29edo: Difference between revisions

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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. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~

[[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.

=== Stacking fifths ===

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it. ~~(''See [[regular temperament]] for more about what all this means and how to use it.'')~~

Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it.

Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].

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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]]: v<sup>3</sup>A1 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 ==

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=== Interval mappings ===

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 116~~

~~| steps = 28.9399661541990~~

~~| step size = 41.4651487014917~~

~~| tempered height = 5.566209~~

~~| pure height = 4.200785~~

~~| integral = 1.000619~~

~~| gap = 14.904418~~

~~| octave = 1202.48931234326~~

~~| consistent = 8~~

~~| distinct = 7~~

}}

== Regular temperament properties ==

~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

{| class="wikitable center-4 center-5 center-6"

|-

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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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: <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 ===

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

* [[Extraclassical tonality]]

* [[Lumatone mapping for 29edo]]

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

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[[Category:IMPORTDEBUG - Change External Images]]

[[Category:3-limit]]

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

[[Category:~~Listen~~]]

[[Category:Subgroup temperaments]]

[[Category:Twentuning]]

[[Category:Negri]]

[[Category:Petrmic]]

[[Category:Porcupine]]

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

[[Category:Listen]]

~~[[Category:Twentuning~~]]

@@ Line 23: / Line 23: @@
 === 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. (''See [[regular temperament]] for more about what all this means and how to use it.'')
+[[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}}
 === Stacking fifths ===
-Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it. (''See [[regular temperament]] for more about what all this means and how to use it.'')
+Another possible use for 29edo is as an equally tempered para-pythagorean scale. Using its fifth as a generator leads to a variant of [[garibaldi temperament]] which is not very accurate but which has relatively low 13-limit complexity. However, it gives the POL2 generator for [[Subgroup temperaments #Edson (2.3.7/5.11/5.13/5 subgroup)|edson temperament]] with essentially perfect accuracy, only 0.034 cents sharp of it.
 Edson is a 2.3.7/5.11/5.13/5 subgroup temperament, and 29 it represents the 2.3.11/5.13/5 subgroup to very high accuracy, and the 2.3.7/5.11/5.13/5 to a lesser but still good accuracy, and so can be used with this subgroup, which is liberally supplied with chords such as the 1-11/7-13/7 (7:11:13) chord, the [[The Archipelago|barbados triad]] 1-13/10-3/2 (10:13:15), the minor barbados triad 1-15/13-3/2, the 1-14/11-3/2 (22:28:33) triad, the 1-13/11-3/2 triad (22:26:33), and the [[petrmic triad]], a 13-limit [[Dyadic chord|essentially tempered dyadic chord]].
@@ Line 44: / 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 359: / 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 436: / 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 ==
@@ Line 491: / Line 492: @@
 === Interval mappings ===
 {{Q-odd-limit intervals|29}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 116
-| steps = 28.9399661541990
-| step size = 41.4651487014917
-| tempered height = 5.566209
-| pure height = 4.200785
-| integral = 1.000619
-| gap = 14.904418
-| octave = 1202.48931234326
-| consistent = 8
-| distinct = 7
-}}
 == Regular temperament properties ==
-''See [[regular temperament]] for more about what all this means and how to use it.''
 {| class="wikitable center-4 center-5 center-6"
 |-
@@ Line 896: / 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 979: / 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 985: / 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 1,004: / 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 1,020: / Line 1,003: @@
 ** "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 1,051: / Line 1,035: @@
 * [[User:Unque/29edo Composition Theory|Unque's approach]]
 * [[Extraclassical tonality]]
-* [[Lumatone mapping for 29edo]]
 * [[67ed5]] – octave-stretched 29edo, improves harmonics 5.7.11.13.17 but damages 2.3
@@ Line 1,061: / Line 1,044: @@
 [[Category:IMPORTDEBUG - Change External Images]]
-[[Category:3-limit]]
+[[Category:3-limit record edos|##]] <!-- 2-digit number -->
-[[Category:Listen]]
+[[Category:Subgroup temperaments]]
+[[Category:Twentuning]]
 [[Category:Negri]]
 [[Category:Petrmic]]
 [[Category:Porcupine]]
-[[Category:Subgroup temperaments]]
+[[Category:Listen]]
-[[Category:Twentuning]]