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| de = 43-EDO
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== Theory ==
== Theory ==
43edo tempers out [[81/80]] in the 5-limit, and as such it is strongly associated with [[meantone]]. Specifically, it is (for all practical purposes) equivalent to [[1/5-comma meantone]], as it tunes the major third sharp of [[5/4]] and perfect fifth flat of [[3/2]] by slightly more than four cents on both of them. It 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.
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]].  
 
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. 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 temperament|Huygens]]. In the 13-limit it supports [[Meantone family #Meridetone|meridetone]], which tempers out [[78/77]], and [[Meantone family #Grosstone|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 }}.
 
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 11/8s stack to a major third (i.e. {{nowrap|(11/8)<sup>5</sup> &rarr; 5/1}}), with [[mos]]es of 15 and 28.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|43}}
{{Harmonics in equal|43|columns=11}}
{{Harmonics in equal|43|start=12|columns=9|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}
{{Harmonics in equal|43|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}
 
Although not [[consistent]], 43edo performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'' (after which the demands on its pitch resolution finally become too great), with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. 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. 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.


43edo has less than 35% relative error (less than 10 cents error) on an impressive 17 of the 19 prime harmonics in the [[67-limit]]. The only ones it misses are 23 and 41. So it can be used as a solid full [[19-limit]] tuning, or as a solid no-23-or-41 67-limit tuning.
=== 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 }}.


It approximates harmonics 31, 37 and 61 close to exactly - within less than a cent (less than 3% relative error). It approximates 3, 13, 43, 53 and 61 slightly flat. It approximates 5, 7, 11, 17, 19, 29, 47, 59 and 67 slightly sharp. Overall this gives 43edo a slightly sharp tendency/feeling, though with the major exception of harmonic 3 (the perfect fifth).
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"
|-
|-
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! Cents
! Cents
! Approximate 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)
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and vd2)
|-
|-
| 0
| 0
| 0.0
| 0.0
| 1/1
| [[1/1]]
| P1
| P1
| perfect unison
| perfect unison
Line 46: Line 45:
| 1
| 1
| 27.9
| 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 53: Line 52:
| 2
| 2
| 55.8
| 55.8
| 26/25, 27/26, 33/32, 40/39, ''49/48''
| [[26/25]], [[27/26]], [[33/32]], [[40/39]], ''[[49/48]]''
| vA1, ^d2
| vA1, ^d2
| downaug unison, updim 2nd
| downaug unison, updim 2nd
Line 60: Line 59:
| 3
| 3
| 83.7
| 83.7
| ''18/17'', 21/20, 22/21, 25/24, ''28/27''
| ''[[18/17]]'', [[21/20]], [[22/21]], [[25/24]], ''[[28/27]]''
| A1, vm2
| A1, vm2
| aug 1sn, downminor 2nd
| aug 1sn, downminor 2nd
Line 67: Line 66:
| 4
| 4
| 111.6
| 111.6
| 15/14, 16/15, 17/16
| [[15/14]], [[16/15]], [[17/16]]
| m2
| m2
| minor 2nd
| minor 2nd
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| 5
| 5
| 139.5
| 139.5
| 12/11, 13/12, 14/13
| [[12/11]], [[13/12]], [[14/13]]
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
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| 6
| 6
| 167.4
| 167.4
| 11/10
| [[11/10]]
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
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| 7
| 7
| 195.3
| 195.3
| 9/8, 10/9
| [[9/8]], [[10/9]]
| M2
| M2
| major 2nd
| major 2nd
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| 8
| 8
| 223.3
| 223.3
| 8/7
| [[8/7]]
| ^M2, d3
| ^M2, d3
| upmajor 2nd, dim 3rd
| upmajor 2nd, dim 3rd
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| 9
| 9
| 251.2
| 251.2
| 15/13
| [[15/13]]
| vA2, ^d3
| vA2, ^d3
| downaug 2nd, updim 3rd
| downaug 2nd, updim 3rd
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| 10
| 10
| 279.1
| 279.1
| 7/6, 13/11
| [[7/6]], [[13/11]], [[20/17]]
| A2, vm3
| A2, vm3
| aug 2nd, downminor 3rd
| aug 2nd, downminor 3rd
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| 11
| 11
| 307.0
| 307.0
| 6/5
| [[6/5]]
| m3
| m3
| minor 3rd
| minor 3rd
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| 12
| 12
| 334.9
| 334.9
| 17/14, ''27/22'', 39/32, 40/33
| [[17/14]], ''[[27/22]]'', [[39/32]], [[40/33]]
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
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| 13
| 13
| 362.8
| 362.8
| ''11/9'', 16/13, 21/17, 26/21
| ''[[11/9]]'', [[16/13]], [[21/17]], [[26/21]]
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
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| 14
| 14
| 390.7
| 390.7
| 5/4
| [[5/4]]
| M3
| M3
| major 3rd
| major 3rd
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| 15
| 15
| 418.6
| 418.6
| ''9/7'', 14/11
| ''[[9/7]]'', [[14/11]]
| ^M3, d4
| ^M3, d4
| upmajor 3rd, dim 4th
| upmajor 3rd, dim 4th
Line 151: Line 150:
| 16
| 16
| 446.5
| 446.5
| 13/10, 22/17
| [[13/10]], [[22/17]]
| vA3, ^d4
| vA3, ^d4
| downaug 3rd, updim 4th
| downaug 3rd, updim 4th
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| 17
| 17
| 474.4
| 474.4
| 21/16
| [[21/16]]
| v4
| v4
| down 4th
| down 4th
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| 18
| 18
| 502.3
| 502.3
| 4/3
| [[4/3]]
| P4
| P4
| perfect 4th
| perfect 4th
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| 19
| 19
| 530.2
| 530.2
| 15/11
| [[15/11]]
| ^4
| ^4
| up 4th
| up 4th
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| 20
| 20
| 558.1
| 558.1
| 11/8, 18/13
| [[11/8]], [[18/13]]
| vA4
| vA4
| downaug 4th
| downaug 4th
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| 21
| 21
| 586.0
| 586.0
| 7/5, 24/17, 45/32
| [[7/5]], [[24/17]], [[45/32]]
| A4, vd5
| A4, vd5
| aug 4th, downdim 5th
| aug 4th, downdim 5th
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| 22
| 22
| 614.0
| 614.0
| 10/7, 17/12, 64/45
| [[10/7]], [[17/12]], [[64/45]]
| ^A4, d5
| ^A4, d5
| upaug 4th, dim 5th
| upaug 4th, dim 5th
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| 23
| 23
| 641.9
| 641.9
| 13/9, 16/11
| [[13/9]], [[16/11]]
| ^d5
| ^d5
| updim 5th
| updim 5th
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| 24
| 24
| 669.8
| 669.8
| 22/15
| [[22/15]]
| v5
| v5
| down 5th
| down 5th
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| 25
| 25
| 697.7
| 697.7
| 3/2
| [[3/2]]
| P5
| P5
| perfect 5th
| perfect 5th
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| 26
| 26
| 725.6
| 725.6
| 32/21
| [[32/21]]
| ^5
| ^5
| up 5th
| up 5th
Line 228: Line 227:
| 27
| 27
| 753.5
| 753.5
| 17/11, 20/13
| [[17/11]], [[20/13]]
| vA5, ^d6
| vA5, ^d6
| downaug 5th, updim 6th
| downaug 5th, updim 6th
Line 235: Line 234:
| 28
| 28
| 781.4
| 781.4
| 11/7, ''14/9''
| [[11/7]], ''[[14/9]]''
| A5, vm6
| A5, vm6
| aug 5th, downminor 6th
| aug 5th, downminor 6th
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| 29
| 29
| 809.3
| 809.3
| 8/5
| [[8/5]]
| m6
| m6
| minor 6th
| minor 6th
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| 30
| 30
| 837.2
| 837.2
| 13/8, ''18/11'', 21/13, 34/21
| [[13/8]], ''[[18/11]]'', [[21/13]], [[34/21]]
| ^m6
| ^m6
| upminor 6th
| upminor 6th
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| 31
| 31
| 865.1
| 865.1
| 28/17, 33/20, ''44/27'', 64/39
| [[28/17]], [[33/20]], ''[[44/27]]'', [[64/39]]
| vM6
| vM6
| downmajor 6th
| downmajor 6th
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| 32
| 32
| 893.0
| 893.0
| 5/3
| [[5/3]]
| M6
| M6
| major 6th
| major 6th
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| 33
| 33
| 920.9
| 920.9
| 12/7, 22/13
| [[12/7]], [[22/13]], [[17/10]]
| ^M6, d7
| ^M6, d7
| upmajor 6th, dim 7th
| upmajor 6th, dim 7th
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| 34
| 34
| 948.8
| 948.8
| 26/15
| [[26/15]]
| vA6, ^d7
| vA6, ^d7
| downaug 6th, updim 7th
| downaug 6th, updim 7th
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| 35
| 35
| 976.7
| 976.7
| 7/4
| [[7/4]]
| A6, vm7
| A6, vm7
| aug 6th, downminor 7th
| aug 6th, downminor 7th
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| 36
| 36
| 1004.7
| 1004.7
| 9/5, 16/9
| [[9/5]], [[16/9]]
| m7
| m7
| minor 7th
| minor 7th
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| 37
| 37
| 1032.6
| 1032.6
| 20/11
| [[20/11]]
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 305: Line 304:
| 38
| 38
| 1060.5
| 1060.5
| 11/6, 13/7, 24/13
| [[11/6]], [[13/7]], [[24/13]]
| vM7
| vM7
| downmajor 7th
| downmajor 7th
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| 39
| 39
| 1088.4
| 1088.4
| 15/8, 28/15, 32/17
| [[15/8]], [[28/15]], [[32/17]]
| M7
| M7
| major 7th
| major 7th
Line 319: Line 318:
| 40
| 40
| 1116.3
| 1116.3
| ''17/9'', 21/11, ''27/14'', 40/21, 48/25
| ''[[17/9]]'', [[21/11]], ''[[27/14]]'', [[40/21]], [[48/25]]
| ^M7, d8
| ^M7, d8
| upmajor 7th, dim 8ve
| upmajor 7th, dim 8ve
Line 326: Line 325:
| 41
| 41
| 1144.2
| 1144.2
| 25/13, 39/20, 52/27, 64/33, ''96/49''
| [[25/13]], [[39/20]], [[52/27]], [[64/33]], ''[[96/49]]''
| vA7, ^d8
| vA7, ^d8
| downaug 7th, updim 8ve
| downaug 7th, updim 8ve
Line 333: Line 332:
| 42
| 42
| 1172.1
| 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 340: Line 339:
| 43
| 43
| 1200.0
| 1200.0
| 2/1
| [[2/1]]
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 353: Line 352:
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.


=== Ups and downs notation ===
=== Stein–Zimmermann–Gould notation ===
In [[ups and downs notation]], the "third-sharp" becomes an up and the "two-thirds-sharp" becomes a downsharp.
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{Sharpness-sharp3-szg}}
{{Ups and downs sharpness}}
 
Or one can use the [[Alternative symbols for ups and downs notation#Sharp-3|alternative ups and downs]]. They use sharps and flats with arrows, borrowed from extended [[Helmholtz-Ellis notation|Helmholtz&ndash;Ellis]] notation:
{{Sharpness-sharp3}}


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 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}}.
Line 366: Line 361:


Double or even triple arrows may arise if the arrows are taken to have their own layer of enharmonic spellings.
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.
{{Ups and downs sharpness}}


=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[36edo#Sagittal notation|36-EDO]].
This notation uses the same sagittal sequence as [[36edo #Sagittal notation|36edo]].


==== Evo flavor ====
==== Evo flavor ====
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</imagemap>
</imagemap>


=== 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 than Kite's [[color notation]].) Now 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♯).
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).
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).
Line 402: Line 402:
=== Interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{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 ==
Line 416: Line 421:
|-
|-
| 2.3
| 2.3
| {{monzo| -68 43 }}
| {{Monzo| -68 43 }}
| {{mapping| 43 68 }}
| {{Mapping| 43 68 }}
| +1.35
| +1.35
| 1.35
| 1.35
Line 424: Line 429:
| 2.3.5
| 2.3.5
| 81/80, 50331648/48828125
| 81/80, 50331648/48828125
| {{mapping| 43 68 100 }}
| {{Mapping| 43 68 100 }}
| +0.27
| +0.27
| 1.88
| 1.88
Line 431: Line 436:
| 2.3.5.7
| 2.3.5.7
| 81/80, 126/125, 17280/16807
| 81/80, 126/125, 17280/16807
| {{mapping| 43 68 100 121 }}
| {{Mapping| 43 68 100 121 }}
| −0.51
| −0.51
| 2.11
| 2.11
Line 438: 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
| {{mapping| 43 68 100 121 149 }}
| {{Mapping| 43 68 100 121 149 }}
| −0.80
| −0.80
| 1.98
| 1.98
Line 445: 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
| {{mapping| 43 68 100 121 149 159 }}
| {{Mapping| 43 68 100 121 149 159 }}
| −0.52
| −0.52
| 1.91
| 1.91
Line 452: 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
| {{mapping| 43 68 100 121 149 159 176 }}
| {{Mapping| 43 68 100 121 149 159 176 }}
| −0.52
| −0.52
| 1.81
| 1.81
Line 459: Line 464:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 78/77, 81/80, 99/98, 120/119, 126/125, 135/133, 144/143
| 78/77, 81/80, 99/98, 120/119, 126/125, 135/133, 144/143
| {{mapping| 43 68 100 121 149 159 176 183 }}
| {{Mapping| 43 68 100 121 149 159 176 183 }}
| −0.87
| −0.87
| 1.77
| 1.77
Line 479: 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 486: 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
Line 493: Line 498:
| 5
| 5
| <abbr title="50331648/48828125">(16 digits)</abbr>
| <abbr title="50331648/48828125">(16 digits)</abbr>
| {{monzo| 24 1 -11 }}
| {{Monzo| 24 1 -11 }}
| 52.50
| 52.50
| Salegu
| Salegu
Line 500: Line 505:
| 5
| 5
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
| 21.51
| 21.51
| Gu
| Gu
Line 507: Line 512:
| 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 514: 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
Line 521: Line 526:
| 7
| 7
| [[59049/57344]]
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| {{Monzo| -13 10 0 -1 }}
| 50.72
| 50.72
| Laru
| Laru
Line 528: Line 533:
| 7
| 7
| [[3645/3584]]
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| {{Monzo| -9 6 1 -1 }}
| 29.22
| 29.22
| Laruyo
| Laruyo
Line 535: Line 540:
| 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
Line 542: Line 547:
| 7
| 7
| [[126/125]]
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| {{Monzo| 1 2 -3 1 }}
| 13.80
| 13.80
| Zotrigu
| Zotrigu
Line 549: 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
Line 556: Line 561:
| 7
| 7
| <abbr title="257298363/256000000">(18 digits)</abbr>
| <abbr title="257298363/256000000">(18 digits)</abbr>
| {{monzo| -14 7 -6 6 }}
| {{Monzo| -14 7 -6 6 }}
| 8.76
| 8.76
| Latribizogu
| Latribizogu
Line 563: Line 568:
| 7
| 7
| [[225/224]]
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
Line 570: 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 577: Line 582:
| 7
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| <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
Line 584: Line 589:
| 11
| 11
| [[1350/1331]]
| [[1350/1331]]
| {{monzo| 1 3 2 0 -3}}
| {{Monzo| 1 3 2 0 -3}}
| 24.54
| 24.54
| Trilu-ayoyo
| Trilu-ayoyo
Line 591: Line 596:
| 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 598: 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 605: 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 612: 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 619: Line 624:
| 11
| 11
| <abbr title="131072/130977">(12 digits)</abbr>
| <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
Line 626: Line 631:
| 11
| 11
| <abbr title="117440512/117406179">(18 digits)</abbr>
| <abbr title="117440512/117406179">(18 digits)</abbr>
| {{monzo| 24 -6 0 1 -5 }}
| {{Monzo| 24 -6 0 1 -5 }}
| 0.51
| 0.51
| Saquinlu-azo
| Saquinlu-azo
Line 633: Line 638:
| 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 640: 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 647: 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
Line 654: Line 659:
| 13
| 13
| <abbr title="373248/371293">(12 digits)</abbr>
| <abbr title="373248/371293">(12 digits)</abbr>
| {{monzo| 9 6 0 0 0 -5 }}
| {{Monzo| 9 6 0 0 0 -5 }}
| 9.09
| 9.09
| Quinthu
| Quinthu
Line 661: Line 666:
| 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
Line 668: Line 673:
| 13
| 13
| [[1001/1000]]
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.73
| 1.73
| Tholozotrigu
| Tholozotrigu
Line 675: Line 680:
| 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
Line 682: Line 687:
| 13
| 13
| [[4096/4095]]
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| 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 696: 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 703: 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 710: 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 717: 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 724: 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
Line 731: Line 736:
| 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 738: 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 745: 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
Line 752: Line 757:
| 19
| 19
| [[210/209]]
| [[210/209]]
| {{monzo| 1 1 1 1 -1 0 0 -1 }}
| {{Monzo| 1 1 1 1 -1 0 0 -1 }}
| 8.26
| 8.26
| Nuluzoyo
| Nuluzoyo
| Spleen comma
| Spleen comma
|}
|}
<references group="note" />


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 770: Line 776:
| 1
| 1
| 1\43
| 1\43
| 27.91
| 27.9
| 64/63
| 64/63
| [[Arch]]
| [[Arch]]
Line 776: 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 788: Line 800:
| 1
| 1
| 5\43
| 5\43
| 139.53
| 139.5
| 13/12
| 13/12
| [[Jerome]]
| [[Jerome]]
Line 794: Line 806:
| 1
| 1
| 6\43
| 6\43
| 167.44
| 167.4
| 11/10
| 11/10
| [[Superpine]]
| [[Superpine]]
Line 800: Line 812:
| 1
| 1
| 7\43
| 7\43
| 195.35
| 195.3
| 28/25
| 28/25
| [[Didacus]]
| [[Didacus]]
Line 806: Line 818:
| 1
| 1
| 8\43
| 8\43
| 223.26
| 223.3
| 8/7
| 8/7
| [[Kumonga]]
| [[Kumonga]]
Line 812: Line 824:
| 1
| 1
| 9\43
| 9\43
| 251.16
| 251.2
| 15/13
| 15/13
| [[Hemimeantone]]
| [[Hemimeantone]]
Line 818: Line 830:
| 1
| 1
| 10\43
| 10\43
| 279.07
| 279.1
| 75/64
| 75/64
| [[Decipentic]]
| [[Decipentic]]
Line 824: Line 836:
| 1
| 1
| 11\43
| 11\43
| 334.88
| 334.9
| 17/14
| 17/14
| [[Cohemimabila]]
| [[Cohemimabila]]
Line 830: 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 842: 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 854: Line 872:
| 1
| 1
| 19\43
| 19\43
| 530.23
| 530.2
| 15/11
| 15/11
| [[Amavil]]
| [[Amavil]]
Line 860: Line 878:
| 1
| 1
| 20\43
| 20\43
| 558.14
| 558.1
| 11/8
| 11/8
| [[Thuja]]
| [[Thuja]]
Line 866: Line 884:
| 1
| 1
| 21\43
| 21\43
| 586.05
| 586.0
| 7/5
| 7/5
| [[Merman]]
| [[Merman]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave


== Detemperaments ==
== Detemperaments ==
Line 973: Line 991:
* Fossa pentatonic scale (approximated from [[catnip]] in [[60edo]]): 5 14 6 6 12
* Fossa pentatonic scale (approximated from [[catnip]] in [[60edo]]): 5 14 6 6 12
* [[Magnetosphere scale]] (approximated from [[Hexany 1728]]): 4 10 11 11 7
* [[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 ==
Line 985: Line 1,011:
* [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)
* [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}}
; {{W|Frédéric Chopin}}
Line 1,009: Line 1,038:
* [https://www.youtube.com/shorts/URUCEOW3Mqo ''43edo improv''] (2025)
* [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/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]]
; [[Cale Gibbard]]
Line 1,030: Line 1,061:
; [[Xotla]]
; [[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
* "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
== 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]]
== Notes ==
<references group="note" />


== References ==
== References ==

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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