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{{Interwiki
| en = 43edo
| de = 43-EDO
}}
{{Infobox ET}}
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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"
|-
|-
Line 34: Line 34:
! 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
Line 74: Line 73:
| 5
| 5
| 139.5
| 139.5
| 12/11, 13/12, 14/13
| [[12/11]], [[13/12]], [[14/13]]
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
Line 81: Line 80:
| 6
| 6
| 167.4
| 167.4
| 11/10
| [[11/10]]
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 88: Line 87:
| 7
| 7
| 195.3
| 195.3
| 9/8, 10/9
| [[9/8]], [[10/9]]
| M2
| M2
| major 2nd
| major 2nd
Line 95: Line 94:
| 8
| 8
| 223.3
| 223.3
| 8/7
| [[8/7]]
| ^M2, d3
| ^M2, d3
| upmajor 2nd, dim 3rd
| upmajor 2nd, dim 3rd
Line 102: Line 101:
| 9
| 9
| 251.2
| 251.2
| 15/13
| [[15/13]]
| vA2, ^d3
| vA2, ^d3
| downaug 2nd, updim 3rd
| downaug 2nd, updim 3rd
Line 109: Line 108:
| 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
Line 130: Line 129:
| 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
Line 137: Line 136:
| 14
| 14
| 390.7
| 390.7
| 5/4
| [[5/4]]
| M3
| M3
| major 3rd
| major 3rd
Line 144: Line 143:
| 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
Line 158: Line 157:
| 17
| 17
| 474.4
| 474.4
| 21/16
| [[21/16]]
| v4
| v4
| down 4th
| down 4th
Line 165: Line 164:
| 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
Line 179: Line 178:
| 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
Line 200: Line 199:
| 23
| 23
| 641.9
| 641.9
| 13/9, 16/11
| [[13/9]], [[16/11]]
| ^d5
| ^d5
| updim 5th
| updim 5th
Line 207: Line 206:
| 24
| 24
| 669.8
| 669.8
| 22/15
| [[22/15]]
| v5
| v5
| down 5th
| down 5th
Line 214: Line 213:
| 25
| 25
| 697.7
| 697.7
| 3/2
| [[3/2]]
| P5
| P5
| perfect 5th
| perfect 5th
Line 221: Line 220:
| 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
Line 242: Line 241:
| 29
| 29
| 809.3
| 809.3
| 8/5
| [[8/5]]
| m6
| m6
| minor 6th
| minor 6th
Line 249: Line 248:
| 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
Line 256: Line 255:
| 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
Line 263: Line 262:
| 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
Line 312: Line 311:
| 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 ====
Line 390: Line 390:
</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}}
* [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}}
; {{W|George Frideric Handel}}
Line 1,002: Line 1,037:
* [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/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]]
; [[Cale Gibbard]]
Line 1,023: 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 ==
Retrieved from "https://en.xen.wiki/w/43edo"