Superpyth: Difference between revisions
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'''Superpyth''', a member of the [[archytas clan]], has 4/3 as a generator, and the Archytas comma 64/63 is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo | '''Superpyth''', a member of the [[archytas clan]], has 4/3 as a generator, and the Archytas comma [[64/63]] is [[tempering out|tempered out]], so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for [[meantone]] and [[12edo]], with the understanding that, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that the [[Wikipedia: Plastic number|plastic number]] has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth. | ||
If the 5th harmonic is used at all, it is mapped to -9 generators through tempering out [[245/243]], so C-D♯ is 5/4. So superpyth is " | Such a temperament without the 5th harmonic is also called '''archy'''. If the 5th harmonic is used at all, it is mapped to -9 generators through tempering out [[245/243]], so C-D♯ is 5/4. So superpyth is the "opposite" of septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex. | ||
If intervals of 11 are desired, the canonical way is to map 11/8 to -16 generators, so 11/8 is a double augmented second (C-Dx), tempering out 100/99. Yet a simpler but reasonable way is to map it to +6 generators, so 11/8 is a diminished fifth (C-G♭), by tempering out 99/98. The latter is called '''supra''', or '''suprapyth'''. The two mappings unite on [[22edo | If intervals of 11 are desired, the canonical way is to map 11/8 to -16 generators, so 11/8 is a double augmented second (C-Dx), tempering out 100/99. Yet a simpler but reasonable way is to map it to +6 generators, so 11/8 is a diminished fifth (C-G♭), by tempering out 99/98. The latter is called '''supra''', or '''suprapyth'''. The two mappings unite on [[22edo]]. | ||
[[MOS scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22. | [[MOS scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22. | ||
== Temperament data == | == Temperament data == | ||
{{ | {{Main| Archytas clan #Superpyth }} | ||
== Interval chains == | == Interval chains == | ||
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* [[Archy12]] | * [[Archy12]] | ||
* [[Supra12]] | * [[Supra12]] | ||
The boundary of propriety is [[17edo | The boundary of propriety is [[17edo]]. | ||
== Tuning spectrum == | == Tuning spectrum == | ||
| Line 315: | Line 315: | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)] | ||
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7] | ||
By [[Joel Grant Taylor]], all in Superpyth[12] in | By [[Joel Grant Taylor]], all in Superpyth[12] in 22edo tuning. | ||
* [https://m.soundcloud.com/lillianhearne/superpyth12-chromatic-riff <nowiki>Superpyth[12] chromatic riff</nowiki>] | * [https://m.soundcloud.com/lillianhearne/superpyth12-chromatic-riff <nowiki>Superpyth[12] chromatic riff</nowiki>] | ||
* [https://m.soundcloud.com/lillianhearne/trio-in-superpyth-temperament-for-irish-whistle-cello-and-piano Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello] | * [https://m.soundcloud.com/lillianhearne/trio-in-superpyth-temperament-for-irish-whistle-cello-and-piano Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello] | ||
Both by [[User:Lhearne|Lillian Hearne]] in | Both by [[User:Lhearne|Lillian Hearne]] in 22edo tuning | ||
== See also == | == See also == | ||
* [[ | * [[Ultrapyth]] | ||
{{IoT}} | |||
[[Category:Superpyth| ]] <!-- main article --> | [[Category:Superpyth| ]] <!-- main article --> | ||
[[Category:Archytas clan]] | [[Category:Archytas clan]] | ||
[[Category:Sensamagic clan]] | [[Category:Sensamagic clan]] | ||
[[Category:Orwellismic temperaments]] | [[Category:Orwellismic temperaments]] | ||
Revision as of 22:27, 4 January 2022
Superpyth, a member of the archytas clan, has 4/3 as a generator, and the Archytas comma 64/63 is tempered out, so two generators represents 7/4 in addition to 16/9. Since 4/3 is a generator we can use the same standard chain-of-fourths notation that is also used for meantone and 12edo, with the understanding that, for example, A♯ is sharper than B♭ (in contrast to meantone where A♯ is flatter than B♭, or 12edo where they are identical). An interesting coincidence is that the plastic number has a value of ~486.822 cents, which, taken as a generator and assuming an octave period, constitutes a variety of superpyth.
Such a temperament without the 5th harmonic is also called archy. If the 5th harmonic is used at all, it is mapped to -9 generators through tempering out 245/243, so C-D♯ is 5/4. So superpyth is the "opposite" of septimal meantone in several different ways: meantone has 4/3 tempered wide so that intervals of 5 are simple and intervals of 7 are complex, while superpyth has 4/3 tempered narrow so that intervals of 7 are simple while intervals of 5 are complex.
If intervals of 11 are desired, the canonical way is to map 11/8 to -16 generators, so 11/8 is a double augmented second (C-Dx), tempering out 100/99. Yet a simpler but reasonable way is to map it to +6 generators, so 11/8 is a diminished fifth (C-G♭), by tempering out 99/98. The latter is called supra, or suprapyth. The two mappings unite on 22edo.
MOS scales of superpyth have cardinalities of 5, 7, 12, 17, or 22.
Temperament data
Interval chains
- Archy (2.3.7)
| 1146.61 | 437.29 | 927.97 | 218.64 | 709.32 | 0 | 490.68 | 981.36 | 272.03 | 762.71 | 53.39 |
| 27/14 | 9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9 | 28/27 |
- Full 7-limit superpyth
| 613.20 | 1102.91 | 392.62 | 882.33 | 172.04 | 661.75 | 1151.46 | 441.16 | 930.87 | 220.58 | 710.29 | 0 | 489.71 | 979.42 | 269.13 | 758.84 | 48.54 | 538.25 | 1027.96 | 317.67 | 807.38 | 97.09 | 586.80 |
| 10/7 | 15/8 | 5/4 | 5/3 | 10/9 | 27/14 | 9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9 | 28/27 | 9/5 | 6/5 | 8/5 | 16/15 | 7/5 |
- Supra (2.3.7.11)
| 857.54 | 150.35 | 643.15 | 1135.96 | 428.77 | 921.58 | 214.38 | 707.19 | 0 | 492.81 | 985.62 | 278.42 | 771.23 | 64.04 | 556.85 | 1049.65 | 342.46 |
| 18/11 | 12/11 | 16/11 | 27/14 | 14/11~9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9~11/7 | 33/32~28/27 | 11/8 | 11/6 | 11/9 |
- Full 11-limit suprapyth
| 604.44 | 1094.94 | 385.45 | 875.96 | 166.46 | 656.97 | 1147.47 | 437.98 | 928.48 | 218.99 | 709.49 | 0 | 490.51 | 981.01 | 271.52 | 762.02 | 52.53 | 543.03 | 1033.54 | 324.04 | 814.55 | 105.06 | 595.56 |
| 10/7 | 15/8 | 5/4 | 18/11~5/3 | 12/11~10/9 | 16/11 | 27/14 | 14/11~9/7 | 12/7 | 9/8~8/7 | 3/2 | 1/1 | 4/3 | 7/4~16/9 | 7/6 | 14/9~11/7 | 33/32~28/27 | 11/8 | 9/5~11/6 | 6/5~11/9 | 8/5 | 16/15 | 7/5 |
Scales
- 5-note MOS (2L 3s, proper)
- 7-note MOS (5L 2s, improper)
In contrast to the meantone diatonic scale, the superpyth diatonic is improper.
- 12-note MOS (5L 7s, borderline improper)
The boundary of propriety is 17edo.
Tuning spectrum
| ET Generator |
Eigenmonzo (Unchanged Interval) |
Generator (¢) |
Comments |
|---|---|---|---|
| 4/3 | 701.955 | Pythagorean tuning | |
| 10\17 | 705.882 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 28/27 | 707.408 | 1/5 comma | |
| 23\39 | 707.692 | ||
| 9/7 | 708.771 | 1/4 comma, 1.3.7.9 minimax | |
| 16/15 | 708.807 | ||
| 13\22 | 709.091 | ||
| 5/4 | 709.590 | 9-odd-limit minimax | |
| 54/49 | 709.745 | 2/7 comma | |
| 25/24 | 710.040 | ||
| 29\49 | 710.204 | ||
| 6/5 | 710.545 | ||
| 7/5 | 710.681 | 7-odd-limit minimax | |
| 7/6 | 711.043 | 1/3 comma, 1.3.7 minimax | |
| 16\27 | 711.111 | ||
| 10/9 | 711.772 | ||
| 49/48 | 712.861 | 2/5 comma | |
| 8/7 | 715.587 | 1/2 comma | |
| 3\5 | 720.000 | Upper bound of 7- and 9-odd-limit diamond monotone |
Music
By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.
Both by Lillian Hearne in 22edo tuning