29edo: Difference between revisions

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}}

== Theory ==

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=== Prime harmonics ===

[[3/1|3]] is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which [[consistent]]ly represents the [[15-odd-limit]]. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: [[250/243]] in the [[5-limit]], [[49/48]] in the [[7-limit]], [[55/54]] in the [[11-limit]], and [[65/64]] in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament.

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! Cents

! Approx. Ratios of the [[13-limit]]

! colspan="3" | [[Ups and ~~Downs Notation~~]]

! colspan="3" | [[Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and ^d2)

! colspan="3" |[[SKULO interval names|SKULO interval names and notation]] (K or S = 1)

|-

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| downminor

| zo

| {a, b, 0, 1}

| [a, b, 0, 1>

| 7/6, 7/4

|-

| minor

| fourthward wa

| {a, b}, b < -1

| [a, b>, b < -1

| 32/27, 16/9

|-

| upminor

| gu

| {a, b, -1}

| [a, b, -1>

| 6/5, 9/5

|-

| "

| ilo

| {a, b, 0, 0, 1}

| [a, b, 0, 0, 1>

| 11/9, 11/6

|-

| downmajor

| lu

| {a, b, 0, 0, -1}

| [a, b, 0, 0, -1>

| 12/11, 18/11

|-

| "

| yo

| {a, b, 1}

| [a, b, 1>

| 5/4, 5/3

|-

| major

| fifthward wa

| {a, b}, b > 1

| [a, b>, b > 1

| 9/8, 27/16

|-

| upmajor

| ru

| {a, b, 0, -1}

| [a, b, 0, -1>

| 9/7, 12/7

|}

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| C upmajor or C up

|}

For a more complete list, see [[Ups and ~~Downs Notation~~ #Chords and Chord Progressions]].

For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].

== Notation ==

=== Standard notation===

=== Standard notation ===

29edo can be notated three different ways. Using only sharps and flats, the chromatic scale from C is:

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* C𝄪 = F𝄫

=== Ups and downs ===

=== Ups and downs notation ===

Since a sharp raises by three steps, 29edo is a good candidate for [[ups and downs notation]], similar to [[22edo]]. ~~Here~~, ~~sharps and flats with arrows from [[Helmholtz–Ellis notation]]~~ can be ~~used:~~

Since a sharp raises by three steps, 29edo is a good candidate for [[ups and downs notation]], similar to [[22edo]]. Spoken as up, downsharp, sharp, etc. Note that downsharp (v#) can be respelled as dup (^^).

Here, sharps and flats with arrows from [[Helmholtz–Ellis notation]] are used:

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If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.

===Sagittal notation===

=== Sagittal notation ===

This notation uses the same sagittal sequence as EDOs [[15edo#Sagittal notation|15]] and [[22edo#Sagittal notation|22]].

~~====Evo flavor====~~

==== Evo flavor ====

File:29-EDO_Evo_Sagittal.svg

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</imagemap>

====Revo flavor====

==== Revo flavor ====

File:29-EDO_Revo_Sagittal.svg

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== Approximation to JI ==

=== Interval mappings ===

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| {{monzo| 46 -29 }}

| {{mapping| 29 46 }}

| ~~−0~~.47

| −0.47

| 0.47

| 1.14

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| 2.3.5

| 250/243, 16875/16384

| [{{~~val~~| 29 46 67 }}]

| {{mapping| 29 46 67 }}

| +1.68

| 3.07

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=== Commas ===

~~29edo~~ [[tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 29 46 67 81 100 107 }}. Cent values are rounded to 5 digits.

29et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 29 46 67 81 100 107 }}. Cent values are rounded to 5 digits.

{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref>~~Ratios longer than 10 digits are presented by placeholders with informative hints~~</ref>

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

! [[Cents]]

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| Minor minthma

|}

~~<references />~~

=== Rank-2 temperaments ===

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* [[Leapfrog]] diatonic [[5L 2s]] 5552552 (17\29, 1\1)

* [[Leapfrog]] chromatic [[5L 7s]] 3232323223232322 (17\29, 1\1)

* [[Leapfrog]] hyperchromatic [[12L 5s]] 21221221221222122122122 (17\29, 1\1)

* [[Porcupine]] [[1L 6s]] 4444445 (4\29, 1\1)

* [[Porcupine]] [[7L 1s]] 44444441 (4\29, 1\1)

* [[Porcupine]] [[7L 8s]] 313131313131311 (4\29, 1\1)

* [[Porcupine]] [[7L 15s]] 2112112112112112112111 (4\29, 1\1)

* [[Negri]] [[1L 8s]] 333333335 (3\29, 1\1)

* [[Negri]] [[9L 1s]] 3333333332 (3\29, 1\1)

* [[Negri]] [[10L 9s]] 2212121212121212121 (3\29 1\1)

* [[Semaphore]] [[4L 1s]] 56666 (6\29, 1\1)

* [[Semaphore]] [[5L 4s]] 551515151 (6\29, 1\1)

* [[Semaphore]] [[5L 9s]] 41411411411411 (6\29, 1\1)

* [[Semaphore]] [[5L 14s]] 3113111311131113111 (6\29, 1\1)

* Pathological [[semaphore]] [[5L 19s]] 211121111211112111121111 (6\29, 1\1)

* [[Nautilus]] [[1L 13s]] 22222222222223 (2\29, 1\1)

* [[Nautilus]] [[14L 1s]] 222222222222221 (2\29, 1\1)

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: Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.

</div>

* [[Lumatone mapping for 29edo]]

== Music ==

=== Modern renderings ===

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* [https://www.youtube.com/watch?v=uGOK7WtVtlM "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

* [https://www.youtube.com/watch?v=jcZaU5PrhvU "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

* [https://www.youtube.com/watch?v=-Sa8IhljHM0 ''BACH - RICERCAR a 6 from the Musical Offering, tuned into 29-EDO'', BWV 1079] (1742-1749) - rendered by Claudi Meneghin (2025)

* [https://www.youtube.com/watch?v=856A7vTqIW8 ''Bach, Art of Fugue: Contrapunctus 11, tuned into 29-edo (harpischord)''] (1740-1746) - rendered by Claudi Meneghin (2025)

; {{W|Nicolaus Bruhns}}

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* [https://www.youtube.com/watch?v=HGQ2b6v0TWE ''Glass Animals - Life Itself''] (2023)

* [https://www.youtube.com/watch?v=ktk0VWbUbDg ''microtonal improvisation in 29edo''] (2023)

* [https://www.youtube.com/shorts/fyPaaW9AyMA ''Homestuck: Pipeorgankind (microtonal cover in 29edo)''] (2024)

* [https://www.youtube.com/shorts/SH5IQOi33Oo ''29edo groove''] (2025)

; [[duckapus]]

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* "Chill Bells" from ''Melancholie'' (2023) [https://open.spotify.com/track/30Ik57efXmIae1YSgTgdIE Spotify] | [https://francium223.bandcamp.com/track/chill-bells Bandcamp] | [https://www.youtube.com/watch?v=Qbz2a4PMnjY YouTube]

* from ''XenRhythms'' (2024)

** "All 29" – [https://francium223.bandcamp.com/track/all-29 Bandcamp] | [https://www.youtube.com/watch?v=C-FFLXHSO_k YouTube]

** "All 29" – [https://open.spotify.com/track/7I0fLgRPKdqCK51PUnt4Oe Spotify] | [https://francium223.bandcamp.com/track/all-29 Bandcamp] | [https://www.youtube.com/watch?v=C-FFLXHSO_k YouTube]

** "Do Not Immerse Yourself In Fire Or Water" – [https://francium223.bandcamp.com/track/do-not-immerse-yourself-in-fire-or-water Bandcamp] | [https://www.youtube.com/watch?v=NHDvNQbBHV0 YouTube] – ~~immunity~~[14] in 29edo tuning

** "Do Not Immerse Yourself In Fire Or Water" – [https://open.spotify.com/track/5FUmlRP1JjLsH99vXbJhXH Spotify] | [https://francium223.bandcamp.com/track/do-not-immerse-yourself-in-fire-or-water Bandcamp] | [https://www.youtube.com/watch?v=NHDvNQbBHV0 YouTube] – in Immunity[14], 29edo tuning

* [https://www.youtube.com/watch?v=di4qn2VFYbs ''Plane Sonatina No. 1''] (2025)

* [https://www.youtube.com/watch?v=ifvvww20XAU ''Strank Running''] (2025)

; [[Igliashon Jones]]

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== See also ==

* [[~~Arto and Tendo~~ Theory]]

* [[User:Unque/29edo Composition Theory|Unque's approach]]

* [[~~Lumatone mapping for 29edo~~]]

* [[Extraclassical tonality]]

* [[67ed5]] - octave-stretched 29edo, improves harmonics 5.7.11.13.17 but damages 2.3

* [[67ed5]] – octave-stretched 29edo, improves harmonics 5.7.11.13.17 but damages 2.3

== Notes ==

== References ==

[[Category:IMPORTDEBUG - Change External Images]]

[[Category:3-limit record edos|##]]

[[Category:3-limit]]

[[Category:Subgroup temperaments]]

[[Category:~~Listen~~]]

[[Category:Twentuning]]

[[Category:Negri]]

[[Category:Petrmic]]

[[Category:Porcupine]]

[[Category:~~Subgroup temperaments]]~~

[[Category:Listen]]

~~[[Category:Twentuning~~]]

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|29}}
+{{ED intro}}
 == Theory ==
@@ Line 24: / Line 24: @@
 === Prime harmonics ===
 [[3/1|3]] is the only harmonic, of the intelligibly low ones anyway, that 29edo approximates very closely, and it does so quite well. Nonetheless, and rather surprisingly, 29 is the smallest equal division which [[consistent]]ly represents the [[15-odd-limit]]. It is able to do this since it has an accurate 3, and the 5, 7, 11 and 13, while not very accurate, are all tuned flatly. Hence it tempers out a succession of fairly large commas: [[250/243]] in the [[5-limit]], [[49/48]] in the [[7-limit]], [[55/54]] in the [[11-limit]], and [[65/64]] in the [[13-limit]]. If using these approximations is desired, 29edo actually shines, and it can be used for such things as an alternative to [[19edo]] for [[negri]], as well as an alternative to [[22edo]] or [[15edo]] for [[porcupine]]. 29edo is also an [[oneirotonic]] tuning with generator 11\29, which generates [[ammonite]] temperament.
 {{Harmonics in equal|29|columns=11}}
@@ Line 45: / Line 44: @@
 ! Cents
 ! Approx. Ratios of the [[13-limit]]
-! colspan="3" | [[Ups and Downs Notation]]
+! colspan="3" | [[Ups and downs notation]]
+([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and ^d2)
 ! colspan="3" |[[SKULO interval names|SKULO interval names and notation]] (K or S = 1)
 |-
@@ Line 360: / Line 360: @@
 | downminor
 | zo
-| {a, b, 0, 1}
+| [a, b, 0, 1>
 | 7/6, 7/4
 |-
 | minor
 | fourthward wa
-| {a, b}, b &lt; -1
+| [a, b>, b &lt; -1
 | 32/27, 16/9
 |-
 | upminor
 | gu
-| {a, b, -1}
+| [a, b, -1>
 | 6/5, 9/5
 |-
 | "
 | ilo
-| {a, b, 0, 0, 1}
+| [a, b, 0, 0, 1>
 | 11/9, 11/6
 |-
 | downmajor
 | lu
-| {a, b, 0, 0, -1}
+| [a, b, 0, 0, -1>
 | 12/11, 18/11
 |-
 | "
 | yo
-| {a, b, 1}
+| [a, b, 1>
 | 5/4, 5/3
 |-
 | major
 | fifthward wa
-| {a, b}, b &gt; 1
+| [a, b>, b &gt; 1
 | 9/8, 27/16
 |-
 | upmajor
 | ru
-| {a, b, 0, -1}
+| [a, b, 0, -1>
 | 9/7, 12/7
 |}
@@ Line 437: / Line 437: @@
 | C upmajor or C up
 |}
-For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]].
+For a more complete list, see [[Ups and downs notation #Chords and Chord Progressions]].
 == Notation ==
-=== Standard notation===
+=== Standard notation ===
 edo can be notated three different ways. Using only sharps and flats, the chromatic scale from C is:
@@ Line 453: / Line 453: @@
 * C𝄪 = F𝄫
-=== Ups and downs ===
+=== Ups and downs notation ===
-Since a sharp raises by three steps, 29edo is a good candidate for [[ups and downs notation]], similar to [[22edo]]. Here, sharps and flats with arrows from [[Helmholtz–Ellis notation]] can be used:
+Since a sharp raises by three steps, 29edo is a good candidate for [[ups and downs notation]], similar to [[22edo]]. Spoken as up, downsharp, sharp, etc. Note that downsharp (v#) can be respelled as dup (^^).
+{{Sharpness-sharp3a}}
+Here, sharps and flats with arrows from [[Helmholtz–Ellis notation]] are used:
 {{Sharpness-sharp3}}
@@ Line 462: / Line 464: @@
 If arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best spelled with double arrows.
-===Sagittal notation===
+=== Sagittal notation ===
 This notation uses the same sagittal sequence as EDOs [[15edo#Sagittal notation|15]] and [[22edo#Sagittal notation|22]].
-====Evo flavor====
+==== Evo flavor ====
 <imagemap>
 File:29-EDO_Evo_Sagittal.svg
@@ Line 475: / Line 477: @@
 </imagemap>
-====Revo flavor====
+==== Revo flavor ====
 <imagemap>
 File:29-EDO_Revo_Sagittal.svg
@@ Line 488: / Line 489: @@
 == Approximation to JI ==
 [[File:29ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|15-odd-limit intervals approximated in 29edo]]
 === Interval mappings ===
 {{Q-odd-limit intervals|29}}
@@ Line 506: / Line 508: @@
 | {{monzo| 46 -29 }}
 | {{mapping| 29 46 }}
-| &minus;0.47
+| −0.47
 | 0.47
 | 1.14
@@ Line 512: / Line 514: @@
 | 2.3.5
 | 250/243, 16875/16384
-| [{{val| 29 46 67 }}]
+| {{mapping| 29 46 67 }}
 | +1.68
 | 3.07
@@ Line 556: / Line 558: @@
 === Commas ===
-edo [[tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 29 46 67 81 100 107 }}. Cent values are rounded to 5 digits.
+et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 29 46 67 81 100 107 }}. Cent values are rounded to 5 digits.
 {| class="commatable wikitable center-all left-3 right-4 left-6"
 |-
 ! [[Harmonic limit|Prime<br>limit]]
-! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
+! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]
 ! [[Cents]]
@@ Line 756: / Line 758: @@
 | Minor minthma
 |}
-<references />
 === Rank-2 temperaments ===
@@ Line 880: / Line 881: @@
 * [[Leapfrog]] diatonic [[5L 2s]] 5552552 (17\29, 1\1)
 * [[Leapfrog]] chromatic [[5L 7s]] 3232323223232322 (17\29, 1\1)
-* [[Leapfrog]] hyperchromatic [[12L 5s]] 21221221221222122122122 (17\29, 1\1)
 * [[Porcupine]] [[1L 6s]] 4444445 (4\29, 1\1)
 * [[Porcupine]] [[7L 1s]] 44444441 (4\29, 1\1)
-* [[Porcupine]] [[7L 8s]] 313131313131311 (4\29, 1\1)
-* [[Porcupine]] [[7L 15s]] 2112112112112112112111 (4\29, 1\1)
 * [[Negri]] [[1L 8s]] 333333335 (3\29, 1\1)
 * [[Negri]] [[9L 1s]] 3333333332 (3\29, 1\1)
-* [[Negri]] [[10L 9s]] 2212121212121212121 (3\29 1\1)
 * [[Semaphore]] [[4L 1s]] 56666 (6\29, 1\1)
 * [[Semaphore]] [[5L 4s]] 551515151 (6\29, 1\1)
 * [[Semaphore]] [[5L 9s]] 41411411411411 (6\29, 1\1)
-* [[Semaphore]] [[5L 14s]] 3113111311131113111 (6\29, 1\1)
-* Pathological [[semaphore]] [[5L 19s]] 211121111211112111121111 (6\29, 1\1)
 * [[Nautilus]] [[1L 13s]] 22222222222223 (2\29, 1\1)
 * [[Nautilus]] [[14L 1s]] 222222222222221 (2\29, 1\1)
@@ Line 963: / Line 958: @@
 : <small>Furthermore, since external images can break, we recommend that you replace the above with a local copy of the image.</small>
 </div>
+* [[Lumatone mapping for 29edo]]
 == Music ==
 === Modern renderings ===
@@ Line 969: / Line 964: @@
 * [https://www.youtube.com/watch?v=uGOK7WtVtlM "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
 * [https://www.youtube.com/watch?v=jcZaU5PrhvU "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
+* [https://www.youtube.com/watch?v=-Sa8IhljHM0 ''BACH - RICERCAR a 6 from the Musical Offering, tuned into 29-EDO'', BWV 1079] (1742-1749) - rendered by Claudi Meneghin (2025)
+* [https://www.youtube.com/watch?v=856A7vTqIW8 ''Bach, Art of Fugue: Contrapunctus 11, tuned into 29-edo (harpischord)''] (1740-1746) - rendered by Claudi Meneghin (2025)
 ; {{W|Nicolaus Bruhns}}
@@ Line 988: / Line 985: @@
 * [https://www.youtube.com/watch?v=HGQ2b6v0TWE ''Glass Animals - Life Itself''] (2023)
 * [https://www.youtube.com/watch?v=ktk0VWbUbDg ''microtonal improvisation in 29edo''] (2023)
+* [https://www.youtube.com/shorts/fyPaaW9AyMA ''Homestuck: Pipeorgankind (microtonal cover in 29edo)''] (2024)
+* [https://www.youtube.com/shorts/SH5IQOi33Oo ''29edo groove''] (2025)
 ; [[duckapus]]
@@ Line 1,001: / Line 1,000: @@
 * "Chill Bells" from ''Melancholie'' (2023) [https://open.spotify.com/track/30Ik57efXmIae1YSgTgdIE Spotify] | [https://francium223.bandcamp.com/track/chill-bells Bandcamp] | [https://www.youtube.com/watch?v=Qbz2a4PMnjY YouTube]
 * from ''XenRhythms'' (2024)
-** "All 29" – [https://francium223.bandcamp.com/track/all-29 Bandcamp] | [https://www.youtube.com/watch?v=C-FFLXHSO_k YouTube]
+** "All 29" – [https://open.spotify.com/track/7I0fLgRPKdqCK51PUnt4Oe Spotify] | [https://francium223.bandcamp.com/track/all-29 Bandcamp] | [https://www.youtube.com/watch?v=C-FFLXHSO_k YouTube]
-** "Do Not Immerse Yourself In Fire Or Water" – [https://francium223.bandcamp.com/track/do-not-immerse-yourself-in-fire-or-water Bandcamp] | [https://www.youtube.com/watch?v=NHDvNQbBHV0 YouTube] – immunity[14] in 29edo tuning
+** "Do Not Immerse Yourself In Fire Or Water" – [https://open.spotify.com/track/5FUmlRP1JjLsH99vXbJhXH Spotify] | [https://francium223.bandcamp.com/track/do-not-immerse-yourself-in-fire-or-water Bandcamp] | [https://www.youtube.com/watch?v=NHDvNQbBHV0 YouTube] – in Immunity[14], 29edo tuning
+* [https://www.youtube.com/watch?v=di4qn2VFYbs ''Plane Sonatina No. 1''] (2025)
+* [https://www.youtube.com/watch?v=ifvvww20XAU ''Strank Running''] (2025)
 ; [[Igliashon Jones]]
@@ Line 1,032: / Line 1,033: @@
 == See also ==
-* [[Arto and Tendo Theory]]
+* [[User:Unque/29edo Composition Theory|Unque's approach]]
-* [[Lumatone mapping for 29edo]]
+* [[Extraclassical tonality]]
-* [[67ed5]] - octave-stretched 29edo, improves harmonics 5.7.11.13.17 but damages 2.3
+* [[67ed5]] – octave-stretched 29edo, improves harmonics 5.7.11.13.17 but damages 2.3
+== Notes ==
+<references group="note" />
+== References ==
+<references />
 [[Category:IMPORTDEBUG - Change External Images]]
+[[Category:3-limit record edos|##]] <!-- 2-digit number -->
-[[Category:3-limit]]
+[[Category:Subgroup temperaments]]
-[[Category:Listen]]
+[[Category:Twentuning]]
 [[Category:Negri]]
 [[Category:Petrmic]]
 [[Category:Porcupine]]
-[[Category:Subgroup temperaments]]
+[[Category:Listen]]
-[[Category:Twentuning]]