Superpyth: Difference between revisions

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m FloraC moved page Archy and superpyth to Superpyth over redirect: Unhighlight the name *archy* since *superpyth* is superior and virtually canonical
Restore old layout and unhighlight archy since superpyth is virtually canonical.
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{{Infobox regtemp|Comma basis=[[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (2.3.5.7)|Edo join 1=5|Edo join 2=22|Generator=3/2|Generator tuning=712.6|Optimization method=DKW|Mapping=1; 1 9 -2
{{Infobox regtemp
| Title = Archy; superpyth
| Subgroups = 2.3.7, 2.3.5.7
| Comma basis = [[64/63]] (2.3.7); <br> [[64/63]], [[245/243]] (2.3.5.7)
| Mapping = 1; 1 9 -2
| Edo join 1 = 5 | Edo join 2 = 22
| Generator = 3/2
| Generator tuning = 712.6
| Optimization method = DKW
| Pergen = (P8, P5)
| Pergen = (P8, P5)
| Color name = Ruti
| Color name = Ruti
|MOS scales=[[2L 3s]], [[5L 2s]], [[5L 7s]]|Subgroups=2.3.7, 2.3.5.7|Title=Archy; superpyth|Odd limit 1=(2.3.7) 9|Mistuning 1=?|Odd limit 2=9|Mistuning 2=?|Complexity 1=12|Complexity 2=27}}'''Archy''' is a [[regular temperament|temperament]] where the [[generator]] is [[4/3]], tuned flat so that stacking two of them gives the interval [[7/4]]. This means that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, two [[3/2]] perfect fifths [[octave reduction|octave-reduced]] gives [[8/7]]. Three fourths reach a subminor third that approximates [[7/6]], while four fifths reach a supermajor third that approximates [[9/7]]. This means that the septimal comma ([[64/63]]) is [[tempering out|tempered out]].
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]]
| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = ? | Complexity 1 = 12
| Odd limit 2 = 9 | Mistuning 2 = ? | Complexity 2 = 27
}}
'''Superpyth''' is a [[regular temperament|temperament]] where the [[generator]] is a [[3/2|perfect fifth]], tuned sharp such that a stack of two perfect fifths [[octave reduction|octave-reduced]] gives a whole tone that represents both [[9/8]] and [[8/7]], [[tempering out]] the septimal comma, [[64/63]]. Likewise, two perfect fourths give a minor seventh that represents both [[7/4]] and [[16/9]], so that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, three fourths reach a minor third that approximates [[7/6]], while four fifths reach a major third that approximates [[9/7]].  


Since the generator is a perfect fourth or perfect fifth, archy can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], 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. [[22edo|13\22]] and [[27edo|16\27]] are typical tunings of the generator.
Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], 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. [[22edo|13\22]] and [[27edo|16\27]] are typical tunings of the generator.


If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ (an augmented second or limma-flat major third) is 5/4, leading to an extension often called '''superpyth'''. Superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.
Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so 5/4 is an augmented second (e.g. C–D♯). Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp 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, or a doubly augmented second (C–Dx), 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 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]].
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]].
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[[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.


For more technical data, see [[Archytas clan #Superpyth|Archytas clan.]]
For more technical data, see [[Archytas clan #Superpyth]].


== Interval chains ==
== Interval chains ==

Revision as of 13:01, 26 April 2025

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). Superpyth is a temperament where the generator is a perfect fifth, tuned sharp such that a stack of two perfect fifths octave-reduced gives a whole tone that represents both 9/8 and 8/7, tempering out the septimal comma, 64/63. Likewise, two perfect fourths give a minor seventh that represents both 7/4 and 16/9, so that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, three fourths reach a minor third that approximates 7/6, while four fifths reach a major third that approximates 9/7.

Since the generator is a perfect fifth, superpyth can be notated using the same standard chain-of-fifths notation that is also used for meantone, 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. 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, the 5th harmonic is mapped to +9 generators through tempering out 245/243, so 5/4 is an augmented second (e.g. C–D♯). Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including 12edo) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp 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, 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], 22edo tuning.

See also

Retrieved from "https://en.xen.wiki/index.php?title=Superpyth&oldid=194090"