View source for Harry Partch's 43-tone scale

[[File:Harry Partch Institute-3.jpg|thumb|right|250px|The [[Quadrangularis Reversum]], one of Partch's instruments featuring the 43-tone scale]] 
The '''43-tone scale''' is a [[just intonation]] scale with 43 pitches in each [[octave]]. It is based on an eleven-limit [[tonality diamond]], similar to the seven-limit diamond previously devised by [[Max Friedrich Meyer]]<ref>[http://www.chrysalis-foundation.org/Meyer-s_Diamond.htm "Musical Mathematics: Meyer's Diamond"], ''Chrysalis-Foundation.org''.</ref> and refined by [[Harry Partch]].<ref>Kassel, R. (2001, January 20). Partch, Harry. [https://www.oxfordmusiconline.com/grovemusic/ ''Grove Music Online''].</ref>

See [[Partch 43]] for the scale as a scala file.

==Ratios of the 11 Limit==
Here are all the ratios within the [[octave]] with odd factors up to and including 11, known as the 11-limit [[tonality diamond]]. Note that the [[Inversion (interval)|inversion]] of every interval is also present, so the set is symmetric about the octave.

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|0||150.6||165.0||182.4||203.9||231.2||266.9||315.6||347.4||386.3||417.5||435.1||498.0||551.3||582.5
|-
|[[Just intonation|Ratio]]
|[[1/1]]||[[12/11]]||[[11/10]]||[[10/9]]||[[9/8]]||[[8/7]]||[[7/6]]||[[6/5]]||[[11/9]]||[[5/4]]||14/11||[[9/7]]||[[4/3]]||[[11/8]]||[[7/5]]
|-
|[[41edo]]
|0.0||5.1||5.6||6.2||7.0||7.9||9.1||10.8||11.9||13.2||14.3||14.9||17.0||18.8||19.9
|}

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|617.5||648.7||702.0||764.9||782.5||813.7||852.6||884.4||933.1||968.8||996.1||1017.6||1035.0||1049.4||1200
|-
|[[Just intonation|Ratio]]
|[[10/7]]||16/11||[[3/2]]||[[14/9]]||[[11/7]]||[[8/5]]||[[18/11]]||[[5/3]]||[[12/7]]||[[7/4]]||[[16/9]]||[[9/5]]||[[20/11]]||[[11/6]]||[[2/1]]
|-
|[[41edo]]
|21.1||22.2||24.0||26.1||26.7||27.8||29.1||30.2||31.9||33.1||34.0||34.8||35.4||35.9||41.0
|}

==Filling in the gaps==
There are two reasons why the 11-limit ratios by themselves would not make a good scale. First, the scale only contains a complete set of chords ([[otonalities]] and [[utonalities]]) based on one [[tonic (music)|tonic]] pitch. Second, it contains large gaps, between the tonic and the two pitches to either side, as well as several other places. Both problems can be solved by filling in the gaps with "multiple-number ratios", or intervals obtained from the product or quotient of other intervals within the 11 limit.

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|0||'''21.5'''||'''53.2'''||'''84.5'''||'''111.7'''||150.6
|-
|[[Just intonation|Ratio]]
|1/1||'''[[81/80]]'''||'''33/32'''||'''[[21/20]]'''||'''[[16/15]]'''||12/11
|}

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|266.9||'''294.1'''||315.6
|-
|[[Just intonation|Ratio]]
|7/6||'''[[32/27]]'''||6/5
|}

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|435.1||'''470.8'''||498.0||'''519.5'''||551.3
|-
|[[Just intonation|Ratio]]
|9/7||'''21/16'''||4/3||'''[[27/20]]'''||11/8
|}

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|648.7||'''680.5'''||702.0||'''729.2'''||764.9
|-
|[[Just intonation|Ratio]]
|16/11||'''[[40/27]]'''||3/2||'''32/21'''||14/9
|}

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|884.4||'''905.9'''||933.1
|-
|[[Just intonation|Ratio]]
|5/3||'''27/16'''||12/7
|}

{| class="wikitable" style="text-align:center;"
|-
|[[Cent (music)|Cents]]
|1049.4||'''1088.3'''||'''1115.5'''||'''1146.8'''||'''1178.5'''||1200
|-
|[[Just intonation|Ratio]]
|11/6||'''[[15/8]]'''||'''40/21'''||'''64/33'''||'''160/81'''||2/1
|}

Together with the 29 ratios of the 11 limit, these 14 multiple-number ratios make up the full 43-tone scale.

[[Erv Wilson]] who worked with Partch has pointed out that these added tones form a constant structure of 41 tones with two variables.<ref name="Anaphoria">"Letter to John from ERV Wilson, 19 October 1964 - SH 5 Chalmers" (PDF). Anaphoria.com. Retrieved 2016-10-28.page 11</ref> A constant structure giving one the property of anytime a ratio appears it will be subtended by the same number of steps. In this way Partch resolved his harmonic and melodic symmetry in one of the best ways possible.<ref name="Anaphoria" />

==Comparison with 41edo==
The 43-note scale is almost [[epimorphic]] under the [[41edo]] [[patent val]]. The only exceptions are the pair {11/10, 10/9} and its octave complement {9/5, 20/11}, which are tempered together in [[41edo]]. Other than those, 41edo does a decent job of representing everything, for an EDO (although of course Partch himself would scoff at such a claim).

{| class="wikitable"
! 41edo steps !! Partch ratio(s) !! Partch cents !! EDO cents !! Error (cents)
|-
| 0 || 1/1 || 0.00 || 0.00 || 0.00
|-
| 1 || 81/80 || 21.51 || 29.27 || +7.76
|-
| 2 || 33/32 || 53.27 || 58.54 || +5.26
|-
| 3 || 21/20 || 84.47 || 87.80 || +3.34
|-
| 4 || 16/15 || 111.73 || 117.07 || +5.34
|-
| 5 || 12/11 || 150.64 || 146.34 || -4.30
|-
| 6 || 11/10, 10/9 || 165.00, 182.40 || 175.61 || 10.61, -6.79
|-
| 7 || 9/8 || 203.91 || 204.88 || +0.97
|-
| 8 || 8/7 || 231.17 || 234.15 || +2.97
|-
| 9 || 7/6 || 266.87 || 263.41 || -3.46
|-
| 10 || 32/27 || 294.13 || 292.68 || -1.45
|-
| 11 || 6/5 || 315.64 || 321.95 || +6.31
|-
| 12 || 11/9 || 347.41 || 351.22 || +3.81
|-
| 13 || 5/4 || 386.31 || 380.49 || -5.83
|-
| 14 || 14/11 || 417.51 || 409.76 || -7.75
|-
| 15 || 9/7 || 435.08 || 439.02 || +3.94
|-
| 16 || 21/16 || 470.78 || 468.29 || -2.49
|-
| 17 || 4/3 || 498.04 || 497.56 || -0.48
|-
| 18 || 27/20 || 519.55 || 526.83 || +7.28
|-
| 19 || 11/8 || 551.32 || 556.10 || +4.78
|-
| 20 || 7/5 || 582.51 || 585.37 || +2.85
|-
| 21 || 10/7 || 617.49 || 614.63 || -2.85
|-
| 22 || 16/11 || 648.68 || 643.90 || -4.78
|-
| 23 || 40/27 || 680.45 || 673.17 || -7.28
|-
| 24 || 3/2 || 701.96 || 702.44 || +0.48
|-
| 25 || 32/21 || 729.22 || 731.71 || +2.49
|-
| 26 || 14/9 || 764.92 || 760.98 || -3.94
|-
| 27 || 11/7 || 782.49 || 790.24 || +7.75
|-
| 28 || 8/5 || 813.69 || 819.51 || +5.83
|-
| 29 || 18/11 || 852.59 || 848.78 || -3.81
|-
| 30 || 5/3 || 884.36 || 878.05 || -6.31
|-
| 31 || 27/16 || 905.87 || 907.32 || +1.45
|-
| 32 || 12/7 || 933.13 || 936.59 || +3.46
|-
| 33 || 7/4 || 968.83 || 965.85 || -2.97
|-
| 34 || 16/9 || 996.09 || 995.12 || -0.97
|-
| 35 || 9/5, 20/11 || 1017.60, 1035.00 || 1024.39 || +6.79, -10.61
|-
| 36 || 11/6 || 1049.36 || 1053.66 || +4.30
|-
| 37 || 15/8 || 1088.27 || 1082.93 || -5.34
|-
| 38 || 40/21 || 1115.53 || 1112.20 || -3.34
|-
| 39 || 64/33 || 1146.73 || 1141.46 || -5.26
|-
| 40 || 160/81 || 1178.49 || 1170.73 || -7.76
|-
| 41 || 2/1 || 1200.00 || 1200.00 || 0.00
|}

== Comparison with 72edo ==
Since [[72edo]] is [[distinctly consistent]] in the 11-limit and is a [[pepper ambiguity]] record in the 11-limit unsurpassed until 270, 72edo fits very well the Harry Partch's 43-tone scale.

The mode is: 1 2 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1

==References==
<references />

==Further reading==
*[http://www.chrysalis-foundation.org/Meyer-s_Diamond.htm "Musical Mathematics: Meyer's Diamond"] at ''Chrysalis-Foundation.org''

[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]