Superpyth: Difference between revisions

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"Superpythagorean" more commonly means any tuning with sharp of just fifths
Improve the readability for the prime-optimized tuning tables. Misc. style cleanup
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'''Superpyth''' is a [[temperament]] where [[~]][[3/2]] is a [[generator]], and the septimal comma ([[64/63]]) is [[tempering out|tempered out]], so that a stack of two perfect fifths [[octave reduction|octave-reduced]] gives a major whole tone that represents both [[9/8]] and [[8/7]] (likewise, two perfect fourths give a minor seventh that represents both [[7/4]] and [[16/9]], so intervals such as A–G and C–B♭ are harmonic sevenths). Since 3/2 is a generator we can use the same standard [[chain-of-fifths notation]] that is also used for [[meantone]] and [[12edo]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[17edo|10\17]], [[22edo|13\22]], and [[27edo|16\27]] are typical tunings of the generator.  
'''Superpyth''' is a [[regular temperament|temperament]] where [[~]][[3/2]] is a [[generator]], and the septimal comma ([[64/63]]) is [[tempering out|tempered out]], so that a stack of two perfect fifths [[octave reduction|octave-reduced]] gives a major whole tone that represents both [[9/8]] and [[8/7]] (likewise, two perfect fourths give a minor seventh that represents both [[7/4]] and [[16/9]], so intervals such as A–G and C–B♭ are harmonic sevenths). Since 3/2 is a generator we can use the same standard [[chain-of-fifths notation]] that is also used for [[meantone]] and [[12edo]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[17edo|10\17]], [[22edo|13\22]], and [[27edo|16\27]] are typical tunings of the generator.  


Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, 5th harmonic 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 (including [[12edo]]) has 3/2 tuned flat so that the 5th harmonic's intervals are simple and the 7th harmonic's intervals are complex, while superpyth has 3/2 tuned sharp so that the 7th harmonic's intervals are simple while the 5th harmonic's intervals are complex.
Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, 5th harmonic 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 (including [[12edo]]) has 3/2 tuned flat so that the 5th harmonic's intervals are simple and the 7th harmonic's intervals are complex, while superpyth has 3/2 tuned sharp so that the 7th harmonic's intervals are simple while the 5th harmonic's intervals are complex.
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== Interval chains ==
== Interval chains ==
In these tables, odd harmonics 1–11 are in '''bold'''.
In these tables, odd harmonics 1–11 are in '''bold'''.


<div><div style="display: inline-grid; margin-right: 25px;">
<div><div style="display: inline-grid; margin-right: 25px;">
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|+ style="font-size: 105%;" | Archy (2.3.7)
|+ style="font-size: 105%;" | Archy (2.3.7)
|-
|-
! &#35; !! Cents* !! Approximate Ratios
! # !! Cents* !! Approximate ratios
|-
|-
| 0 || 0.0 || '''1/1'''
| 0 || 0.0 || '''1/1'''
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| 7 || 167.2 || 54/49
| 7 || 167.2 || 54/49
|}
|}
<nowiki />* In 2.3.7-subgroup [[CTE]] tuning
<nowiki/>* In 2.3.7-subgroup [[CTE]] tuning
</div>
</div>
<div style="display: inline-grid; margin-right: 25px;">
<div style="display: inline-grid; margin-right: 25px;">
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|+ style="font-size: 105%;" | Supra (2.3.7.11)
|+ style="font-size: 105%;" | Supra (2.3.7.11)
|-
|-
! &#35; !! Cents* !! Approximate Ratios
! # !! Cents* !! Approximate ratios
|-
|-
| 0 || 0.0 || '''1/1'''
| 0 || 0.0 || '''1/1'''
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| 7 || 159.2 || 12/11
| 7 || 159.2 || 12/11
|}
|}
<nowiki />* In 2.3.7.11-subgroup CTE tuning
<nowiki/>* In 2.3.7.11-subgroup CTE tuning
</div>
</div>
<div style="display: inline-grid;">
<div style="display: inline-grid;">
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|+ style="font-size: 105%;" | Full 7-limit superpyth
|+ style="font-size: 105%;" | Full 7-limit superpyth
|-
|-
! rowspan="3" | &#35; !! rowspan="3" | Cents* !! colspan="3" | Approximate Ratios
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
|-
! rowspan="2" | 7-limit !! colspan="2" | 11-limit Extension
! rowspan="2" | 7-limit !! colspan="2" | 11-limit extension
|-
|-
! Superpyth !! Suprapyth
! Superpyth !! Suprapyth
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| 17 || 63.0 || 25/24 || 22/21, 33/32 || 45/44
| 17 || 63.0 || 25/24 || 22/21, 33/32 || 45/44
|}
|}
<nowiki />* In 7-limit CTE tuning
<nowiki/>* In 7-limit CTE tuning
</div></div>
</div></div>


== Scales ==
== Scales ==
; 5-note MOS ([[2L 3s]], proper)
; 5-note mos ([[2L 3s]], proper)
* [[Archy5]] &ndash; archy in 472edo tuning
* [[Archy5]] – archy in 472edo tuning


; 7-note MOS ([[5L 2s]], improper)
; 7-note mos ([[5L 2s]], improper)
* [[Archy7]] &ndash; archy in 472edo tuning
* [[Archy7]] – archy in 472edo tuning
* [[Supra7]] &ndash; supra in 56edo tuning
* [[Supra7]] – supra in 56edo tuning
In contrast to the meantone diatonic scale, the superpyth diatonic is improper.
In contrast to the meantone diatonic scale, the superpyth diatonic is improper.


; 12-note MOS ([[5L 7s]], borderline improper)
; 12-note mos ([[5L 7s]], borderline improper)
* [[Archy12]] &ndash; archy in 472edo tuning
* [[Archy12]] – archy in 472edo tuning
* [[Supra12]] &ndash; supra in 56edo tuning
* [[Supra12]] – supra in 56edo tuning
* [[12-22a]] &ndash; superpyth in 22edo tuning
* [[12-22a]] – superpyth in 22edo tuning
The boundary of propriety is [[17edo]].
The boundary of propriety is [[17edo]].


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|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7 Subgroup Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7 Subgroup Prime-Optimized Tunings
|-
|-
! Weight-skew\Order !! Euclidean
! rowspan="2" |
! colspan="2" | Euclidean
|-
|-
| Tenney || CTE: ~3/2 = 709.5948¢
! Unskewed
! Skewed
|-
|-
| Weil || CWE: ~3/2 = 709.3901¢
! Equilateral
| CEE: ~3/2 = 712.8606¢<br>(2/5-comma)
| CSEE: ~3/2 = 711.9997¢<br>(7/19-comma)
|-
|-
| Equilateral || CEE: ~3/2 = 712.8606¢ (2/5-comma tuning)
! Tenney
| CTE: ~3/2 = 709.5948¢
| CWE: ~3/2 = 709.3901¢
|-
|-
| Skewed-equilateral || CSEE: ~3/2 = 711.9997¢ (7/19-comma tuning)
! Benedetti, <br>Wilson
|-
| CBE: ~3/2 = 707.7286¢<br>(18/85-comma)
| Benedetti/Wilson || CBE: ~3/2 = 707.7286¢ (18/85-comma tuning)
| CSBE: ~3/2 = 707.9869¢<br>(25/113-comma)
|-
| Skewed-Benedetti/Wilson || CSBE: ~3/2 = 707.9869¢ (25/113-comma tuning)
|}
|}


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|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-Optimized Tunings
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit Prime-Optimized Tunings
|-
|-
! Weight-skew\Order !! Euclidean
! rowspan="2" |  
|-
! colspan="2" | Euclidean
| Tenney || CTE: ~3/2 = 709.5907¢
|-
| Weil || CWE: ~3/2 = 710.1193¢
|-
|-
| Equilateral || CEE: ~3/2 = 709.7805¢
! Unskewed
! Skewed
|-
|-
| Skewed-equilateral || CSEE: ~3/2 = 710.2428¢
! Equilateral
| CEE: ~3/2 = 709.7805¢
| CSEE: ~3/2 = 710.2428¢
|-
|-
| Benedetti/Wilson || CBE: ~3/2 = 709.4859¢
! Tenney
| CTE: ~3/2 = 709.5907¢
| CWE: ~3/2 = 710.1193¢
|-
|-
| Skewed-Benedetti/Wilson || CSBE: ~3/2 = 710.0321¢
! Benedetti, <br>Wilson
| CBE: ~3/2 = 709.4859¢
| CSBE: ~3/2 = 710.0321¢
|}
|}


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{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| Full comma
| Full comma
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5 Superpyth): ~2 = 1\1, ~3/2 = 709.758
* [[DKW theory|DKW]] (2.3.5 superpyth): ~2 = 1200.000, ~3/2 = 709.758
* DKW (2.3.7 Archy): ~3/2 = 712.585
* DKW (2.3.7 archy): ~2 = 1200.000, ~3/2 = 712.585


== Music ==
== Music ==
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* [https://soundcloud.com/lillianhearne/superpyth12-chromatic-riff ''Superpyth{{lbrack}}12{{rbrack}} chromatic riff''] (2015)
* [https://soundcloud.com/lillianhearne/superpyth12-chromatic-riff ''Superpyth{{lbrack}}12{{rbrack}} chromatic riff''] (2015)
* [https://soundcloud.com/lillianhearne/trio-in-superpyth-temperament-for-irish-whistle-cello-and-piano ''Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello''] (2015)
* [https://soundcloud.com/lillianhearne/trio-in-superpyth-temperament-for-irish-whistle-cello-and-piano ''Trio in Superpyth Temperament for Irish Whistle, Piano, and Cello''] (2015)
Both in 22edo tuning
: Both in 22edo tuning


; [[Joel Grant Taylor]]
; [[Joel Grant Taylor]]
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* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}
All in superpyth[12] in 22edo tuning.
: All in Superpyth[12] in 22edo tuning.


== See also ==
== See also ==
* [[Ultrapyth]]
* Alternative extensions:
** [[Quasisuper]]
** [[Ultrapyth]]
** [[Quasiultra]]


[[Category:Temperaments]]
[[Category:Temperaments]]

Revision as of 11:25, 29 December 2024

Superpyth is a temperament where ~3/2 is a generator, and the septimal comma (64/63) is tempered out, so that a stack of two perfect fifths octave-reduced gives a major whole tone that represents both 9/8 and 8/7 (likewise, two perfect fourths give a minor seventh that represents both 7/4 and 16/9, so intervals such as A–G and C–B♭ are harmonic sevenths). Since 3/2 is a generator we can use the same standard chain-of-fifths notation that is also used for meantone and 12edo, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in Pythagorean tuning, in contrast to meantone where sharps are flatter than or equal to the corresponding flats. 10\17, 13\22, and 16\27 are typical tunings of the generator.

Such a temperament without the 5th harmonic is also called archy. If intervals of 5 are desired, 5th harmonic 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 (including 12edo) has 3/2 tuned flat so that the 5th harmonic's intervals are simple and the 7th harmonic's intervals are complex, while superpyth has 3/2 tuned sharp so that the 7th harmonic's intervals are simple while the 5th harmonic's intervals are complex.

If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–D𝄪), tempering out 100/99. Yet a simpler but reasonable way is to map it to −6 generators, or 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 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out 31213/31104.

Mos scales of superpyth have cardinalities of 5, 7, 12, 17, or 22.

For more technical data, see Archytas clan #Superpyth.

Interval chains

In these tables, odd harmonics 1–11 are in bold.

Archy (2.3.7)
# Cents* Approximate ratios
0 0.0 1/1
1 709.6 3/2
2 219.2 8/7, 9/8
3 928.8 12/7
4 438.4 9/7
5 1148.0 27/14
6 657.6 72/49, 81/56
7 167.2 54/49

* In 2.3.7-subgroup CTE tuning

Supra (2.3.7.11)
# Cents* Approximate ratios
0 0.0 1/1
1 708.5 3/2
2 216.9 8/7, 9/8
3 925.4 12/7
4 433.8 9/7, 14/11
5 1142.3 21/11, 27/14, 64/33
6 650.7 16/11
7 159.2 12/11

* In 2.3.7.11-subgroup CTE tuning

Full 7-limit superpyth
# Cents* Approximate ratios
7-limit 11-limit extension
Superpyth Suprapyth
0 0.0 1/1
1 709.6 3/2
2 219.2 8/7, 9/8
3 928.8 12/7
4 438.4 9/7 14/11
5 1148.0 27/14, 35/18 88/45 21/11, 64/33
6 657.5 35/24, 40/27 22/15 16/11
7 167.1 10/9 11/10 12/11
8 876.7 5/3 33/20 18/11
9 386.3 5/4 27/22
10 1095.9 15/8, 40/21
11 605.5 10/7
12 115.1 15/14
13 824.7 45/28 44/27
14 334.3 60/49 11/9 40/33
15 1043.9 50/27 11/6 20/11
16 553.5 25/18 11/8 15/11
17 63.0 25/24 22/21, 33/32 45/44

* In 7-limit CTE tuning

Scales

5-note mos (2L 3s, proper)
  • Archy5 – archy in 472edo tuning
7-note mos (5L 2s, improper)
  • Archy7 – archy in 472edo tuning
  • Supra7 – supra in 56edo tuning

In contrast to the meantone diatonic scale, the superpyth diatonic is improper.

12-note mos (5L 7s, borderline improper)
  • Archy12 – archy in 472edo tuning
  • Supra12 – supra in 56edo tuning
  • 12-22a – superpyth in 22edo tuning

The boundary of propriety is 17edo.

Tunings

The plastic number has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming an octave period, constitutes a variety of superpyth. This can be explained since superpyth equates 21/16 and 4/3, making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping ~9 + ~12 = ~21 the generator becomes the plastic number.

Prime-optimized tunings

2.3.7 Subgroup Prime-Optimized Tunings
Euclidean
Unskewed Skewed
Equilateral CEE: ~3/2 = 712.8606¢
(2/5-comma) 		CSEE: ~3/2 = 711.9997¢
(7/19-comma)
Tenney CTE: ~3/2 = 709.5948¢ CWE: ~3/2 = 709.3901¢
Benedetti,
Wilson 		CBE: ~3/2 = 707.7286¢
(18/85-comma) 		CSBE: ~3/2 = 707.9869¢
(25/113-comma)
7-limit Prime-Optimized Tunings
Euclidean
Unskewed Skewed
Equilateral CEE: ~3/2 = 709.7805¢ CSEE: ~3/2 = 710.2428¢
Tenney CTE: ~3/2 = 709.5907¢ CWE: ~3/2 = 710.1193¢
Benedetti,
Wilson 		CBE: ~3/2 = 709.4859¢ CSBE: ~3/2 = 710.0321¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
3/2 701.955 Pythagorean tuning
10\17 705.882 Lower bound of 7- and 9-odd-limit diamond monotone
81/56 706.499 1/6 comma
27/14 707.408 1/5 comma
23\39 707.692 39cd val
9/7 708.771 1/4 comma, {1, 3, 7, 9} minimax
15/8 708.807
13\22 709.091
5/4 709.590 9-odd-limit minimax
49/27 709.745 2/7 comma
42\71 709.859 71d val
15/14 709.954
25/24 710.040
29\49 710.204
45\76 710.526 76bcd val
5/3 710.545
7/5 710.681 7-odd-limit minimax
7/6 711.043 1/3 comma, {1, 3, 7} minimax
16\27 711.111
21/20 711.553
9/5 711.772
19\32 712.500 32c val
55/32 712.544 Suprapyth mapping
49/48 712.861 2/5 comma, 2.3.7 subgroup CEE tuning
22\37 713.514 37cc val
25\42 714.286 42cc val
7/4 715.587 1/2 comma
3\5 720.000 Upper bound of 7- and 9-odd-limit diamond monotone
21/16 729.219 Full comma

* Besides the octave

Other tunings

  • DKW (2.3.5 superpyth): ~2 = 1200.000, ~3/2 = 709.758
  • DKW (2.3.7 archy): ~2 = 1200.000, ~3/2 = 712.585

Music

Lillian Hearne
Both in 22edo tuning
Joel Grant Taylor
All in Superpyth[12] in 22edo tuning.

See also

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