Harry Partch's 43-tone scale: Difference between revisions

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 [[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===
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 [[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
+|}
 ==References==