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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|43}}
{{ED intro}}


== History ==
== History ==
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{{Harmonics in equal|43}}
{{Harmonics in equal|43}}
{{Harmonics in equal|43|start=12|columns=9|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)}}
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 32: 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
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|-
|-
| 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 50: 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 57: 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 64: 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
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|-
|-
| 5
| 5
| 139.535
| 139.5
| 12/11, 13/12, 14/13
| 12/11, 13/12, 14/13
| ^m2
| ^m2
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|-
|-
| 6
| 6
| 167.442
| 167.4
| 11/10
| 11/10
| vM2
| vM2
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|-
|-
| 7
| 7
| 195.349
| 195.3
| 9/8, 10/9
| 9/8, 10/9
| M2
| M2
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|-
|-
| 8
| 8
| 223.256
| 223.3
| 8/7
| 8/7
| ^M2, d3
| ^M2, d3
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|-
|-
| 9
| 9
| 251.163
| 251.2
| 15/13
| 15/13
| vA2, ^d3
| vA2, ^d3
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|-
|-
| 10
| 10
| 279.070
| 279.1
| 7/6, 13/11
| 7/6, 13/11
| A2, vm3
| A2, vm3
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|-
|-
| 11
| 11
| 306.977
| 307.0
| 6/5
| 6/5
| m3
| m3
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|-
|-
| 12
| 12
| 334.884
| 334.9
| 39/32, 17/14
| 17/14, ''27/22'', 39/32, 40/33
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
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|-
|-
| 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
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|-
|-
| 14
| 14
| 390.698
| 390.7
| 5/4
| 5/4
| M3
| M3
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|-
|-
| 15
| 15
| 418.605
| 418.6
| ''9/7'', 14/11
| ''9/7'', 14/11
| ^M3, d4
| ^M3, d4
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|-
|-
| 16
| 16
| 446.512
| 446.5
| 13/10
| 13/10, 22/17
| vA3, ^d4
| vA3, ^d4
| downaug 3rd, updim 4th
| downaug 3rd, updim 4th
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|-
|-
| 17
| 17
| 474.419
| 474.4
| 21/16
| 21/16
| v4
| v4
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|-
|-
| 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
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|-
|-
| 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
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|-
|-
| 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
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|-
|-
| 25
| 25
| 697.674
| 697.7
| 3/2
| 3/2
| P5
| P5
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|-
|-
| 26
| 26
| 725.581
| 725.6
| 32/21
| 32/21
| ^5
| ^5
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|-
|-
| 27
| 27
| 753.488
| 753.5
| 20/13
| 17/11, 20/13
| vA5, ^d6
| vA5, ^d6
| downaug 5th, updim 6th
| downaug 5th, updim 6th
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|-
|-
| 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
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|-
|-
| 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
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|-
|-
| 35
| 35
| 976.744
| 976.7
| 7/4
| 7/4
| A6, vm7
| A6, vm7
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|-
|-
| 36
| 36
| 1004.651
| 1004.7
| 16/9, 9/5
| 9/5, 16/9
| m7
| m7
| minor 7th
| minor 7th
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|-
|-
| 37
| 37
| 1032.558
| 1032.6
| 20/11
| 20/11
| ^m7
| ^m7
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|-
|-
| 38
| 38
| 1060.465
| 1060.5
| 11/6, 24/13, 13/7
| 11/6, 13/7, 24/13
| vM7
| vM7
| downmajor 7th
| downmajor 7th
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|-
|-
| 39
| 39
| 1088.372
| 1088.4
| 15/8, 28/15, 32/17
| 15/8, 28/15, 32/17
| M7
| M7
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|-
|-
| 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
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|-
|-
| 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
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|-
|-
| 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
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|-
|-
| 43
| 43
| 1200.000
| 1200.0
| 2/1
| 2/1
| P8
| P8
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|}
|}


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


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


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}}
{{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.
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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 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 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).
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'' unambiguous red-note/blue-note notation for [[45edo]], which is another meantone (actually, a [[flattone]]) system.
If {{colored note|red|C♭}} and {{colored note|blue|B♯}} (and {{colored note|red|F♭}}&nbsp;/&nbsp;{{colored note|blue|E♯}}) are instead forced to be distinct, but the requirement that all notes be equally spaced is maintained, then we end up with a ''completely'' single-name red-note/blue-note notation for [[45edo]], which is another meantone (actually, a [[flattone]]) system.
 
== Approximation to JI ==
[[File:43ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 43edo]]
 
=== Interval mappings ===
{{Q-odd-limit intervals}}


== Regular temperament properties ==
== Regular temperament properties ==
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| 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 749: Line 757:
| Spleen comma
| Spleen comma
|}
|}
<references group="note"/>


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave
== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[196zpi]]
| 43.0264994477693
| 27.8897892090130
| 6.166118
| 1.035628
| 15.545919
| 43edo
| 1199.26093598756
| 8
| 8
|}


== Detemperaments ==
== Detemperaments ==
Line 910: Line 885:
=== Harmonic scales ===
=== Harmonic scales ===
43edo represents the first 16 overtones of the [[harmonic series]] well (written as a ratio of 8:9:10:11:12:13:14:15:16 in [[just intonation]]) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.
43edo represents the first 16 overtones of the [[harmonic series]] well (written as a ratio of 8:9:10:11:12:13:14:15:16 in [[just intonation]]) with degrees 0, 7, 14, 20, 25, 30, 35, 39, and 43, and scale steps of 7, 7, 6, 5, 5, 5, 4, and 4.
* 7\43 (195.349¢) stands in for frequency ratio [[9/8]] (203.910¢) and [[10/9]] (182.404¢).
* 7\43 (195.3¢) stands in for frequency ratio [[9/8]] (203.9¢) and [[10/9]] (182.4¢).
* 6\43 (156.522¢) stands in for [[11/10]] (165.004¢)
* 6\43 (156.5¢) stands in for [[11/10]] (165.0¢).
* 5\46 (130.435¢) stands in for [[12/11]] (150.637¢), [[13/12]] (138.573¢), and [[14/13]] (128.298¢).
* 5\46 (130.4¢) stands in for [[12/11]] (150.6¢), [[13/12]] (138.6¢), and [[14/13]] (128.3¢).
* 4\43 (111.628¢) stands in for [[15/14]] (119.443¢) and [[16/15]] (111.731¢).
* 4\43 (111.6¢) stands in for [[15/14]] (119.4¢) and [[16/15]] (111.7¢).


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 1,026: Line 1,001:
; [[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
Retrieved from "https://en.xen.wiki/w/43edo"