Superpyth: Difference between revisions

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| ja =

}}

'''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.

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== Interval chains ==

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

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

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|+ style="font-size: 105%;" | Archy (2.3.7)

|-

! &#~~35;~~ !! Cents* !! Approximate ~~Ratios~~

! # !! Cents* !! Approximate ratios

|-

| 0 || 0.0 || '''1/1'''

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| 7 || 167.2 || 54/49

|}

<nowiki />* In 2.3.7-subgroup [[CTE]] tuning

<nowiki/>* In 2.3.7-subgroup [[CTE]] tuning

</div>

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|+ style="font-size: 105%;" | Supra (2.3.7.11)

|-

! &#~~35;~~ !! Cents* !! Approximate ~~Ratios~~

! # !! Cents* !! Approximate ratios

|-

| 0 || 0.0 || '''1/1'''

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| 7 || 159.2 || 12/11

|}

<nowiki />* In 2.3.7.11-subgroup CTE tuning

<nowiki/>* In 2.3.7.11-subgroup CTE tuning

</div>

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|+ 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

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| 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>

== Scales ==

; 5-note ~~MOS~~ ([[2L 3s]], proper)

; 5-note mos ([[2L 3s]], proper)

* [[Archy5]] ~~–~~ archy in 472edo tuning

* [[Archy5]] – archy in 472edo tuning

; 7-note ~~MOS~~ ([[5L 2s]], improper)

; 7-note mos ([[5L 2s]], improper)

* [[Archy7]] ~~–~~ archy in 472edo tuning

* [[Archy7]] – archy in 472edo tuning

* [[Supra7]] ~~–~~ supra in 56edo 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)

; 12-note mos ([[5L 7s]], borderline improper)

* [[Archy12]] ~~–~~ archy in 472edo tuning

* [[Archy12]] – archy in 472edo tuning

* [[Supra12]] ~~–~~ supra in 56edo tuning

* [[Supra12]] – supra in 56edo tuning

* [[12-22a]] ~~–~~ superpyth in 22edo tuning

* [[12-22a]] – superpyth in 22edo tuning

The boundary of propriety is [[17edo]].

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|+ 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

|-

! ~~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"

|-

! Edo<br />~~Generator~~

! Edo<br>generator

! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*

! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*

! Generator (¢)

! Comments

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| Full comma

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

=== 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 ==

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* [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)

Both in 22edo tuning

: Both in 22edo tuning

; [[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/12of22study7.mp3 12of22study7]{{dead link}}

All in ~~superpyth~~[12] in 22edo tuning.

: All in Superpyth[12] in 22edo tuning.

== See also ==

* [[Ultrapyth]]

* Alternative extensions:

** [[Quasisuper]]

** [[Ultrapyth]]

** [[Quasiultra]]

[[Category:Temperaments]]

@@ Line 5: / Line 5: @@
 | ja =
 }}
-'''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.
@@ Line 18: / Line 18: @@
 == Interval chains ==
-In these tables, odd harmonics 1&ndash;11 are in '''bold'''.
+In these tables, odd harmonics 1–11 are in '''bold'''.
 <div><div style="display: inline-grid; margin-right: 25px;">
@@ Line 24: / Line 24: @@
 |+ style="font-size: 105%;" | Archy (2.3.7)
 |-
-! &#35; !! Cents* !! Approximate Ratios
+! # !! Cents* !! Approximate ratios
 |-
 | 0 || 0.0 || '''1/1'''
@@ Line 42: / Line 42: @@
 | 7 || 167.2 || 54/49
 |}
-<nowiki />* In 2.3.7-subgroup [[CTE]] tuning
+<nowiki/>* In 2.3.7-subgroup [[CTE]] tuning
 </div>
 <div style="display: inline-grid; margin-right: 25px;">
@@ Line 48: / Line 48: @@
 |+ style="font-size: 105%;" | Supra (2.3.7.11)
 |-
-! &#35; !! Cents* !! Approximate Ratios
+! # !! Cents* !! Approximate ratios
 |-
 | 0 || 0.0 || '''1/1'''
@@ Line 66: / Line 66: @@
 | 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 style="display: inline-grid;">
@@ Line 72: / Line 72: @@
 |+ 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
@@ Line 114: / Line 114: @@
 | 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>
 == 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.
-; 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]].
@@ Line 139: / Line 139: @@
 |+ 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)
 |}
@@ Line 157: / Line 161: @@
 |+ 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¢
 |}
@@ Line 175: / Line 183: @@
 {| class="wikitable center-all left-4"
 |-
-! Edo<br />Generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
 ! Generator (¢)
 ! Comments
@@ Line 325: / Line 333: @@
 | Full comma
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
 === 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 ==
@@ Line 335: / Line 343: @@
 * [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)
-Both in 22edo tuning
+: Both in 22edo tuning
 ; [[Joel Grant Taylor]]
@@ Line 342: / Line 350: @@
 * [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}}
-All in superpyth[12] in 22edo tuning.
+: All in Superpyth[12] in 22edo tuning.
 == See also ==
-* [[Ultrapyth]]
+* Alternative extensions:
+** [[Quasisuper]]
+** [[Ultrapyth]]
+** [[Quasiultra]]
 [[Category:Temperaments]]

Superpyth: Difference between revisions

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