35edo: Difference between revisions

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==== Caveats of dual-fifth ====

However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. If we try to map~~, say,~~ the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.

However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. Their dyadic consonance relies on the compositeness of the number 9 as 3 × 3, and here the mapping breaks down when we try to use two different mappings of harmonic 3. For example, if we try to map the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.

Additionally, many structures present in systems with a single fifth do not work well in 35edo. For example, the perfect fifth generates several [[mos scale]], such as the traditional [[diatonic]] scale. The diatonic mos scale does not exist in 35edo, with the 20\35 whitewood fifth generating an [[equalized]] version of the scale, while the 21\35 fifth generates a [[collapsed]] version of the scale. Since 35edo does not have a diatonic scale, [[chain-of-fifths notation]] also does not work in 35edo. However, there are scales such as 6 6 2 6 6 6 3 which sound similar to diatonic, and this particular scale can be obtained by alternately stacking 21\35 and 20\35 fifths, or [[Hobbled scale|hobbling]] a [[34edo]] or [[36edo]] diatonic scale.

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=== Modern renderings ===

; {{W|Frederick Chopin}}

* [https://www.youtube.com/watch?v=1odAmqiQaz0 ''CHOPIN Waltz op 64 #2''] (1847 ~~—~~ rendered ~~by [[Claudi Meneghin]]~~ in 35-edo with alternating sharp and flat fifths, 2025)

* [https://www.youtube.com/watch?v=1odAmqiQaz0 ''CHOPIN Waltz op 64 #2''] (1847) – rendered in 35-edo with alternating sharp and flat fifths by [[Claudi Meneghin]] (2025)

; {{W|Gesualdo}}

; {{W|Carlo Gesualdo}}

* [https://~~youtu~~.be/idUG-x8kT3o ''~~Gesualdo's~~ Dolcissima mia vita in three comparative tunings ~~after translation to english'']~~ (5:~~05 to 10~~:05) ~~—~~ rendered by [[Chris Vaisvil]], 2025

* [https://www.youtube.com/watch?v=idUG-x8kT3o&t=305 ''Dolcissima mia vita''] – in three comparative tunings including 35edo (5:05–10:05), rendered by [[Chris Vaisvil]] (2025)

=== 21st century ===

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/6X1-692axAk ''35edo''] (2025)

* ''Whistling Like An Oberon - 35edo'' (2026) [https://www.youtube.com/shorts/rTkr2YHDvZM <nowiki>[short 1]</nowiki>], [https://www.youtube.com/shorts/AvIGI8TG9_8 <nowiki>[short 2]</nowiki>], [https://m.youtube.com/watch?v=zPRYktfbJj8 <nowiki>[full piece]</nowiki>]

* ''Whistling Like An Oberon - 35edo'' (2026)

** [https://www.youtube.com/shorts/rTkr2YHDvZM <nowiki>[short 1]</nowiki>]

** [https://www.youtube.com/shorts/AvIGI8TG9_8 <nowiki>[short 2]</nowiki>]

** [https://m.youtube.com/watch?v=zPRYktfbJj8 <nowiki>[full piece]</nowiki>]

* [https://www.youtube.com/watch?v=x8doWEgXMCY ''35edo improv''] (2026)

; [[dotuXil]]

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; [[Francium]]

* "What Kind Of Things" from ''TOTMC 2025'' (2025) – [https://francium223.bandcamp.com/track/what-kind-of-things Bandcamp] | [https://www.youtube.com/watch?v=WaRm0dlUqQU YouTube]

; [[groundfault]]

* "Sakura Blade Minivan", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/sakura-blade-minivan-27-35edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1436 YouTube (23:56–27:58)] – in part, the rest being in 27edo

; [[JUMBLE]]

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; [[Budjarn Lambeth]]

* [https://www.youtube.com/watch?v=ZPXaMTdTSgw ''Lighting the Jack-o'-lanterns''] (2025, uses meta-monsoon scale{{idio}} from 6/5-[[~~blackwood~~]][20])

* [https://www.youtube.com/watch?v=ZPXaMTdTSgw ''Lighting the Jack-o'-lanterns''] (2025, uses meta-monsoon scale{{idio}} from 6/5-[[Blackwood]][20])

; [[Chuckles McGee]]

* [https://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 Self-Destructing Mechanical Forest] (in Secund[9])

* [https://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 ''Self-Destructing Mechanical Forest''] – in Secund[9], 35edo tuning

; [[Claudi Meneghin]]

* [https://web.archive.org/web/20190412163316/http://soonlabel.com/xenharmonic/archives/2348'' Little Prelude &~~amp;~~ Fugue, "The Bijingle"''] (2014)

* [https://web.archive.org/web/20190412163316/http://soonlabel.com/xenharmonic/archives/2348'' Little Prelude & Fugue, "The Bijingle"''] (2014)

* [https://www.youtube.com/watch?v=JPie2YDwA8I ''MicroFugue on Happy Birthday for Baroque Ensemble''] (2023)

* [https://www.youtube.com/shorts/c9rCrQwF1HI ''NEOBAROQUE CANON, 3-in-1 without Bass in 35-edo for Baroque Consort: Oboe, Recorder, Violin''] (2025)

@@ Line 26: / Line 26: @@
 ==== Caveats of dual-fifth ====
-However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. If we try to map, say, the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.
+However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. Their dyadic consonance relies on the compositeness of the number 9 as 3 × 3, and here the mapping breaks down when we try to use two different mappings of harmonic 3. For example, if we try to map the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.
 Additionally, many structures present in systems with a single fifth do not work well in 35edo. For example, the perfect fifth generates several [[mos scale]], such as the traditional [[diatonic]] scale. The diatonic mos scale does not exist in 35edo, with the 20\35 whitewood fifth generating an [[equalized]] version of the scale, while the 21\35 fifth generates a [[collapsed]] version of the scale. Since 35edo does not have a diatonic scale, [[chain-of-fifths notation]] also does not work in 35edo. However, there are scales such as 6 6 2 6 6 6 3 which sound similar to diatonic, and this particular scale can be obtained by alternately stacking 21\35 and 20\35 fifths, or [[Hobbled scale|hobbling]] a [[34edo]] or [[36edo]] diatonic scale.
@@ Line 1,019: / Line 1,019: @@
 === Modern renderings ===
 ; {{W|Frederick Chopin}}
-* [https://www.youtube.com/watch?v=1odAmqiQaz0 ''CHOPIN Waltz op 64 #2''] (1847 &mdash; rendered by [[Claudi Meneghin]] in 35-edo with alternating sharp and flat fifths, 2025)
+* [https://www.youtube.com/watch?v=1odAmqiQaz0 ''CHOPIN Waltz op 64 #2''] (1847) – rendered in 35-edo with alternating sharp and flat fifths by [[Claudi Meneghin]] (2025)
-; {{W|Gesualdo}}
+; {{W|Carlo Gesualdo}}
-* [https://youtu.be/idUG-x8kT3o ''Gesualdo's Dolcissima mia vita in three comparative tunings after translation to english''] (5:05 to 10:05) &mdash; rendered by [[Chris Vaisvil]], 2025
+* [https://www.youtube.com/watch?v=idUG-x8kT3o&t=305 ''Dolcissima mia vita''] – in three comparative tunings including 35edo (5:05–10:05), rendered by [[Chris Vaisvil]] (2025)
 === 21st century ===
 ; [[Bryan Deister]]
 * [https://www.youtube.com/shorts/6X1-692axAk ''35edo''] (2025)
-* ''Whistling Like An Oberon - 35edo'' (2026) [https://www.youtube.com/shorts/rTkr2YHDvZM <nowiki>[short 1]</nowiki>], [https://www.youtube.com/shorts/AvIGI8TG9_8 <nowiki>[short 2]</nowiki>], [https://m.youtube.com/watch?v=zPRYktfbJj8 <nowiki>[full piece]</nowiki>]
+* ''Whistling Like An Oberon - 35edo'' (2026)
+** [https://www.youtube.com/shorts/rTkr2YHDvZM <nowiki>[short 1]</nowiki>]
+** [https://www.youtube.com/shorts/AvIGI8TG9_8 <nowiki>[short 2]</nowiki>]
+** [https://m.youtube.com/watch?v=zPRYktfbJj8 <nowiki>[full piece]</nowiki>]
+* [https://www.youtube.com/watch?v=x8doWEgXMCY ''35edo improv''] (2026)
 ; [[dotuXil]]
@@ Line 1,037: / Line 1,041: @@
 ; [[Francium]]
 * "What Kind Of Things" from ''TOTMC 2025'' (2025) – [https://francium223.bandcamp.com/track/what-kind-of-things Bandcamp] | [https://www.youtube.com/watch?v=WaRm0dlUqQU YouTube]
+; [[groundfault]]
+* "Sakura Blade Minivan", from ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/track/sakura-blade-minivan-27-35edo-2 Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0&t=1436 YouTube (23:56–27:58)] – in part, the rest being in 27edo
 ; [[JUMBLE]]
@@ Line 1,042: / Line 1,049: @@
 ; [[Budjarn Lambeth]]
-* [https://www.youtube.com/watch?v=ZPXaMTdTSgw ''Lighting the Jack-o'-lanterns''] (2025, uses meta-monsoon scale{{idio}} from 6/5-[[blackwood]][20])
+* [https://www.youtube.com/watch?v=ZPXaMTdTSgw ''Lighting the Jack-o'-lanterns''] (2025, uses meta-monsoon scale{{idio}} from 6/5-[[Blackwood]][20])
 ; [[Chuckles McGee]]
-* [https://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 Self-Destructing Mechanical Forest] (in Secund[9])
+* [https://www.archive.org/download/Transcendissonance/05Self-destructingMechanicalForest-CityOfTheAsleep.mp3 ''Self-Destructing Mechanical Forest''] – in Secund[9], 35edo tuning
 ; [[Claudi Meneghin]]
-* [https://web.archive.org/web/20190412163316/http://soonlabel.com/xenharmonic/archives/2348'' Little Prelude &amp; Fugue, "The Bijingle"''] (2014)
+* [https://web.archive.org/web/20190412163316/http://soonlabel.com/xenharmonic/archives/2348'' Little Prelude & Fugue, "The Bijingle"''] (2014)
 * [https://www.youtube.com/watch?v=JPie2YDwA8I ''MicroFugue on Happy Birthday for Baroque Ensemble''] (2023)
 * [https://www.youtube.com/shorts/c9rCrQwF1HI ''NEOBAROQUE CANON, 3-in-1 without Bass in 35-edo for Baroque Consort: Oboe, Recorder, Violin''] (2025)