monzo| 17 -5 0 -2 -1 }}
| 1.26
| 1.26
| Salururu
| Salururu
| Olympia comma
| [[Olympia]]
|-
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{Monzo| 24 -6 0 1 -5 }}
| 0.51
| Saquinlu-azo
| [[Quartisma]]
|-
|-
| 13
| 13
| [[78/77]]
| [[78/77]]
| {{monzo| 1 1 0 -1 -1 1 }}
| {{Monzo| 1 1 0 -1 -1 1 }}
| 22.34
| 22.34
| Tholuru
| Tholuru
Line 674: Line 645:
| 13
| 13
| [[144/143]]
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| {{Monzo| 4 2 0 0 -1 -1 }}
| 12.06
| 12.06
| Thulu
| Thulu
Line 681: Line 652:
| 13
| 13
| [[169/168]]
| [[169/168]]
| {{monzo| -3 -1 0 -1 0 2 }}
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| 10.27
| Thothoru
| Thothoru
| Buzurgisma, dhanvantarisma
| Buzurgisma, dhanvantarisma
|-
| 13
| <abbr title="373248/371293">(12 digits)</abbr>
| {{Monzo| 9 6 0 0 0 -5 }}
| 9.09
| Quinthu
| [[Glacier comma]]
|-
|-
| 13
| 13
| [[364/363]]
| [[364/363]]
| {{monzo| 2 -1 0 1 -2 1 }}
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| 4.76
| Tholuluzo
| Tholuluzo
| Minor minthma
| Minor minthma
|-
| 13
| [[1001/1000]]
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
|-
| 13
| 13
| [[2080/2079]]
| [[2080/2079]]
| {{monzo| 5 -3 1 -1 -1 1 }}
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| 0.83
| Tholuruyo
| Tholuruyo
| Ibnsinma, sinaisma  
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
|-
| 17
| 17
| [[120/119]]
| [[120/119]]
| {{monzo| 3 1 1 -1 0 0 -1 }}
| {{Monzo| 3 1 1 -1 0 0 -1 }}
| 14.49
| 14.49
| Suruyo
| Suruyo
Line 709: Line 701:
| 17
| 17
| [[221/220]]
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
| {{Monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| 7.85
| Sotholugu
| Sotholugu
Line 716: Line 708:
| 17
| 17
| [[256/255]]
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| {{Monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| 6.78
| Sugu
| Sugu
Line 723: Line 715:
| 17
| 17
| [[273/272]]
| [[273/272]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 6.35
| 6.35
| Suthozo
| Suthozo
Line 730: Line 722:
| 17
| 17
| [[715/714]]
| [[715/714]]
| {{monzo| -1 -1 1 -1 1 1 -1 }}
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| 2.42
| 2.42
| Sutholoruyo
| Sutholoruyo
Line 737: Line 729:
| 19
| 19
| [[96/95]]
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| 18.13
| Nugu
| Nugu
| 19th Partial chroma
| 19th-partial chroma
|-
|-
| 19
| 19
| [[153/152]]
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| {{Monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| 11.35
| Nuso
| Nuso
Line 751: Line 743:
| 19
| 19
| [[171/170]]
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| {{Monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| 10.15
| Nosugu
| Nosugu
Line 758: Line 750:
| 19
| 19
| [[209/208]]
| [[209/208]]
| {{monzo| -4 0 0 0 1 -1 0 1 }}
| {{Monzo| -4 0 0 0 1 -1 0 1 }}
| 8.30
| 8.30
| Nothulo
| Nothulo
| Yama comma
| Yama comma
|-
|-
| 23
| 19
| 256/253
| [[210/209]]
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| 20.41
| 8.26
| Twethulu
| Nuluzoyo
|
| Spleen comma
|-
| 23
| [[323/322]]
| {{monzo| -1 0 0 -1 0 0 1 1 -1 }}
| 5.37
| Twethunosoru
| Major semivicema
|-
| 23
| [[392/391]]
| {{monzo| 3 0 0 2 0 0 -1 0 -1 }}
| 4.42
| Twethusuzozo
| Minor semivicema
|-
| 23
| [[484/483]]
| {{monzo| 2 -1 0 -1 2 0 0 0 -1 }}
| 3.58
| Twethuloloru
| Pittsburghisma
|-
| 23
| [[736/735]]
| {{monzo| 5 -1 -1 -2 0 0 0 0 1 }}
| 2.35
| Twethorurugu
| Harvardisma
|-
| 29
| 672/667
| {{monzo| 5 1 0 1 0 0 0 0 -1 -1 }}
| 12.93
| Twenutwethuzo
|
|-
| 29
| 204/203
| {{monzo| 2 1 0 -1 0 0 1 0 0 -1 }}
| 8.51
| Twenusoru
|
|-
| 29
| 320/319
| {{monzo| 6 0 1 0 -1 0 0 0 0 -1}}
| 5.42
| Twenuluyo
|
|-
| 29
| [[552/551]]
| {{monzo| 3 1 0 0 0 0 0 -1 1 -1 }}
| 3.14
| Twenutwethonu
| Marmosarubra
|-
| 29
| [[9251/9248]]
| {{monzo| -5 0 0 0 1 0 -2 0 0 2 }}
| 0.56
| Bitwenosu-alo
| Helevenisma
|-
| 31
| <abbr title="448630/447051">(12 digits)</abbr>
| {{monzo| 1 -1 1 1 -1 1 1 -1 -1 1 -1 }}
| 6.10
| Thiwutwenotwethunusotholuzoyo
| [[Junebug comma]]
|-
| 31
| [[621/620]]
| {{monzo| -2 3 -1 0 0 0 0 0 1 0 -1 }}
| 2.79
| Thiwutwethogu
| Owowhatsthisma
|-
| 31
| [[1024/1023]]
| {{monzo| 10 -1 0 0 -1 0 0 0 0 0 -1 }}
| 1.69
| Thiwulu
| Kilobyte comma
|-
| 31
| [[3969/3968]]
| {{monzo| -7 4 0 2 0 0 0 0 0 0 -1 }}
| 0.44
| Lathiwuzozo
| Yunzee comma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br> per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated Ratio<br>(Reduced)
! Associated<br>ratio*
! Temperament
! Temperaments
|-
|-
| 1
| 1
| 1\43
| 1\43
| 27.91
| 27.9
| 64/63
| 64/63
| [[Arch]]
| [[Arch]]
Line 880: Line 782:
| 1
| 1
| 2\43
| 2\43
| 55.81
| 55.8
| 33/32
| 33/32
| [[Escapade]]
| [[Escapade]]
|-
| 1
| 3\43
| 83.7
| 21/20
| [[Marvolo]]
|-
|-
| 1
| 1
| 4\43
| 4\43
| 111.63
| 111.6
| 16/15
| 16/15
| [[Vavoom]]
| [[Vavoom]]
Line 892: Line 800:
| 1
| 1
| 5\43
| 5\43
| 139.53
| 139.5
| 13/12
| 13/12
| [[Jerome]]
| [[Jerome]]
Line 898: Line 806:
| 1
| 1
| 6\43
| 6\43
| 167.44
| 167.4
| 11/10
| 11/10
| [[Superpine]]
| [[Superpine]]
Line 904: Line 812:
| 1
| 1
| 7\43
| 7\43
| 195.35
| 195.3
| 28/25
| 28/25
| [[Didacus]]
| [[Didacus]]
Line 910: Line 818:
| 1
| 1
| 8\43
| 8\43
| 223.26
| 223.3
| 8/7
| 8/7
| [[Kumonga]]
| [[Kumonga]]
Line 916: Line 824:
| 1
| 1
| 9\43
| 9\43
| 251.16
| 251.2
| 15/13
| 15/13
| [[Hemimeantone]]
| [[Hemimeantone]]
Line 922: Line 830:
| 1
| 1
| 10\43
| 10\43
| 279.07
| 279.1
| 75/64
| 75/64
| [[Decipentic]]
| [[Decipentic]]
Line 928: Line 836:
| 1
| 1
| 11\43
| 11\43
| 334.88
| 334.9
| 17/14
| 17/14
| [[Cohemimabila]]
| [[Cohemimabila]]
Line 934: Line 842:
| 1
| 1
| 13\43
| 13\43
| 362.79
| 362.8
| 16/13
| 16/13
| [[Submajor]] (43e) / interpental (43)
| [[Demibuzzard]] / interpental
|-
|-
| 1
| 1
| 14\43
| 14\43
| 390.70
| 390.7
| 5/4
| 5/4
| [[Amigo]]
| [[Amigo]]
Line 946: Line 854:
| 1
| 1
| 16\43
| 16\43
| 446.51
| 446.5
| 13/10
| 13/10
| [[Supersensi]]
| [[Supersensi]]
|-
| 1
| 17\43
| 474.4
| 21/16
| [[Buzzard]] (2.3.7)
|-
|-
| 1
| 1
| 18\43
| 18\43
| 502.33
| 502.3
| 4/3
| 4/3
| [[Meantone]]
| [[Meantone]]
Line 958: Line 872:
| 1
| 1
| 19\43
| 19\43
| 530.23
| 530.2
| 15/11
| 15/11
| [[Amavil]]
| [[Amavil]]
Line 964: Line 878:
| 1
| 1
| 20\43
| 20\43
| 558.14
| 558.1
| 11/8
| 11/8
| [[Thuja]]
| [[Thuja]]
Line 970: Line 884:
| 1
| 1
| 21\43
| 21\43
| 586.05
| 586.0
| 7/5
| 7/5
| [[Merman]]
| [[Merman]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave


== Detemperaments ==
== Detemperaments ==
Line 986: Line 901:


== Scales ==
== Scales ==
{{main|5- to 10-tone scales in 43edo}}
=== Harmonic scales ===
=== Harmonic scales ===
43edo represents the first 16 overtones of the [[harmonic series]] well (written as a ratio of 8:9:10:11:12:13:14:15:16 in [[just intonation]]) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.
43edo represents the first 16 overtones of the [[harmonic series]] well (written as a ratio of 8:9:10:11:12:13:14:15:16 in [[just intonation]]) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.
*7\43 (195.349¢) stands in for frequency ratio [[9/8]] (203.910¢) and [[10/9]] (182.404¢).
* 7\43 (195.3¢) stands in for frequency ratio [[9/8]] (203.9¢) and [[10/9]] (182.4¢).
*6\43 (156.522¢) stands in for [[11/10]] (165.004¢)
* 6\43 (156.5¢) stands in for [[11/10]] (165.0¢).
* 5\46 (130.435¢) stands in for [[12/11]] (150.637¢), [[13/12]] (138.573¢), and [[14/13]] (128.298¢).
* 5\46 (130.4¢) stands in for [[12/11]] (150.6¢), [[13/12]] (138.6¢), and [[14/13]] (128.3¢).
*4\43 (111.628¢) stands in for [[15/14]] (119.443¢) and [[16/15]] (111.731¢).
* 4\43 (111.6¢) stands in for [[15/14]] (119.4¢) and [[16/15]] (111.7¢).
 
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
Line 1,009: Line 923:
|-
|-
! 7
! 7
| style="font-size: 16px;" | B{{flatdown|40x40px}}
| style="font-size: 16px;" | A♯, B{{flatdown|36}}
|-
|-
! 9
! 9
Line 1,015: Line 929:
|-
|-
! 11
! 11
| style="font-size: 16px;" | F{{sharpdown|40x40px}}, F{{naturalup2|40x40px}}
| style="font-size: 16px;" | E𝄪, F{{sharpdown|36}}, F{{naturalup2|36}}
|-
|-
! 13
! 13
| style="font-size: 16px;" | A{{flatup|40x40px}}
| style="font-size: 16px;" | B♭♭♭, A{{flatup|36}}
|-
|-
! 15
! 15
Line 1,024: Line 938:
|}
|}


==Music==
=== Mos scales ===
===Modern renderings===
{{Main| List of MOS scales in 43edo }}
;{{W|Johann Sebastian Bach}}
* Meantone[5]: 7 7 11 7 11
*[https://www.youtube.com/watch?v=E7W-t2KDeSs "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* Meantone[7]: 7 7 4 7 7 7 4
 
=== Other meantone scales ===
; Major scales
* Ionian Pentatonic: 14 4 7 14 4
 
; Minor scales
* Minor Harmonic: 7 4 7 7 4 10 4
* Minor Harmonic Pentatonic: 7 4 14 14 4
* Minor Hexatonic: 7 4 7 7 11 7
* Minor Melodic: 7 4 7 7 7 7 4
 
; Modal scales
* Mixolydian Harmonic: 14 4 7 4 7 7
* Mixolydian Pentatonic: 14 4 7 11 7
* Phrygian Dominant: 4 10 4 7 4 7 7
* Phrygian Dominant Hexatonic: 4 10 4 7 11 7
* Phrygian Dominant Pentatonic: 14 4 7 4 14
* Phrygian Pentatonic: 4 7 14 4 14
 
; Blues scales
* Blues Aeolian Hexatonic: 11 7 4 3 4 14
* Blues Aeolian Pentatonic I: 11 7 7 4 14
* Blues Aeolian Pentatonic II: 11 14 4 7 7
* Blues Bright Double Harmonic: 4 10 4 7 4 7 3 4
* Blues Dark Double Harmonic: 7 4 7 4 3 4 10 4
* Blues Dorian Hexatonic: 11 7 7 7 4 7
* Blues Dorian Pentatonic: 11 14 7 4 7
* Blues Dorian Septatonic: 11 7 4 3 7 4 7
* Blues Harmonic Hexatonic: 7 4 7 7 14 4
* Blues Harmonic Septatonic: 11 7 4 3 4 10 4
* Blues Leading: 11 7 4 3 11 3 4
* Blues Minor: 11 7 4 3 11 7
* Blues Minor Maj7: 11 7 4 3 14 4
* Blues Pentachordal: 7 4 7 4 3 18
* Hyperblue Dorian: 11 7 2 5 9 2 7
* Hyperblue Harmonic: 11 7 2 5 3 12 3
 
; Others
* Akebono I: 7 4 14 7 11
* Dominant Pentatonic: 7 7 11 11 7
* Double Harmonic: 4 10 4 7 4 10 4
* Hirajoshi: 7 4 14 4 14
* Javanese Pentachordal: 4 7 11 3 18
* Picardy Hexatonic: 7 7 4 7 4 14
* Picardy Pentatonic: 7 7 11 4 14


;{{W|Nicolaus Bruhns}}
=== Other notable scales ===
*[https://www.youtube.com/watch?v=GkuUVQYpjo4 ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* Fossa pentatonic scale (approximated from [[catnip]] in [[60edo]]): 5 14 6 6 12
* [[Magnetosphere scale]] (approximated from [[Hexany 1728]]): 4 10 11 11 7


===21st century===
== Instruments ==
;[[Bryan Deister]]
*[[Lumatone mapping for 43edo]]
*[https://www.youtube.com/watch?v=pALxebjbhZo ''microtonal improvisation in 43edo''] (2023)
*[[Skip fretting system 43 2 9]]


;[[Peter Kosmorsky]]
=== Keyboards ===
*[[:File:43_edo_counterpoint.mid|43 edo counterpoint.mid]] [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3 mp3]{{dead link}} – in meantone (late 2011)
A possible isomorphic keyboard layout for 43edo:
[[File:Fifth Comma Meantone Keyboard Layout.svg|800px|none|thumb]]


;[[MortisTheneRd]]
== Music ==
*[https://www.youtube.com/watch?v=WyU38ESUBqo Shiro Sagisu - ''Pensées Intimes''] (2024)
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=v3mBkctQ4SI ''Prelude in C minor'', BWV 999] (1717–1723) – transposed into E minor, arranged for Organ and rendered by [[Claudi Meneghin]] (2024)
* [https://www.youtube.com/watch?v=u3ss3H2x_QA "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=E7W-t2KDeSs "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=qyWfguU0iQM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)


;[[Juhan Puhm]] ([http://juhanpuhmmusic.ca site])
; {{W|Nicolaus Bruhns}}
*[https://www.youtube.com/watch?v=I68hwh45CyQ ''Meantone Suite V in D Minor''] [http://juhanpuhmmusic.ca/Juhan-Puhm-Meantone-Suite-V-D-Minor.pdf score] (2017)
* [https://www.youtube.com/watch?v=GkuUVQYpjo4 ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [https://www.youtube.com/watch?v=UYaZZXUrGeA ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)


;[[Randy Wells]]
; {{W|John Bull (composer)|John Bull}}
*[https://www.youtube.com/watch?v=xN-QzI6OgGs ''Time Travel''] (2021)
* [https://www.youtube.com/watch?v=hkW5aqnhaSc ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2026)


==Articles==
; {{W|Frédéric Chopin}}
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Harmonic-Resources-43Et-EMT-43EBMT.pdf ''Harmonic Resources of 43Et EMT and 43EBMT''] by Juhan Puhm (2018)
* [https://www.youtube.com/watch?v=VyEKLxAtWm4 ''Prelude'', Op. 28, No. 4] (1838) – arranged for organ, rendered by Claudi Meneghin (2021)
* ''"Waterfall" Étude from 12 Études, op. 10'' (1829–1832)
** [https://www.youtube.com/shorts/m408V08QAMI Sine wave version] &mdash; rendered by Claudi Meneghin (2025)
** [https://www.youtube.com/shorts/oZiYri-sDYo Fortepiano version] &mdash; rendered by Claudi Meneghin (2025)


==Diagrams==
; {{W|Gabriel Fauré}}
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keys-and-Modes-of-43Et.pdf ''Keys and Modes of 43Et''] by Juhan Puhm (2016)
* [https://www.youtube.com/watch?v=72YtMnmu_sw ''Pavane, Op. 50''] (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2026)
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keyboard-Mapping-for-43Et.pdf ''Keyboard Mapping for 43Et''] by Juhan Puhm (2017)
*[http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-43Et.pdf ''Mapping Range for 43Et''] by Juhan Puhm (2017)


==Instruments==
; {{W|Toby Fox}}
*[[Lumatone mapping for 43edo]]
* [https://www.youtube.com/shorts/r8gvT-CZLEI ''Shop 3''] via ''{{W|Deltarune}} Chapter 1'' (2018) – microtonal cover in 43edo by [[Bryan Deister]] (2026)
*[[Skip fretting system 43 2 9]]
 
; {{W|George Frideric Handel}}
* [https://www.youtube.com/watch?v=l5g9XvUNaVg ''Suite in D minor HWV 428 for Harpsichord - Allemande''] (1720) – rendered by Claudi Meneghin (2024)
 
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=nV_jmn31Kiw ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
 
; {{W|Shirō Sagisu}}
* [https://www.youtube.com/watch?v=WyU38ESUBqo ''Pensées Intimes''] – rendered by [[MortisTheneRd]] (2024)
 
=== 21st century ===
;; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=t83BNX1g6Jg ''DOUBLE FUGUE on «Old Mc Donald» + «Shave & a Haircut», tuned into E43 (1/5-comma meantone)''] (2024)
 
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=pALxebjbhZo ''microtonal improvisation in 43edo''] (2023)
* [https://www.youtube.com/shorts/URUCEOW3Mqo ''43edo improv''] (2025)
* [https://www.youtube.com/shorts/f0zt-iBln44 ''Being for the Benefit of Mr. Kite! - The Beatles (microtonal cover in 43edo)''] (2025)
* [https://www.youtube.com/shorts/Qh5rjmsfwE0 ''43edo improv''] (2026)
* [https://www.youtube.com/watch?v=j5qbzEPRUUY ''Waltz in 43edo''] (2026)
 
; [[Cale Gibbard]]
* [https://www.youtube.com/watch?v=nUoTzgi8FtM 43edo fun with A, Bbb, Cbbb] (2023)
 
; [[Peter Kosmorsky]]
* [[:File:43_edo_counterpoint.mid|43 edo counterpoint.mid]] [http://micro.soonlabel.com/gene_ward_smith/Others/Kosmorsky/43%20edo%20counterpoint.mp3 mp3]{{dead link}} (late 2011) – in meantone
 
; [[Budjarn Lambeth]]
* ''Gamelan-Inspired Improvisation in 43edo, Fossa Scale'' (Nov 2024) - [https://www.youtube.com/watch?v=KsB9T9_6cGk YouTube]
 
; [[Juhan Puhm]] ([http://juhanpuhmmusic.ca site])
* ''Meantone Suite V in D Minor'' (2017) – [https://www.youtube.com/watch?v=I68hwh45CyQ YouTube] | [http://juhanpuhmmusic.ca/Juhan-Puhm-Meantone-Suite-V-D-Minor.pdf score]
 
; [[Sevish]]
* Mystify (2025) – [https://www.youtube.com/watch?v=NgXXTMS5YPc Youtube] | [https://sevish.bandcamp.com/track/mystify Bandcamp]
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=xN-QzI6OgGs ''Time Travel''] (2021)
 
; [[Xotla]]
* "Beebounce" from ''Jazzbeetle'' (2023) – [https://open.spotify.com/track/4PzANNtxXsNEsdApnYKgHK Spotify] | [https://xotla.bandcamp.com/track/beebounce-43edo Bandcamp] | [https://youtu.be/EZIg5fojFfE YouTube] – jazzy big band electronic hybrid


==References==
== References ==
<references />
<references />
== External links ==
=== Articles ===
* [http://tonalsoft.com/enc/m/meride.aspx méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Harmonic-Resources-43Et-EMT-43EBMT.pdf ''Harmonic Resources of 43Et EMT and 43EBMT''] by Juhan Puhm (2018)
=== Diagrams ===
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keys-and-Modes-of-43Et.pdf ''Keys and Modes of 43Et''] by Juhan Puhm (2016)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Keyboard-Mapping-for-43Et.pdf ''Keyboard Mapping for 43Et''] by Juhan Puhm (2017)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-43Et.pdf ''Mapping Range for 43Et''] by Juhan Puhm (2017)


[[Category:Meantone]]
[[Category:Meantone]]
Retrieved from "https://en.xen.wiki/w/43edo"