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]], with the understanding that, for example, A# is sharper than Bb (in contrast to meantone where A# is flatter than Bb, 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.
'''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 Bb (in contrast to meantone where A# is flatter than Bb, 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, 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 the 5th harmonic is used at all, it is mapped to -9 generators, 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.
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== Temperament data ==
== Temperament data ==
{{main| Archytas clan #Superpyth }}
{{main| Archytas clan #Superpyth }}
=== 7-limit superpyth ===
Subgroup: 2.3.5.7
Comma list: 64/63, 245/243
Mapping: [{{val| 1 0 -12 6 }}, {{val| 0 1 9 -2 }}]
Wedgie: {{wedgie| 1 9 -2 12 -6 -30 }}
{{Val list|legend=1| 5, 17, 22, 27, 49 }}
Badness: 0.0323
=== 11-limit superpyth ===
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 245/243
Mapping: [{{val| 1 0 -12 6 -22 }}, {{val| 0 1 9 -2 16 }}]
{{Val list|legend=1| 22, 27e, 49 }}
Badness: 0.0250
=== 13-limit superpyth ===
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 91/90, 100/99
Mapping: [{{val| 1 0 -12 6 -22 -17 }}, {{val| 0 1 9 -2 16 13 }}]
{{Val list|legend=1| 22, 27e, 49, 76bcde }}
Badness: 0.0247
=== 11-limit suprapyth ===
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 99/98
Mapping: [{{val| 1 0 -12 6 13 }}, {{val| 0 1 9 -2 -6 }}]
{{Val list|legend=1| 17, 22 }}
Badness: 0.0328
=== 13-limit suprapyth ===
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 64/63, 65/63, 99/98
Mapping: [{{val| 1 0 -12 6 13 18 }}, {{val| 0 1 9 -2 -6 -9 }}]
{{Val list|legend=1| 17, 22, 83cdf }}
Badness: 0.0363


== Interval chains ==
== Interval chains ==
; Archy (2.3.7)
; Archy (2.3.7)


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
| | 1146.61
| 1146.61
| | 437.29
| 437.29
| | 927.97
| 927.97
| | 218.64
| 218.64
| | 709.32
| 709.32
| | 0
| 0
| | 490.68
| 490.68
| | 981.36
| 981.36
| | 272.03
| 272.03
| | 762.71
| 762.71
| | 53.39
| 53.39
|-
|-
| | 27/14
| 27/14
| | 9/7
| 9/7
| | 12/7
| 12/7
| | 9/8~8/7
| 9/8~8/7
| | 3/2
| 3/2
| | 1/1
| 1/1
| | 4/3
| 4/3
| | 7/4~16/9
| 7/4~16/9
| | 7/6
| 7/6
| | 14/9
| 14/9
| | 28/27
| 28/27
|}
|}


; Full 7-limit superpyth
; Full 7-limit superpyth


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
| | 613.20
| 613.20
| | 1102.91
| 1102.91
| | 392.62
| 392.62
| | 882.33
| 882.33
| | 172.04
| 172.04
| | 661.75
| 661.75
| | 1151.46
| 1151.46
| | 441.16
| 441.16
| | 930.87
| 930.87
| | 220.58
| 220.58
| | 710.29
| 710.29
| | 0
| 0
| | 489.71
| 489.71
| | 979.42
| 979.42
| | 269.13
| 269.13
| | 758.84
| 758.84
| | 48.54
| 48.54
| | 538.25
| 538.25
| | 1027.96
| 1027.96
| | 317.67
| 317.67
| | 807.38
| 807.38
| | 97.09
| 97.09
| | 586.80
| 586.80
|-
|-
| | 10/7
| 10/7
| | 15/8
| 15/8
| | 5/4
| 5/4
| | 5/3
| 5/3
| | 10/9
| 10/9
| |  
|  
| | 27/14
| 27/14
| | 9/7
| 9/7
| | 12/7
| 12/7
| | 9/8~8/7
| 9/8~8/7
| | 3/2
| 3/2
| | 1/1
| 1/1
| | 4/3
| 4/3
| | 7/4~16/9
| 7/4~16/9
| | 7/6
| 7/6
| | 14/9
| 14/9
| | 28/27
| 28/27
| |  
|  
| | 9/5
| 9/5
| | 6/5
| 6/5
| | 8/5
| 8/5
| | 16/15
| 16/15
| | 7/5
| 7/5
|}
|}


; Supra (2.3.7.11)
; Supra (2.3.7.11)


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
| | 857.54
| 857.54
| | 150.35
| 150.35
| | 643.15
| 643.15
| | 1135.96
| 1135.96
| | 428.77
| 428.77
| | 921.58
| 921.58
| | 214.38
| 214.38
| | 707.19
| 707.19
| | 0
| 0
| | 492.81
| 492.81
| | 985.62
| 985.62
| | 278.42
| 278.42
| | 771.23
| 771.23
| | 64.04
| 64.04
| | 556.85
| 556.85
| | 1049.65
| 1049.65
| | 342.46
| 342.46
|-
|-
| | 18/11
| 18/11
| | 12/11
| 12/11
| | 16/11
| 16/11
| | 27/14
| 27/14
| | 14/11~9/7
| 14/11~9/7
| | 12/7
| 12/7
| | 9/8~8/7
| 9/8~8/7
| | 3/2
| 3/2
| | 1/1
| 1/1
| | 4/3
| 4/3
| | 7/4~16/9
| 7/4~16/9
| | 7/6
| 7/6
| | 14/9~11/7
| 14/9~11/7
| | 33/32~28/27
| 33/32~28/27
| | 11/8
| 11/8
| | 11/6
| 11/6
| | 11/9
| 11/9
|}
|}


; Full 11-limit suprapyth
; Full 11-limit suprapyth


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
| | 604.44
| 604.44
| | 1094.94
| 1094.94
| | 385.45
| 385.45
| | 875.96
| 875.96
| | 166.46
| 166.46
| | 656.97
| 656.97
| | 1147.47
| 1147.47
| | 437.98
| 437.98
| | 928.48
| 928.48
| | 218.99
| 218.99
| | 709.49
| 709.49
| | 0
| 0
| | 490.51
| 490.51
| | 981.01
| 981.01
| | 271.52
| 271.52
| | 762.02
| 762.02
| | 52.53
| 52.53
| | 543.03
| 543.03
| | 1033.54
| 1033.54
| | 324.04
| 324.04
| | 814.55
| 814.55
| | 105.06
| 105.06
| | 595.56
| 595.56
|-
|-
| | 10/7
| 10/7
| | 15/8
| 15/8
| | 5/4
| 5/4
| | 18/11~5/3
| 18/11~5/3
| | 12/11~10/9
| 12/11~10/9
| | 16/11
| 16/11
| | 27/14
| 27/14
| | 14/11~9/7
| 14/11~9/7
| | 12/7
| 12/7
| | 9/8~8/7
| 9/8~8/7
| | 3/2
| 3/2
| | 1/1
| 1/1
| | 4/3
| 4/3
| | 7/4~16/9
| 7/4~16/9
| | 7/6
| 7/6
| | 14/9~11/7
| 14/9~11/7
| | 33/32~28/27
| 33/32~28/27
| | 11/8
| 11/8
| | 9/5~11/6
| 9/5~11/6
| | 6/5~11/9
| 6/5~11/9
| | 8/5
| 8/5
| | 16/15
| 16/15
| | 7/5
| 7/5
|}
|}


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! Eigenmonzo
! Eigenmonzo
! Generator
! Generator
! Comment
! Comments
|-
|-
| 4/3
| 4/3

Revision as of 08:41, 26 May 2021

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 Bb (in contrast to meantone where A# is flatter than Bb, 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.

If the 5th harmonic is used at all, it is mapped to -9 generators, 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 simplest reasonable way is to map 11/8 to 6 generators (so 11/8 is a "diminished fifth"), by tempering out 99/98. This temperament is called "supra", or "suprapyth".

MOSes include 5, 7, 12, 17, and 22.

Temperament data

Main article: Archytas clan § Superpyth

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

MOS scales

5-note (LsLss, proper)
See 2L 3s.
7-note (LLLsLLs, improper)
See 5L 2s. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.
12-note (LsLsLssLsLss, borderline improper)
See 5L 7s. The boundary of propriety is 17edo.

Spectrum of superpyth tunings

Eigenmonzo Generator Comments
4/3 701.955
(10\17) 705.882
28/27 707.408 1/5 comma
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

Music

By Joel Grant Taylor, all in Superpyth[12] in 22edo tuning.

Both by Lillian Hearne in 22edo tuning

Retrieved from "https://en.xen.wiki/index.php?title=Superpyth&oldid=70752"