43edo: Difference between revisions

{{ED intro}}

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]].  

{{Harmonics in equal|43|columns=11}}

{{Harmonics in equal|43|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}

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 }}.

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)⁵ → 5/1}}), with [[mos scale]]s of 15 and 28.

=== Subsets and supersets ===

! Approximate ratios*

! colspan="3" | [[Ups and downs notation]]<br>([[Enharmonic unisons in ups and downs notation|EUs]]: v³A1 and vd2)

| 0.0

| [[1/1]]

| 27.9

| ''[[36/35]]'', [[50/49]], [[64/63]], [[65/64]], [[66/65]]

| 55.8

| [[26/25]], [[27/26]], [[33/32]], [[40/39]], ''[[49/48]]''

| 83.7

| ''[[18/17]]'', [[21/20]], [[22/21]], [[25/24]], ''[[28/27]]''

| 111.6

| [[15/14]], [[16/15]], [[17/16]]

| 139.5

| [[12/11]], [[13/12]], [[14/13]]

| 167.4

| [[11/10]]

| 195.3

| [[9/8]], [[10/9]]

| 223.3

| [[8/7]]

| 251.2

| [[15/13]]

| 279.1

| [[7/6]], [[13/11]], [[20/17]]

| 307.0

| [[6/5]]

| 334.9

| [[17/14]], ''[[27/22]]'', [[39/32]], [[40/33]]

| 362.8

| ''[[11/9]]'', [[16/13]], [[21/17]], [[26/21]]

| 390.7

| [[5/4]]

| 418.6

| ''[[9/7]]'', [[14/11]]

| 446.5

| [[13/10]], [[22/17]]

| 474.4

| [[21/16]]

| 502.3

| [[4/3]]

| 530.2

| [[15/11]]

| 558.1

| [[11/8]], [[18/13]]

| 586.0

| [[7/5]], [[24/17]], [[45/32]]

| 614.0

| [[10/7]], [[17/12]], [[64/45]]

| 641.9

| [[13/9]], [[16/11]]

| 669.8

| [[22/15]]

| 697.7

| [[3/2]]

| 725.6

| [[32/21]]

| 753.5

| [[17/11]], [[20/13]]

| 781.4

| [[11/7]], ''[[14/9]]''

| 809.3

| [[8/5]]

| 837.2

| [[13/8]], ''[[18/11]]'', [[21/13]], [[34/21]]

| 865.1

| [[28/17]], [[33/20]], ''[[44/27]]'', [[64/39]]

| 893.0

| [[5/3]]

| 920.9

| [[12/7]], [[22/13]], [[17/10]]

| 948.8

| [[26/15]]

| 976.7

| [[7/4]]

| 1004.7

| [[9/5]], [[16/9]]

| 1032.6

| [[20/11]]

| 1060.5

| [[11/6]], [[13/7]], [[24/13]]

| 1088.4

| [[15/8]], [[28/15]], [[32/17]]

| 1116.3

| ''[[17/9]]'', [[21/11]], ''[[27/14]]'', [[40/21]], [[48/25]]

| 1144.2

| [[25/13]], [[39/20]], [[52/27]], [[64/33]], ''[[96/49]]''

| 1172.1

| ''[[35/18]]'', [[49/25]], [[63/32]], [[65/33]], [[128/65]]

| 1200.0

| [[2/1]]

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

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

 

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 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.

 

Note that downsharp can be respelled as dup (double-up), and upflat as dud.

{{Ups and downs sharpness}}

 

 

 

default [[File:43-EDO_Revo_Sagittal.svg]]

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).

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]]

=== 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.

! rowspan="2" | [[Comma list]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

| {{Monzo| -68 43 }}

| {{Mapping| 43 68 }}

| {{Mapping| 43 68 100 }}

| {{Mapping| 43 68 100 121 }}

| {{Mapping| 43 68 100 121 149 }}

| {{Mapping| 43 68 100 121 149 159 }}

| {{Mapping| 43 68 100 121 149 159 176 }}

| {{Mapping| 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 }}.

! [[Harmonic limit|Prime<br>limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

| {{Monzo| -68 43 }}

| {{Monzo| 20 5 -12 }}

| {{Monzo| 24 1 -11 }}

| {{Monzo| -4 4 -1 }}

| Syntonic comma, Didymus' comma, meantone comma

| {{Monzo| 32 -7 -9 }}

| {{Monzo| -68 18 17 }}

| [[Vavoom comma]]

| {{Monzo| -13 10 0 -1 }}

| {{Monzo| -9 6 1 -1 }}

| {{Monzo| 5 -1 7 -7 }}

| [[Mermisma]]

| {{Monzo| 1 2 -3 1 }}

| {{Monzo| 21 -5 -2 -3 }}

| {{Monzo| -14 7 -6 6 }}

| {{Monzo| -5 2 2 -1 }}

| Marvel comma

| {{Monzo| 6 0 -5 2 }}

| {{Monzo| -11 2 7 -3 }}

| [[Meter]]

| {{Monzo| 1 3 2 0 -3}}

| Large tetracot diesis

| {{Monzo| -1 2 0 -2 1 }}

| {{Monzo| 4 0 -2 -1 1 }}

| {{Monzo| -3 2 -1 2 -1 }}

| {{Monzo| 5 -1 3 0 -3}}

| {{Monzo| 17 -5 0 -2 -1 }}

| [[Olympia]]

| {{Monzo| 24 -6 0 1 -5 }}

| {{Monzo| 1 1 0 -1 -1 1 }}

| {{Monzo| 4 2 0 0 -1 -1 }}

| {{Monzo| -3 -1 0 -1 0 2 }}

| {{Monzo| 9 6 0 0 0 -5 }}

| {{Monzo| 2 -1 0 1 -2 1 }}

| {{Monzo| -3 0 -3 1 1 1 }}

| {{Monzo| 5 -3 1 -1 -1 1 }}

| {{Monzo| 12 -2 -1 -1 0 -1 }}

| Minisma

| {{Monzo| 3 1 1 -1 0 0 -1 }}

| {{Monzo| -2 0 -1 0 -1 1 1 }}

| {{Monzo| 8 -1 -1 0 0 0 -1 }}

| {{Monzo| 5 1 -1 0 0 0 0 -1 }}

| {{Monzo| -1 -1 1 -1 1 1 -1 }}

| {{Monzo| 5 1 -1 0 0 0 0 -1 }}

| 19th-partial chroma

| {{Monzo| -3 2 0 0 0 0 1 -1}}

| {{Monzo| -1 2 -1 0 0 0 -1 1 }}

| {{Monzo| -4 0 0 0 1 -1 0 1 }}

| {{Monzo| 1 1 1 1 -1 0 0 -1 }}

<references group="note" />

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods<br>per 8ve

! Generator*

! Cents*

! Associated<br>ratio*

! Temperaments

| 27.9

| 55.8

| 111.6

| 139.5

| 167.4

| 195.3

| 223.3

| 251.2

| 279.1

| 334.9

| 362.8

| [[Demibuzzard]] / interpental

| 390.7

| 446.5

| 502.3

| 530.2

| 558.1

| 586.0

* 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¢).

 

=== Mos scales ===

; Major scales

; Minor scales

; Modal scales

; Blues scales

; Others

== References ==