Harry Partch's 43-tone scale: Difference between revisions
Expansion. |
More edo comparisons. |
||
| Line 93: | Line 93: | ||
[[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" /> | [[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 various equal tunings == | |||
==Comparison with 41edo== | ===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). | 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). | ||
| Line 185: | Line 185: | ||
|} | |} | ||
== Comparison with 72edo == | === Comparison with 46edo === | ||
Like 41edo, [[46edo]] maps the 43 note scale to its patent val in a consistent way. However, this time, 12/11 and 11/10 (and their octave compliments) are tempered together rather than 11/10 and 10/9. In addition, several other intervals that are tempered closer together than JI in 41 edo are tempered further apart here, making the scale more uneven sounding in general. | |||
The step sizes in this tuning are: 1 1 1 1 2 0 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 0 2 1 1 1 1 | |||
=== Comparison with 53edo === | |||
While [[53edo]] is excellently tuned in the [[5-limit]] and has more than enough steps to represent this scale without conflating any notes, it's poor representation of the 11th harmonic causes problems. Not only are 12/11 and 11/10 once again tempered together, but 14/11 is conflated with 9/7 in the patent val despite actually being closer to 81/64, which would mean only 39 of the scales notes can be cleanly defined under this system. Using the 53e val separates these notes, but now 11/10 and 10/9 are tempered together, which means it also does not cleanly map all 43 notes to different steps. | |||
=== Comparison with 58edo === | |||
As the first edo that is distinctly consistent in the full [[11-odd-limit]], [[58edo]] is also the first edo that can clearly represent every interval in this scale with no conflations or discrepancies between the patent val and the direct mapping. However, it is strongly sharp trending, which means compound harmonics stack their error, with 33/32 and its octave compliment the worst tuned intervals at 8.8 cents out. | |||
The step sizes in this tuning are: 1 2 1 1 2 1 1 1 1 2 1 1 2 2 1 1 2 1 1 2 1 2 1 2 1 1 2 1 1 2 2 1 1 2 1 1 1 1 2 1 1 2 1 | |||
=== 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 Harry Partch's 43-tone scale very well, with a maximum error of 5.9 cents, with the worst-tuned ratios the result of stacking the compound flatness of multiple 3/2's. | Since [[72edo]] is [[distinctly consistent]] in the 11-limit and is a [[pepper ambiguity]] record in the 11-limit unsurpassed until 270, 72edo fits Harry Partch's 43-tone scale very well, with a maximum error of 5.9 cents, with the worst-tuned ratios the result of stacking the compound flatness of multiple 3/2's. | ||
The | The step sizes in this tuning are: 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 | ||
== Comparison with 270edo == | === Comparison with 270edo === | ||
In [[270edo]] the step sizes are somewhat more uneven, but far more accurate, with a maximum error of approximately 0.8 cents from JI. | In [[270edo]] the step sizes are somewhat more uneven, but far more accurate, with a maximum error of approximately 0.8 cents from JI. | ||
The step sizes in this tuning are 5 7 7 6 9 3 4 5 6 8 6 5 7 9 7 4 8 6 5 7 7 8 7 7 5 6 8 4 7 9 7 5 6 8 6 5 4 3 9 6 7 7 5 | The step sizes in this tuning are: 5 7 7 6 9 3 4 5 6 8 6 5 7 9 7 4 8 6 5 7 7 8 7 7 5 6 8 4 7 9 7 5 6 8 6 5 4 3 9 6 7 7 5 | ||
== Music == | == Music == | ||
;[[Chris Ranier]] | ;[[Chris Ranier]] | ||
* [https://chrisrainier.bandcamp.com/album/chris-rainier-sings-the-music-of-harry-partch ''Chris Ranier sings the music of Harry Partch](2024) | * [https://chrisrainier.bandcamp.com/album/chris-rainier-sings-the-music-of-harry-partch ''Chris Ranier sings the music of Harry Partch](2024) | ||
| Line 208: | Line 225: | ||
*[http://www.chrysalis-foundation.org/Meyer-s_Diamond.htm "Musical Mathematics: Meyer's Diamond"] at ''Chrysalis-Foundation.org'' | *[http://www.chrysalis-foundation.org/Meyer-s_Diamond.htm "Musical Mathematics: Meyer's Diamond"] at ''Chrysalis-Foundation.org'' | ||
[[Category:Just intonation scales]] | [[Category:Just intonation scales]] [[Category:Harry Partch]] [[Category:11-limit]] | ||