Superpyth: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
+tuning spectrum
m General cleanup
Line 1: Line 1:
<span style="display: block; text-align: right;">[[:de:Superpyth|Deutsch]]</span>
{{interwiki
| de = Superpyth
| en = Superpyth
| es =
| ja =  
}}
'''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.


__FORCETOC__
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.
-----
'''Superpyth''', a member of the [[Archytas_clan|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|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 [http://en.wikipedia.org/wiki/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 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'''".  
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'''".  
Line 11: Line 13:
MOSes include 5, 7, 12, 17, and 22.
MOSes include 5, 7, 12, 17, and 22.


=Superpyth=
== Temperament data ==
{{main| Archytas clan #Superpyth }}
 
=== Superpyth ===
Commas: 64/63, 245/243
Commas: 64/63, 245/243


Line 24: Line 29:
Badness: 0.0323
Badness: 0.0323


==11-limit==
==== 11-limit ====
Commas: 64/63, 100/99, 245/243
Commas: 64/63, 100/99, 245/243


Line 35: Line 40:
Badness: 0.0250
Badness: 0.0250


==13-limit==
==== 13-limit ====
Commas: 64/63, 78/77, 91/90, 100/99
Commas: 64/63, 78/77, 91/90, 100/99


Line 46: Line 51:
Badness: 0.0247
Badness: 0.0247


=Suprapyth=
=== Suprapyth ===
Commas: 55/54, 64/63, 99/98
Commas: 55/54, 64/63, 99/98


Line 57: Line 62:
Badness: 0.0328
Badness: 0.0328


=Interval of superpyth=
== Interval chains ==
==Interval chains==
; Basic superpyth (2.3.7)
===Basic superpyth (2.3.7)===


{| class="wikitable"
{| class="wikitable"
Line 88: Line 92:
|}
|}


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


{| class="wikitable"
{| class="wikitable"
Line 141: Line 145:
|}
|}


===Supra (2.3.7.11)===
; Supra (2.3.7.11)


{| class="wikitable"
{| class="wikitable"
Line 182: Line 186:
|}
|}


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


{| class="wikitable"
{| class="wikitable"
Line 235: Line 239:
|}
|}


==MOSes==
== MOS scales ==
 
; 5-note (LsLss, proper)
===5-note (LsLss, proper)===
: See [[2L 3s]].
See [[2L_3s|2L 3s]].


===7-note (LLLsLLs, improper)===
; 7-note (LLLsLLs, improper)
See [[5L_2s|5L 2s]]. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.
: See [[5L 2s]]. In contrast to the meantone diatonic scale, the superpyth diatonic is slightly improper.


===12-note (LsLsLssLsLss, borderline improper)===
; 12-note (LsLsLssLsLss, borderline improper)
See [[5L_7s|5L 7s]]. The boundary of propriety is [[17edo]].
: See [[5L 7s]]. The boundary of propriety is [[17edo]].


= Spectrum of superpyth tunings =
== Spectrum of superpyth tunings ==
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-3"
! Eigenmonzo
! Eigenmonzo
Line 321: Line 324:
|}
|}


=Music=
== Music ==
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]
 
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]
* [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/12of22study3.mp3 12of22study3 (children's story)]
By [[Joel Grant Taylor]], all in Superpyth[12] in 22edo tuning.
 
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]


By [[Joel_Grant_Taylor|Joel Grant Taylor]], all in Superpyth[12] in 22edo tuning.
[[Category:Temperament]]
[[Category:archytas]]
[[Category:Superpyth]]
[[Category:todo:add_definition]]
[[Category:Archytas]]
[[Category:todo:intro]]

Revision as of 10:52, 13 February 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

Superpyth

Commas: 64/63, 245/243

POTE generator: ~3/2 = 710.291

Map: [<1 0 -12 6|, <0 1 9 -2|]

Wedgie: <<1 9 -2 12 -6 -30||

EDOs: 5, 17, 22, 27, 49

Badness: 0.0323

11-limit

Commas: 64/63, 100/99, 245/243

POTE generator: ~3/2 = 710.175

Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]

EDOs: 22, 27e, 49

Badness: 0.0250

13-limit

Commas: 64/63, 78/77, 91/90, 100/99

POTE generator: ~3/2 = 710.479

Map: [<1 0 -12 6 -22 -17|, <0 1 9 -2 16 13|]

EDOs: 22, 27e, 49, 76bcde

Badness: 0.0247

Suprapyth

Commas: 55/54, 64/63, 99/98

POTE generator: ~3/2 = 709.495

Map: [<1 0 -12 6 13|, <0 1 9 -2 -6|]

EDOs: 5, 17, 22

Badness: 0.0328

Interval chains

Basic superpyth (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 Comment
4/3 701.955
(10\17) 705.882
28/27 707.408 1/5 comma
9/7 708.771 1/4 comma, 2.3.7.9 minimax
16/15 708.807
(13\22) 709.091
5/4 709.590 9-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-limit minimax
7/6 711.043 1/3 comma, 2.3.7 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.

Retrieved from "https://en.xen.wiki/index.php?title=Superpyth&oldid=61623"