Superpyth: Difference between revisions

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~~<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'''".

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

=Superpyth=

== Temperament data ==

=== Superpyth ===

Commas: 64/63, 245/243

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Badness: 0.0323

==11-limit==

==== 11-limit ====

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

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Badness: 0.0250

==13-limit==

==== 13-limit ====

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

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Badness: 0.0247

=Suprapyth=

=== Suprapyth ===

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

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Badness: 0.0328

~~=Interval of superpyth=~~

== Interval chains ==

==Interval chains==

; Basic superpyth (2.3.7)

~~===~~Basic superpyth (2.3.7)~~===~~

{| class="wikitable"

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|}

~~===~~Full 7-limit superpyth~~===~~

; Full 7-limit superpyth

{| class="wikitable"

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|}

~~===~~Supra (2.3.7.11)~~===~~

; Supra (2.3.7.11)

{| class="wikitable"

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|}

~~===~~Full 11-limit suprapyth~~===~~

; Full 11-limit suprapyth

{| class="wikitable"

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|}

==~~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"

! Eigenmonzo

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|}

=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~~]]

@@ 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'''".
@@ Line 11: / Line 13: @@
 MOSes include 5, 7, 12, 17, and 22.
-=Superpyth=
+== Temperament data ==
+{{main| Archytas clan #Superpyth }}
+=== Superpyth ===
 Commas: 64/63, 245/243
@@ Line 24: / Line 29: @@
 Badness: 0.0323
-==11-limit==
+==== 11-limit ====
 Commas: 64/63, 100/99, 245/243
@@ Line 35: / Line 40: @@
 Badness: 0.0250
-==13-limit==
+==== 13-limit ====
 Commas: 64/63, 78/77, 91/90, 100/99
@@ Line 46: / Line 51: @@
 Badness: 0.0247
-=Suprapyth=
+=== Suprapyth ===
 Commas: 55/54, 64/63, 99/98
@@ Line 57: / Line 62: @@
 Badness: 0.0328
-=Interval of superpyth=
+== Interval chains ==
-==Interval chains==
+; Basic superpyth (2.3.7)
-===Basic superpyth (2.3.7)===
 {| class="wikitable"
@@ Line 88: / Line 92: @@
 |}
-===Full 7-limit superpyth===
+; Full 7-limit superpyth
 {| class="wikitable"
@@ Line 141: / Line 145: @@
 |}
-===Supra (2.3.7.11)===
+; Supra (2.3.7.11)
 {| class="wikitable"
@@ Line 182: / Line 186: @@
 |}
-===Full 11-limit suprapyth===
+; Full 11-limit suprapyth
 {| 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"
 ! 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]]