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43edo: Difference between revisions

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→Prime harmonics: Since 43EDO is supposed to perform very well at higher limits, let's add some more prime harmonics
m no need for 3 digit precision here
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|43}}
{{ED intro}}


== History ==
== History ==
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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|43|columns=12}}
{{Harmonics in equal|43}}
{{Harmonics in equal|43|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}
{{Harmonics in equal|43|start=12|columns=9|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}
{{Harmonics in equal|43|columns=12|start=25|collapsed=true|title=Approximation of prime harmonics in 43edo (continued even further)}}
 
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.
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.


Line 33: Line 33:
! #
! #
! Cents
! Cents
! Approximate 17-limit Ratios
! Approximate ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
([[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
Line 44: Line 45:
|-
|-
| 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
Line 51: Line 52:
|-
|-
| 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 58: Line 59:
|-
|-
| 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''
| A1, vm2
| A1, vm2
| aug 1sn, downminor 2nd
| aug 1sn, downminor 2nd
Line 65: Line 66:
|-
|-
| 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 72: Line 73:
|-
|-
| 5
| 5
| 139.535
| 139.5
| 12/11, 13/12, 14/13
| 12/11, 13/12, 14/13
| ^m2
| ^m2
Line 79: Line 80:
|-
|-
| 6
| 6
| 167.442
| 167.4
| 11/10
| 11/10
| vM2
| vM2
Line 86: Line 87:
|-
|-
| 7
| 7
| 195.349
| 195.3
| 9/8, 10/9
| 9/8, 10/9
| M2
| M2
Line 93: Line 94:
|-
|-
| 8
| 8
| 223.256
| 223.3
| 8/7
| 8/7
| ^M2, d3
| ^M2, d3
Line 100: Line 101:
|-
|-
| 9
| 9
| 251.163
| 251.2
| 15/13
| 15/13
| vA2, ^d3
| vA2, ^d3
Line 107: Line 108:
|-
|-
| 10
| 10
| 279.070
| 279.1
| 7/6, 13/11
| 7/6, 13/11
| A2, vm3
| A2, vm3
Line 114: Line 115:
|-
|-
| 11
| 11
| 306.977
| 307.0
| 6/5
| 6/5
| m3
| m3
Line 121: Line 122:
|-
|-
| 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 128: Line 129:
|-
|-
| 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 135: Line 136:
|-
|-
| 14
| 14
| 390.698
| 390.7
| 5/4
| 5/4
| M3
| M3
Line 142: Line 143:
|-
|-
| 15
| 15
| 418.605
| 418.6
| ''9/7'', 14/11
| ''9/7'', 14/11
| ^M3, d4
| ^M3, d4
Line 149: Line 150:
|-
|-
| 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 156: Line 157:
|-
|-
| 17
| 17
| 474.419
| 474.4
| 21/16
| 21/16
| v4
| v4
Line 163: Line 164:
|-
|-
| 18
| 18
| 502.326
| 502.3
| 4/3
| 4/3
| P4
| P4
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|-
|-
| 19
| 19
| 530.233
| 530.2
| 15/11
| 15/11
| ^4
| ^4
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|-
|-
| 20
| 20
| 558.140
| 558.1
| 11/8, 18/13
| 11/8, 18/13
| vA4
| vA4
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|-
|-
| 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 191: Line 192:
|-
|-
| 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 198: Line 199:
|-
|-
| 23
| 23
| 641.860
| 641.9
| 16/11, 13/9
| 13/9, 16/11
| ^d5
| ^d5
| updim 5th
| updim 5th
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|-
|-
| 24
| 24
| 669.767
| 669.8
| 22/15
| 22/15
| v5
| v5
Line 212: Line 213:
|-
|-
| 25
| 25
| 697.674
| 697.7
| 3/2
| 3/2
| P5
| P5
Line 219: Line 220:
|-
|-
| 26
| 26
| 725.581
| 725.6
| 32/21
| 32/21
| ^5
| ^5
Line 226: Line 227:
|-
|-
| 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 233: Line 234:
|-
|-
| 28
| 28
| 781.395
| 781.4
| ''14/9'', 11/7
| 11/7, ''14/9''
| A5, vm6
| A5, vm6
| aug 5th, downminor 6th
| aug 5th, downminor 6th
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|-
|-
| 29
| 29
| 809.302
| 809.3
| 8/5
| 8/5
| m6
| m6
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|-
|-
| 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 254: Line 255:
|-
|-
| 31
| 31
| 865.116
| 865.1
| 64/39, 28/17
| 28/17, 33/20, ''44/27'', 64/39
| vM6
| vM6
| downmajor 6th
| downmajor 6th
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|-
|-
| 32
| 32
| 893.023
| 893.0
| 5/3
| 5/3
| M6
| M6
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|-
|-
| 33
| 33
| 920.930
| 920.9
| 12/7, 22/13
| 12/7, 22/13
| ^M6, d7
| ^M6, d7
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|-
|-
| 34
| 34
| 948.837
| 948.8
| 26/15
| 26/15
| vA6, ^d7
| vA6, ^d7
Line 282: Line 283:
|-
|-
| 35
| 35
| 976.744
| 976.7
| 7/4
| 7/4
| A6, vm7
| A6, vm7
Line 289: Line 290:
|-
|-
| 36
| 36
| 1004.651
| 1004.7
| 16/9, 9/5
| 9/5, 16/9
| m7
| m7
| minor 7th
| minor 7th
Line 296: Line 297:
|-
|-
| 37
| 37
| 1032.558
| 1032.6
| 20/11
| 20/11
| ^m7
| ^m7
Line 303: Line 304:
|-
|-
| 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 310: Line 311:
|-
|-
| 39
| 39
| 1088.372
| 1088.4
| 15/8, 28/15, 32/17
| 15/8, 28/15, 32/17
| M7
| M7
Line 317: Line 318:
|-
|-
| 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, d8
| ^M7, d8
| upmajor 7th, dim 8ve
| upmajor 7th, dim 8ve
Line 324: Line 325:
|-
|-
| 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 331: Line 332:
|-
|-
| 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
Line 338: Line 339:
|-
|-
| 43
| 43
| 1200.000
| 1200.0
| 2/1
| 2/1
| P8
| P8
Line 345: Line 346:
|}
|}


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


== JI approximation ==
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].
[[File:43ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 43edo]]
=== Interval mappings ===
{{Q-odd-limit intervals|43}}


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


For people who aren't colorblind, a red-note/blue-note system (similar to that proposed for [[36edo]]) can 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♯).
=== Ups and downs notation ===
In [[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.
{{sharpness-sharp3a}}


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 unambiguous names 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).
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}}
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'' unambiguous red-note/blue-note notation for [[45edo]], which is another meantone (actually, a [[flattone]]) system.


=== Ups and downs notation ===
The third-sharps and third-flats can also be notated using [[ups and downs notation]] and extended [[Helmholtz-Ellis notation|Helmholtz&ndash;Ellis]] accidentals:
{{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}}.


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 above, all notes in 43edo therefore have unambiguous names except for B{{sharpdown}}&nbsp;/&nbsp;C{{flatup}} and E{{sharpdown}}&nbsp;/&nbsp;F{{flatup}}.
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}}.


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.


=== Sagittal ===
=== Sagittal notation ===
The following table shows [[sagittal notation]] accidentals in one apotome for 43do.  
This notation uses the same sagittal sequence as [[36edo#Sagittal notation|36-EDO]].
 
==== Evo flavor ====
<imagemap>
File:43-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
default [[File:43-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:43-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
default [[File:43-EDO_Revo_Sagittal.svg]]
</imagemap>
 
=== Red-Blue Notation ===
For people who are not colorblind, a red-note/blue-note system (similar to that proposed for [[36edo]]) can also be used. (Note that this is different 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♯).
 
The diatonic semitone is four steps, so for the region between B and C, we can use: {{colored note|B}}, {{colored note|C♭}}, {{colored note|blue|B♯}}&nbsp;/&nbsp;{{colored note|red|C♭}} (they are enharmonic equivalents), {{colored note|B♯}}, and {{colored note|C}}. All of the notes in 43edo therefore have only one name except for {{colored note|blue|B♯}}&nbsp;/&nbsp;{{colored note|red|C♭}}, and {{colored note|blue|E♯}}&nbsp;/&nbsp;{{colored note|red|F♭}}. It might also be possible to design special symbols for those two notes (resembling a cross between the letters B and C in the former case, and E and F in the latter).
 
If {{colored note|red|C♭}} and {{colored note|blue|B♯}} (and {{colored note|red|F♭}}&nbsp;/&nbsp;{{colored note|blue|E♯}}) are instead forced to be distinct, but the requirement that all notes be equally spaced is maintained, then we end up with a ''completely'' single-name red-note/blue-note notation for [[45edo]], which is another meantone (actually, a [[flattone]]) system.
 
== Approximation to JI ==
[[File:43ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 43edo]]


{| class="wikitable center-all"
=== Interval mappings ===
! Steps
{{Q-odd-limit intervals}}
| 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 ==
Line 394: Line 409:
! rowspan="2" | [[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
|-
|-
Line 402: Line 417:
| 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 409: Line 424:
| 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 416: Line 431:
| 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 }}
| &minus;0.51
| −0.51
| 2.11
| 2.11
| 7.56
| 7.56
Line 423: Line 438:
| 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 }}
| &minus;0.80
| −0.80
| 1.98
| 1.98
| 7.08
| 7.08
Line 430: Line 445:
| 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 }}
| &minus;0.52
| −0.52
| 1.91
| 1.91
| 6.85
| 6.85
Line 437: Line 452:
| 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 }}
| &minus;0.52
| −0.52
| 1.81
| 1.81
| 6.49
| 6.49
Line 444: Line 459:
| 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
| [{{val| 43 68 100 121 149 159 176 183 }}]
| {{mapping| 43 68 100 121 149 159 176 183 }}
| &minus;0.87
| −0.87
| 1.77
| 1.77
| 6.34
| 6.34
Line 451: Line 466:


=== Commas ===
=== Commas ===
This is a partial list of the 19-limit [[commas]] that 43edo [[tempers out]] with its patent [[val]], {{val| 43 68 100 121 149 159 176 183 }}.
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 group="note">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 488: Line 503:
| 21.51
| 21.51
| Gu
| Gu
| Syntonic comma, Didymus comma, meantone comma
| Syntonic comma, Didymus' comma, meantone comma
|-
|-
| 5
| 5
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| 2.52
| 2.52
| Quinla-seyo
| Quinla-seyo
| [[Vavoom family|Vavoom comma]]
| [[Vavoom comma]]
|-
|-
| 7
| 7
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| 20.46
| 20.46
| Sepruyo
| Sepruyo
| [[Merman|Mermisma]]
| [[Mermisma]]
|-
|-
| 7
| 7
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| 7.71
| 7.71
| Ruyoyo
| Ruyoyo
| Septimal kleisma, Marvel comma
| Marvel comma
|-
|-
| 7
| 7
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| 1.63
| 1.63
| Latriru-asepyo
| Latriru-asepyo
| [[Meter comma]]
| [[Meter]]
|-
|-
| 11
| 11
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| 24.54
| 24.54
| Trilu-ayoyo
| Trilu-ayoyo
| Large Tetracot diesis
| Large tetracot diesis
|-
|-
| 11
| 11
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| 1.26
| 1.26
| Salururu
| Salururu
| [[Olympic comma]]
| [[Olympia]]
|-
|-
| 11
| 11
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| 18.13
| 18.13
| Nugu
| Nugu
| 19th Partial chroma
| 19th-partial chroma
|-
|-
| 19
| 19
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| Spleen comma
| Spleen comma
|}
|}
<references group="note"/>


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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|+ style="font-size: 105%;" | 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*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Merman]]
| [[Merman]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave


== Detemperaments ==
== Detemperaments ==
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== Scales ==
== Scales ==
=== MOS scales ===
{{main|List of MOS scales in 43edo}}
=== Harmonic scales ===
=== Harmonic scales ===
43edo represents the first 16 overtones of the [[harmonic series]] well (written as a ratio of 8:9:10:11:12:13:14:15:16 in [[just intonation]]) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.
43edo represents the first 16 overtones of the [[harmonic series]] well (written as a ratio of 8:9:10:11:12:13:14:15:16 in [[just intonation]]) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.
* 7\43 (195.349¢) stands in for frequency ratio [[9/8]] (203.910¢) and [[10/9]] (182.404¢).
* 7\43 (195.3¢) stands in for frequency ratio [[9/8]] (203.9¢) and [[10/9]] (182.4¢).
* 6\43 (156.522¢) stands in for [[11/10]] (165.004¢)
* 6\43 (156.5¢) stands in for [[11/10]] (165.0¢).
* 5\46 (130.435¢) stands in for [[12/11]] (150.637¢), [[13/12]] (138.573¢), and [[14/13]] (128.298¢).
* 5\46 (130.4¢) stands in for [[12/11]] (150.6¢), [[13/12]] (138.6¢), and [[14/13]] (128.3¢).
* 4\43 (111.628¢) stands in for [[15/14]] (119.443¢) and [[16/15]] (111.731¢).
* 4\43 (111.6¢) stands in for [[15/14]] (119.4¢) and [[16/15]] (111.7¢).
 
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
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|}
|}


=== Subsets of Meantone ===
=== 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


'''Major scales'''
=== Other meantone scales ===
; Major scales
* Ionian Pentatonic: 14 4 7 14 4
* Ionian Pentatonic: 14 4 7 14 4
* Major: 7 7 4 7 7 7 4
* Major Pentatonic: 7 7 11 7 11


'''Minor scales'''
; Minor scales
* Minor: 7 4 7 7 4 7 7
* Minor Harmonic: 7 4 7 7 4 10 4
* Minor Harmonic: 7 4 7 7 4 10 4
* Minor Harmonic Pentatonic: 7 4 14 14 4
* Minor Harmonic Pentatonic: 7 4 14 14 4
* Minor Hexatonic: 7 4 7 7 11 7
* Minor Hexatonic: 7 4 7 7 11 7
* Minor Melodic: 7 4 7 7 7 7 4
* Minor Melodic: 7 4 7 7 7 7 4
* Minor Pentatonic: 11 7 7 11 7


'''Modal scales'''
; Modal scales
* Dorian: 7 4 7 7 7 4 7
* Locrian: 4 7 7 4 7 7 7
* Lydian: 7 7 7 4 7 7 4
* Mixolydian: 7 7 4 7 7 4 7
* Mixolydian Harmonic: 14 4 7 4 7 7
* Mixolydian Harmonic: 14 4 7 4 7 7
* Mixolydian Pentatonic: 14 4 7 11 7
* Mixolydian Pentatonic: 14 4 7 11 7
* Phrygian: 4 7 7 7 4 7 7
* Phrygian Dominant: 4 10 4 7 4 7 7
* Phrygian Dominant: 4 10 4 7 4 7 7
* Phrygian Dominant Hexatonic: 4 10 4 7 11 7
* Phrygian Dominant Hexatonic: 4 10 4 7 11 7
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* Phrygian Pentatonic: 4 7 14 4 14
* Phrygian Pentatonic: 4 7 14 4 14


 
; Blues scales
'''Blues scales'''
* Blues Aeolian Hexatonic: 11 7 4 3 4 14
* Blues Aeolian Hexatonic: 11 7 4 3 4 14
* Blues Aeolian Pentatonic I: 11 7 7 4 14
* Blues Aeolian Pentatonic I: 11 7 7 4 14
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* Hyperblue Harmonic: 11 7 2 5 3 12 3
* Hyperblue Harmonic: 11 7 2 5 3 12 3


 
; Others
'''Others'''
* Akebono I: 7 4 14 7 11
* Akebono I: 7 4 14 7 11
* Dominant Pentatonic: 7 7 11 11 7
* Dominant Pentatonic: 7 7 11 11 7
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; {{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=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)
* [https://www.youtube.com/watch?v=qyWfguU0iQM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
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; [[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)
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}} (late 2011) – in meantone
* [[: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
Line 1,014: Line 1,019:
* "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==
== Instruments ==
*[[Lumatone mapping for 43edo]]
*[[Lumatone mapping for 43edo]]
*[[Skip fretting system 43 2 9]]
*[[Skip fretting system 43 2 9]]


===Keyboards===
=== Keyboards ===
A possible isomorphic keyboard layout for 43edo:
A possible isomorphic keyboard layout for 43edo:
[[File:Fifth Comma Meantone Keyboard Layout.svg|800px|none|thumb]]
[[File:Fifth Comma Meantone Keyboard Layout.svg|800px|none|thumb]]


==References==
== Notes ==
<references group="note" />
 
== References ==
<references />
<references />


Retrieved from "https://en.xen.wiki/w/43edo"