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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}}


== Theory ==
== History ==
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].
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 meantone suites for fortepiano and prefers it to [[31edo]].
The composer [[Juhan Puhm]] uses 43edo in some of his fortepiano suites and prefers it to [[31edo]].


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.
== Theory ==
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 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.
=== 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)}}


=== Prime harmonics ===
=== As a tuning for other temperaments ===
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]].
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 }}.


{{Harmonics in equal|43}}
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.


=== Divisors ===
=== Subsets and supersets ===
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 36: 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 43: 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 50: 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 64: 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 71: Line 79:
|-
|-
| 6
| 6
| 167.442
| 167.4
| 11/10
| [[11/10]]
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 78: Line 86:
|-
|-
| 7
| 7
| 195.349
| 195.3
| 9/8, 10/9
| [[9/8]], [[10/9]]
| M2
| M2
| major 2nd
| major 2nd
Line 85: 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 99: 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 113: 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 120: 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 127: Line 135:
|-
|-
| 14
| 14
| 390.698
| 390.7
| 5/4
| [[5/4]]
| M3
| M3
| major 3rd
| major 3rd
Line 134: 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 148: Line 156:
|-
|-
| 17
| 17
| 474.419
| 474.4
| 21/16
| [[21/16]]
| v4
| v4
| down 4th
| down 4th
Line 155: Line 163:
|-
|-
| 18
| 18
| 502.326
| 502.3
| 4/3
| [[4/3]]
| P4
| P4
| perfect 4th
| perfect 4th
Line 162: Line 170:
|-
|-
| 19
| 19
| 530.233
| 530.2
| 15/11
| [[15/11]]
| ^4
| ^4
| up 4th
| up 4th
Line 169: Line 177:
|-
|-
| 20
| 20
| 558.140
| 558.1
| 11/8, 18/13
| [[11/8]], [[18/13]]
| vA4
| vA4
| downaug 4th
| downaug 4th
Line 176: 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 183: 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 190: Line 198:
|-
|-
| 23
| 23
| 641.860
| 641.9
| 16/11, 13/9
| [[13/9]], [[16/11]]
| ^d5
| ^d5
| updim 5th
| updim 5th
Line 197: Line 205:
|-
|-
| 24
| 24
| 669.767
| 669.8
| 22/15
| [[22/15]]
| v5
| v5
| down 5th
| down 5th
Line 204: Line 212:
|-
|-
| 25
| 25
| 697.674
| 697.7
| 3/2
| [[3/2]]
| P5
| P5
| perfect 5th
| perfect 5th
Line 211: Line 219:
|-
|-
| 26
| 26
| 725.581
| 725.6
| 32/21
| [[32/21]]
| ^5
| ^5
| up 5th
| up 5th
Line 218: 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 225: 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 239: 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 246: 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 253: Line 261:
|-
|-
| 32
| 32
| 893.023
| 893.0
| 5/3
| [[5/3]]
| M6
| M6
| major 6th
| major 6th
Line 260: 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 274: 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 288: Line 296:
|-
|-
| 37
| 37
| 1032.558
| 1032.6
| 20/11
| [[20/11]]
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 295: 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 302: 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 309: 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 323: 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 330: Line 338:
|-
|-
| 43
| 43
| 1200.000
| 1200.0
| 2/1
| [[2/1]]
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 337: 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]].
=== 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"
== Notation ==
|+style=white-space:nowrap| Direct mapping (even if inconsistent)
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.
|-
 
! Interval, complement
=== Stein–Zimmermann–Gould notation ===
! Error (abs, [[Cent|¢]])
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
! Error (rel, [[Relative cent|%]])
{{Sharpness-sharp3-szg}}
|-
 
| [[16/15]], [[15/8]]
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}}.
| 0.103
 
| 0.4
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}}.
|-
 
| [[13/12]], [[24/13]]
Double or even triple arrows may arise if the arrows are taken to have their own layer of enharmonic spellings.
| 0.962
 
| 3.4
=== Kite's ups and downs notation ===
|-
In [[Kite's ups and downs notation]], the "third-sharp" becomes an up and the "two-thirds-sharp" becomes a downsharp.
| [[14/11]], [[11/7]]
Note that downsharp can be respelled as dup (double-up), and upflat as dud.
| 1.097
{{Ups and downs sharpness}}
| 3.9
 
|-
=== Sagittal notation ===
| [[11/10]], [[20/11]]
This notation uses the same sagittal sequence as [[36edo #Sagittal notation|36edo]].
| 2.438
 
| 8.7
==== Evo flavor ====
|-
<imagemap>
| '''[[16/13]], [[13/8]]'''
File:43-EDO_Evo_Sagittal.svg
| '''3.318'''
desc none
| '''11.9'''
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| [[15/13]], [[26/15]]
rect 20 80 120 106 [[64/63]]
| 3.422
default [[File:43-EDO_Evo_Sagittal.svg]]
| 12.3
</imagemap>
|-
 
| [[7/5]], [[10/7]]
==== Revo flavor ====
| 3.534
<imagemap>
| 12.7
File:43-EDO_Revo_Sagittal.svg
|-
desc none
| '''[[4/3]], [[3/2]]'''
rect 80 0 300 50 [[Sagittal_notation]]
| '''4.281'''
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| '''15.3'''
rect 20 80 120 106 [[64/63]]
|-
default [[File:43-EDO_Revo_Sagittal.svg]]
| '''[[5/4]], [[8/5]]'''
</imagemap>
| '''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 ==
=== Red-Blue notation ===
=== 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♯).


Because 43edo is a meantone system, this makes it easier to adapt traditional Western notation to it than to some other tunings. A# and Bb 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.
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).


Alternatively, 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#, Bb, red Bb, 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 Bb is actually just A#, for instance).
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.


The diatonic semitone is four steps, so for the region between B and C (or, E and F), we can use: B, Cb, red Cb/blue B# (''they are enharmonic equivalents''), B#, and C. All of the notes in 43edo therefore have unambiguous names except for two: red Cb/blue B#, and red Fb/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).
== Approximation to JI ==
[[File:43ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 43edo]]


If red Cb and blue B# (and red Fb/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.
=== Interval mappings ===
{{Q-odd-limit intervals}}


=== Sagittal ===
=== Higher-limit JI ===
The following table shows [[sagittal notation]] accidentals in one apotome for 43do.  
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.  


{| class="wikitable center-all"
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.
! Steps
| 0
| 1
| 2
| 3
|-
! Symbol
| [[File:Sagittal natural.png]]
| [[File:Sagittal tai.png]]
| [[File:Sagittal sharp tao.png]]
| [[File:Sagittal sharp.png]]
|}


== 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 489: 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 497: 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 504: 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 511: 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 518: 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 525: 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 }}.
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 545: 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
| 43-comma
| 43-comma
|-
|-
| 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
|  
| [[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 573: 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
| [[Vavoom family|Vavoom comma]]
| [[Vavoom comma]]
|-
| 7
| [[59049/57344]]
| {{Monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[3645/3584]]
| {{Monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| 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
|  
| [[Mermisma]]
|-
|-
| 7
| 7
| [[126/125]]
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| {{Monzo| 1 2 -3 1 }}
| 13.80
| 13.80
| Zotrigu
| Zotrigu
Line 594: 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 608: Line 575:
| 7
| 7
| [[3136/3125]]
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| {{Monzo| 6 0 -5 2 }}
| 6.08
| 6.08
| Zozoquingu
| Zozoquingu
| Hemimean comma
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[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 622: 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 629: 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
| Werckisma
| Werckisma
|-
| 11
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3}}
| 3.03
| Triluyo
| Wizardharry comma, pine comma
|-
| 11
| <abbr title="131072/130977">(12 digits)</abbr>
| {{Monzo| 17 -5 0 -2 -1 }}
| 1.26
| Salururu
| [[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 643: 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 650: 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
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Tholuluzo
| Minor minthma
|-
| 13
| [[1001/1000]]
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[2080/2079]]
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| Tholuruyo
| 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 664: 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 671: 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
| Charisma, septendecimal kleisma
| Charisma, septendecimal kleisma
|-
| 17
| [[273/272]]
| {{Monzo| 5 1 -1 0 0 0 0 -1 }}
| 6.35
| Suthozo
| Tannisma
|-
| 17
| [[715/714]]
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| 2.42
| Sutholoruyo
| September comma
|-
|-
| 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
| [[153/152]]
| {{Monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{Monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
|-
| 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
Line 691: Line 756:
|-
|-
| 19
| 19
| [[273/272]]
| [[210/209]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| 6.35
| 8.26
| Suthozo
| Nuluzoyo
| Tannisma
| Spleen 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 716: 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 728: Line 800:
| 1
| 1
| 5\43
| 5\43
| 139.53
| 139.5
| 13/12
| 13/12
| [[Jerome]]
| [[Jerome]]
|-
| 1
| 6\43
| 167.4
| 11/10
| [[Superpine]]
|-
|-
| 1
| 1
| 7\43
| 7\43
| 195.35
| 195.3
| 28/25
| 28/25
| [[Didacus]]
| [[Didacus]]
Line 740: Line 818:
| 1
| 1
| 8\43
| 8\43
| 223.26
| 223.3
| 8/7
| 8/7
| [[Kumonga]]
| [[Kumonga]]
Line 746: Line 824:
| 1
| 1
| 9\43
| 9\43
| 251.16
| 251.2
| 15/13
| 15/13
| [[Hemimeantone]]
| [[Hemimeantone]]
Line 752: Line 830:
| 1
| 1
| 10\43
| 10\43
| 279.07
| 279.1
| 75/64
| 75/64
| [[Decipentic]]
| [[Decipentic]]
Line 758: Line 836:
| 1
| 1
| 11\43
| 11\43
| 334.88
| 334.9
| 17/14
| 17/14
| [[Cohemimabila]]
| [[Cohemimabila]]
Line 764: 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 776: 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 788: Line 872:
| 1
| 1
| 19\43
| 19\43
| 530.23
| 530.2
| 15/11
| 15/11
| [[Amavil]]
| [[Amavil]]
Line 794: Line 878:
| 1
| 1
| 20\43
| 20\43
| 558.14
| 558.1
| 11/8
| 11/8
| [[Thuja]]
| [[Thuja]]
Line 800: 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 816: Line 901:


== Scales ==
== Scales ==
* [[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 (195.3¢) stands in for frequency ratio [[9/8]] (203.9¢) and [[10/9]] (182.4¢).
* 6\43 (156.5¢) stands in for [[11/10]] (165.0¢).
* 5\46 (130.4¢) stands in for [[12/11]] (150.6¢), [[13/12]] (138.6¢), and [[14/13]] (128.3¢).
* 4\43 (111.6¢) stands in for [[15/14]] (119.4¢) and [[16/15]] (111.7¢).


* 7\43 (195.349¢) stands in for frequency ratio [[9/8]] (203.910¢) and [[10/9]] (182.404¢).
{| class="wikitable center-all"
* 6\43 (156.522¢) stands in for [[11/10]] (165.004¢)
|-
* 5\46 (130.435¢) stands in for [[12/11]] (150.637¢), [[13/12]] (138.573¢), and [[14/13]] (128.298¢).
! Harmonic
* 4\43 (111.628¢) stands in for [[15/14]] (119.443¢) and [[16/15]] (111.731¢).
! Note (starting from C)
|-
! 1
| style="font-size: 16px;" | C
|-
! 3
| style="font-size: 16px;" | G
|-
! 5
| style="font-size: 16px;" | E
|-
! 7
| style="font-size: 16px;" | A♯, B{{flatdown|36}}
|-
! 9
| style="font-size: 16px;" | D
|-
! 11
| style="font-size: 16px;" | E𝄪, F{{sharpdown|36}}, F{{naturalup2|36}}
|-
! 13
| style="font-size: 16px;" | B♭♭♭, A{{flatup|36}}
|-
! 15
| style="font-size: 16px;" | B
|}
 
=== Mos scales ===
{{Main| List of MOS scales in 43edo }}
* Meantone[5]: 7 7 11 7 11
* 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
 
=== Other notable scales ===
* Fossa pentatonic scale (approximated from [[catnip]] in [[60edo]]): 5 14 6 6 12
* [[Magnetosphere scale]] (approximated from [[Hexany 1728]]): 4 10 11 11 7
 
== Instruments ==
*[[Lumatone mapping for 43edo]]
*[[Skip fretting system 43 2 9]]
 
=== Keyboards ===
A possible isomorphic keyboard layout for 43edo:
[[File:Fifth Comma Meantone Keyboard Layout.svg|800px|none|thumb]]


== Music ==
== Music ==
=== Modern renderings ===
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
; {{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=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)


; {{W|Nicolaus Bruhns}}
; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=GkuUVQYpjo4 ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)
* [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)
; {{W|John Bull (composer)|John Bull}}
* [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)
; {{W|Frédéric Chopin}}
* [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)
; {{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 ===
=== 21st century ===
;; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=t83BNX1g6Jg ''DOUBLE FUGUE on «Old Mc Donald» + «Shave & a Haircut», tunedo into E43 (1/5-comma meantone)''] (2024)
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=pALxebjbhZo ''microtonal improvisation in 43edo''] (2023)
* [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]]
; [[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}} – in meantone (late 2011)
* [[: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


; [[MortisTheneRd]]
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=WyU38ESUBqo Shiro Sagisu - ''Pensées Intimes''] (2024)
* ''Gamelan-Inspired Improvisation in 43edo, Fossa Scale'' (Nov 2024) - [https://www.youtube.com/watch?v=KsB9T9_6cGk YouTube]


; [[Juhan Puhm]] ([http://juhanpuhmmusic.ca site])
; [[Juhan Puhm]] ([http://juhanpuhmmusic.ca site])
* [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)
* ''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]]
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=xN-QzI6OgGs ''Time Travel''] (2021)
* [https://www.youtube.com/watch?v=xN-QzI6OgGs ''Time Travel''] (2021)


== Articles ==
; [[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 />
 
== 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)
* [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 ==
=== 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-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-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)
* [http://juhanpuhmmusic.ca/Juhan-Puhm-Compendium-Musica-Mapping-Range-for-43Et.pdf ''Mapping Range for 43Et''] by Juhan Puhm (2017)
== Instruments ==
* [[Lumatone mapping for 43edo]]
* [[Skip fretting system 43 2 9]]
== References ==
<references/>


[[Category:Meantone]]
[[Category:Meantone]]

Latest revision as of 01:42, 30 May 2026

← 42edo 43edo 44edo →
Prime factorization 43 (prime)
Step size 27.907 ¢ 
Fifth 25\43 (697.674 ¢)
Semitones (A1:m2) 3:4 (83.72 ¢ : 111.6 ¢)
Consistency limit 7
Distinct consistency limit 7

43 equal divisions of the octave (abbreviated 43edo or 43ed2), also called 43-tone equal temperament (43tet) or 43 equal temperament (43et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 43 equal parts of about 27.9 ¢ each. Each step represents a frequency ratio of 21/43, or the 43rd root of 2.

History

The French Baroque acoustician 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.[1]

The composer Juhan Puhm uses 43edo in some of his fortepiano suites and prefers it to 31edo.

Theory

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-, 11-, and 13-limit.

Prime harmonics

Approximation of prime harmonics in 43edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -4.3 +4.4 +7.9 +6.8 -3.3 +6.7 +9.5 +13.6 +3.0 -0.8
Relative (%) +0.0 -15.3 +15.7 +28.4 +24.4 -11.9 +23.9 +33.9 +48.7 +10.7 -3.0
Steps
(reduced) 		43
(0) 		68
(25) 		100
(14) 		121
(35) 		149
(20) 		159
(30) 		176
(4) 		183
(11) 		195
(23) 		209
(37) 		213
(41)
Approximation of prime harmonics in 43edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -0.2 -10.5 -9.2 +4.3 -8.4 +1.3 -0.6 +4.4 -12.3 -4.5 -1.7
Relative (%) -0.6 -37.5 -32.9 +15.3 -30.1 +4.6 -2.2 +15.8 -43.9 -16.2 -6.3
Steps
(reduced) 		224
(9) 		230
(15) 		233
(18) 		239
(24) 		246
(31) 		253
(38) 		255
(40) 		261
(3) 		264
(6) 		266
(8) 		271
(13)

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 ⟨0 1 4 10 18 27], for which 43 supplies the optimal patent val for, and grosstone ⟨0 1 4 10 18 -16].

43edo's patent 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. (11/8)5 → 5/1), with mos scales of 15 and 28.

Subsets and supersets

43edo is the 14th prime edo, following 41edo and coming before 47edo.

Intervals

The distance from C to C♯ is 3 edosteps (or keys, frets). Thus one edostep equals one third of a sharp.

# Cents Approximate ratios* Ups and downs notation
(EUs: v3A1 and vd2)
0 0.0 1/1 P1 perfect unison D
1 27.9 36/35, 50/49, 64/63, 65/64, 66/65 ^1, d2 up unison, dim 2nd ^D, Ebb
2 55.8 26/25, 27/26, 33/32, 40/39, 49/48 vA1, ^d2 downaug unison, updim 2nd vD#, ^Ebb
3 83.7 18/17, 21/20, 22/21, 25/24, 28/27 A1, vm2 aug 1sn, downminor 2nd D#, vEb
4 111.6 15/14, 16/15, 17/16 m2 minor 2nd Eb
5 139.5 12/11, 13/12, 14/13 ^m2 upminor 2nd ^Eb
6 167.4 11/10 vM2 downmajor 2nd vE
7 195.3 9/8, 10/9 M2 major 2nd E
8 223.3 8/7 ^M2, d3 upmajor 2nd, dim 3rd ^E, Fb
9 251.2 15/13 vA2, ^d3 downaug 2nd, updim 3rd vE#, ^Fb
10 279.1 7/6, 13/11, 20/17 A2, vm3 aug 2nd, downminor 3rd E#, vF
11 307.0 6/5 m3 minor 3rd F
12 334.9 17/14, 27/22, 39/32, 40/33 ^m3 upminor 3rd ^F
13 362.8 11/9, 16/13, 21/17, 26/21 vM3 downmajor 3rd vF#
14 390.7 5/4 M3 major 3rd F#
15 418.6 9/7, 14/11 ^M3, d4 upmajor 3rd, dim 4th ^F#, Gb
16 446.5 13/10, 22/17 vA3, ^d4 downaug 3rd, updim 4th vFx, ^Gb
17 474.4 21/16 v4 down 4th vG
18 502.3 4/3 P4 perfect 4th G
19 530.2 15/11 ^4 up 4th ^G
20 558.1 11/8, 18/13 vA4 downaug 4th vG#
21 586.0 7/5, 24/17, 45/32 A4, vd5 aug 4th, downdim 5th G#, ^Ab
22 614.0 10/7, 17/12, 64/45 ^A4, d5 upaug 4th, dim 5th ^G#, Ab
23 641.9 13/9, 16/11 ^d5 updim 5th ^Ab
24 669.8 22/15 v5 down 5th vA
25 697.7 3/2 P5 perfect 5th A
26 725.6 32/21 ^5 up 5th ^A
27 753.5 17/11, 20/13 vA5, ^d6 downaug 5th, updim 6th vA#, ^Bbb
28 781.4 11/7, 14/9 A5, vm6 aug 5th, downminor 6th A#, vBb
29 809.3 8/5 m6 minor 6th Bb
30 837.2 13/8, 18/11, 21/13, 34/21 ^m6 upminor 6th ^Bb
31 865.1 28/17, 33/20, 44/27, 64/39 vM6 downmajor 6th vB
32 893.0 5/3 M6 major 6th B
33 920.9 12/7, 22/13, 17/10 ^M6, d7 upmajor 6th, dim 7th ^B, Cb
34 948.8 26/15 vA6, ^d7 downaug 6th, updim 7th vB#, ^Cb
35 976.7 7/4 A6, vm7 aug 6th, downminor 7th B#, vC
36 1004.7 9/5, 16/9 m7 minor 7th C
37 1032.6 20/11 ^m7 upminor 7th ^C
38 1060.5 11/6, 13/7, 24/13 vM7 downmajor 7th vC#
39 1088.4 15/8, 28/15, 32/17 M7 major 7th C#
40 1116.3 17/9, 21/11, 27/14, 40/21, 48/25 ^M7, d8 upmajor 7th, dim 8ve ^C#, Db
41 1144.2 25/13, 39/20, 52/27, 64/33, 96/49 vA7, ^d8 downaug 7th, updim 8ve vCx, ^Db
42 1172.1 35/18, 49/25, 63/32, 65/33, 128/65 A7, v8 aug 7th, down 8ve Cx, vD
43 1200.0 2/1 P8 perfect 8ve D

* As a 17-limit system

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and downs notation #Chords and chord progressions.

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.

Stein–Zimmermann–Gould notation

Stein–Zimmermann–Gould notation uses sharps and flats with arrows:

Step offset 0 1 2 3 4 5 6 7
Sharp symbol
Flat symbol

The notes between A and B can then be notated as A, A⁠ ⁠, A⁠ ⁠, A♯, B♭, B⁠ ⁠, B⁠ ⁠, B. Note that A♯ is enharmonic to B⁠ ⁠, and B♭ is enharmonic to A⁠ ⁠.

The notes from B to C are B, C♭, B⁠ ⁠ / C⁠ ⁠, B♯, and C. Similarily, the notes from E to F are E, F♭, E⁠ ⁠ / F⁠ ⁠, E♯, and F. As with the red/blue note system described below, all notes in 43edo therefore have only one name, except for B⁠ ⁠ / C⁠ ⁠ and E⁠ ⁠ / F⁠ ⁠.

Double or even triple arrows may arise if the arrows are taken to have their own layer of enharmonic spellings.

Kite's ups and downs notation

In Kite's ups and downs notation, the "third-sharp" becomes an up and the "two-thirds-sharp" becomes a downsharp. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

Step offset 0 1 2 3 4 5 6 7
Sharp symbol   
 
   
  
   
   
  
   
Flat symbol
 
   
  
   
   
  
   

Sagittal notation

This notation uses the same sagittal sequence as 36edo.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/63

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation64/63

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:  A ,  A ,  A♯ ,  A♯ ,  B♭ ,  B♭ ,  B ,  B . (Note that red sharps or blue flats are enharmonically equivalent to simpler notes:  A♯  is enharmonic to B♭, and  B♭  is actually just A♯).

The diatonic semitone is four steps, so for the region between B and C, we can use:  B ,  C♭ ,  B♯  /  C♭  (they are enharmonic equivalents),  B♯ , and  C . All of the notes in 43edo therefore have only one name except for  B♯  /  C♭ , and  E♯  /  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  C♭  and  B♯  (and  F♭  /  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

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Selected 19-limit intervals approximated in 43edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 43edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 43edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/8, 16/15 0.103 0.4
13/12, 24/13 0.962 3.4
11/7, 14/11 1.097 3.9
11/10, 20/11 2.438 8.7
13/8, 16/13 3.318 11.9
15/13, 26/15 3.422 12.3
7/5, 10/7 3.534 12.7
3/2, 4/3 4.281 15.3
5/4, 8/5 4.384 15.7
13/9, 18/13 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
7/4, 8/7 7.918 28.4
9/8, 16/9 8.561 30.7
5/3, 6/5 8.665 31.0
13/11, 22/13 10.140 36.3
11/6, 12/11 11.102 39.8
13/7, 14/13 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
9/5, 10/9 12.945 46.4
15-odd-limit intervals in 43edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
15/8, 16/15 0.103 0.4
13/12, 24/13 0.962 3.4
11/7, 14/11 1.097 3.9
11/10, 20/11 2.438 8.7
13/8, 16/13 3.318 11.9
15/13, 26/15 3.422 12.3
7/5, 10/7 3.534 12.7
3/2, 4/3 4.281 15.3
5/4, 8/5 4.384 15.7
13/9, 18/13 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
7/4, 8/7 7.918 28.4
9/8, 16/9 8.561 30.7
5/3, 6/5 8.665 31.0
13/11, 22/13 10.140 36.3
11/6, 12/11 11.102 39.8
13/7, 14/13 11.237 40.3
7/6, 12/7 12.199 43.7
9/5, 10/9 12.945 46.4
11/9, 18/11 15.383 55.1
9/7, 14/9 16.479 59.1

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 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 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-68 43⟩ [⟨43 68]] +1.35 1.35 4.84
2.3.5 81/80, 50331648/48828125 [⟨43 68 100]] +0.27 1.88 6.75
2.3.5.7 81/80, 126/125, 17280/16807 [⟨43 68 100 121]] −0.51 2.11 7.56
2.3.5.7.11 81/80, 99/98, 126/125, 864/847 [⟨43 68 100 121 149]] −0.80 1.98 7.08
2.3.5.7.11.13 78/77, 81/80, 99/98, 126/125, 144/143 [⟨43 68 100 121 149 159]] −0.52 1.91 6.85
2.3.5.7.11.13.17 78/77, 81/80, 99/98, 120/119, 126/125, 144/143 [⟨43 68 100 121 149 159 176]] −0.52 1.81 6.49
2.3.5.7.11.13.17.19 78/77, 81/80, 99/98, 120/119, 126/125, 135/133, 144/143 [⟨43 68 100 121 149 159 176 183]] −0.87 1.77 6.34

Commas

This is a partial list of the 19-limit commas that 43et tempers out with its patent val, ⟨43 68 100 121 149 159 176 183].

Prime
limit 		Ratio[note 1] Monzo Cents Color name Name(s)
3 (42 digits) [-68 43⟩ 184.07 Tribilawa 43-comma
5 (18 digits) [20 5 -12⟩ 74.01 Saquadtrigu Hypovishnuzma
5 (16 digits) [24 1 -11⟩ 52.50 Salegu Magus comma
5 81/80 [-4 4 -1⟩ 21.51 Gu Syntonic comma, Didymus' comma, meantone comma
5 (20 digits) [32 -7 -9⟩ 9.49 Sasa-tritrigu Escapade comma
5 (42 digits) [-68 18 17⟩ 2.52 Quinla-seyo Vavoom comma
7 59049/57344 [-13 10 0 -1⟩ 50.72 Laru Harrison's comma
7 3645/3584 [-9 6 1 -1⟩ 29.22 Laruyo Schismean comma
7 (14 digits) [5 -1 7 -7⟩ 20.46 Sepruyo Mermisma
7 126/125 [1 2 -3 1⟩ 13.80 Zotrigu Starling comma
7 (14 digits) [21 -5 -2 -3⟩ 11.12 Satriru-agugu Bronzisma
7 (18 digits) [-14 7 -6 6⟩ 8.76 Latribizogu Historisma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Marvel comma
7 3136/3125 [6 0 -5 2⟩ 6.08 Zozoquingu Hemimean comma
7 (12 digits) [-11 2 7 -3⟩ 1.63 Latriru-asepyo Meter
11 1350/1331 [1 3 2 0 -3⟩ 24.54 Trilu-ayoyo Large tetracot diesis
11 99/98 [-1 2 0 -2 1⟩ 17.58 Loruru Mothwellsma
11 176/175 [4 0 -2 -1 1⟩ 9.86 Lorugugu Valinorsma
11 441/440 [-3 2 -1 2 -1⟩ 3.93 Luzozogu Werckisma
11 4000/3993 [5 -1 3 0 -3⟩ 3.03 Triluyo Wizardharry comma, pine comma
11 (12 digits) [17 -5 0 -2 -1⟩ 1.26 Salururu Olympia
11 (18 digits) [24 -6 0 1 -5⟩ 0.51 Saquinlu-azo Quartisma
13 78/77 [1 1 0 -1 -1 1⟩ 22.34 Tholuru Negustma
13 144/143 [4 2 0 0 -1 -1⟩ 12.06 Thulu Grossma
13 169/168 [-3 -1 0 -1 0 2⟩ 10.27 Thothoru Buzurgisma, dhanvantarisma
13 (12 digits) [9 6 0 0 0 -5⟩ 9.09 Quinthu Glacier comma
13 364/363 [2 -1 0 1 -2 1⟩ 4.76 Tholuluzo Minor minthma
13 1001/1000 [-3 0 -3 1 1 1⟩ 1.73 Tholozotrigu Fairytale comma, sinbadma
13 2080/2079 [5 -3 1 -1 -1 1⟩ 0.83 Tholuruyo Ibnsinma, sinaisma
13 4096/4095 [12 -2 -1 -1 0 -1⟩ 0.42 Sathurugu Minisma
17 120/119 [3 1 1 -1 0 0 -1⟩ 14.49 Suruyo Lynchisma
17 221/220 [-2 0 -1 0 -1 1 1⟩ 7.85 Sotholugu Minor naiadma
17 256/255 [8 -1 -1 0 0 0 -1⟩ 6.78 Sugu Charisma, septendecimal kleisma
17 273/272 [5 1 -1 0 0 0 0 -1⟩ 6.35 Suthozo Tannisma
17 715/714 [-1 -1 1 -1 1 1 -1⟩ 2.42 Sutholoruyo September comma
19 96/95 [5 1 -1 0 0 0 0 -1⟩ 18.13 Nugu 19th-partial chroma
19 153/152 [-3 2 0 0 0 0 1 -1⟩ 11.35 Nuso Ganassisma
19 171/170 [-1 2 -1 0 0 0 -1 1⟩ 10.15 Nosugu Malcolmisma
19 209/208 [-4 0 0 0 1 -1 0 1⟩ 8.30 Nothulo Yama comma
19 210/209 [1 1 1 1 -1 0 0 -1⟩ 8.26 Nuluzoyo Spleen comma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 1\43 27.9 64/63 Arch
1 2\43 55.8 33/32 Escapade
1 3\43 83.7 21/20 Marvolo
1 4\43 111.6 16/15 Vavoom
1 5\43 139.5 13/12 Jerome
1 6\43 167.4 11/10 Superpine
1 7\43 195.3 28/25 Didacus
1 8\43 223.3 8/7 Kumonga
1 9\43 251.2 15/13 Hemimeantone
1 10\43 279.1 75/64 Decipentic
1 11\43 334.9 17/14 Cohemimabila
1 13\43 362.8 16/13 Demibuzzard / interpental
1 14\43 390.7 5/4 Amigo
1 16\43 446.5 13/10 Supersensi
1 17\43 474.4 21/16 Buzzard (2.3.7)
1 18\43 502.3 4/3 Meantone
1 19\43 530.2 15/11 Amavil
1 20\43 558.1 11/8 Thuja
1 21\43 586.0 7/5 Merman

* Octave-reduced form, reduced to the first half-octave

Detemperaments

Ringer 43

The metaphorical color palette that the intervals of 43edo present can be quite appealing for various reasons such as being meantone and splitting 4/3 into 6 equal parts and 3/2 into 5 equal parts, but the accuracy leaves one wanting in many cases, which is why an excellent alternative (given the unambiguity of mappings of all primes in the 109-limit except 71 and 89) is Ringer 43, a Ringer scale with 43 notes per octave period: 

55:56:57:58:59:60:61:62:63:64:65:66:67:68:69:70:72:73:74:75:76:78:79:80:82:83:84:86:87:88:90:91:92:94:96:97:98:100:102:104:106:108:109:110

Or equivalently in the form of reduced, rooted intervals:

65/64, 33/32, 67/64, 17/16, 69/64, 35/32, 9/8, 73/64, 37/32, 75/64, 19/16, 39/32, 79/64, 5/4, 41/32, 83/64, 21/16, 43/32, 87/64, 11/8, 45/32, 91/64, 23/16, 47/32, 3/2, 97/64, 49/32, 25/16, 51/64, 13/8, 53/32, 27/16, 109/64, 55/32, 7/4, 57/32, 29/16, 59/32, 15/8, 61/32, 31/16, 63/32, 2/1

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.

  • 7\43 (195.3¢) stands in for frequency ratio 9/8 (203.9¢) and 10/9 (182.4¢).
  • 6\43 (156.5¢) stands in for 11/10 (165.0¢).
  • 5\46 (130.4¢) stands in for 12/11 (150.6¢), 13/12 (138.6¢), and 14/13 (128.3¢).
  • 4\43 (111.6¢) stands in for 15/14 (119.4¢) and 16/15 (111.7¢).
Harmonic Note (starting from C)
1 C
3 G
5 E
7 A♯, B⁠ ⁠
9 D
11 E𝄪, F⁠ ⁠, F⁠ ⁠
13 B♭♭♭, A⁠ ⁠
15 B

Mos scales

Main article: List of MOS scales in 43edo
  • Meantone[5]: 7 7 11 7 11
  • 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

Other notable scales

Instruments

Keyboards

A possible isomorphic keyboard layout for 43edo:

Music

Modern renderings

Johann Sebastian Bach
Nicolaus Bruhns
John Bull
Frédéric Chopin
George Frideric Handel
Scott Joplin
  • Maple Leaf Rag (1899) – arranged for harpsichord and rendered by Claudi Meneghin (2024)
Shirō Sagisu

21st century

Claudi Meneghin
Bryan Deister
Cale Gibbard
Peter Kosmorsky
Budjarn Lambeth
  • Gamelan-Inspired Improvisation in 43edo, Fossa Scale (Nov 2024) - YouTube
Juhan Puhm (site)
Sevish
Randy Wells
Xotla

References

  1. ↑ Stichting Huygens-Fokker: Logarithmic Interval Measures

External links

Articles

Diagrams

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