35edo - Revision history

Lucius Chiaraviglio: /* 21st century */ Add Bryan Deister's ''35edo improv'' (2026)

2026-06-09T02:54:40Z

21st century: Add Bryan Deister's ''35edo improv'' (2026)

← Older revision		Revision as of 02:54, 9 June 2026
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	; [[Bryan Deister]]		; [[Bryan Deister]]
	* [https://www.youtube.com/shorts/6X1-692axAk ''35edo''] (2025)		* [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]]		; [[dotuXil]]

FloraC: /* Music */ +1

2026-05-07T14:42:01Z

Music: +1

← Older revision		Revision as of 14:42, 7 May 2026
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	; [[Francium]]		; [[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]		* "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]]		; [[JUMBLE]]
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	; [[Chuckles McGee]]		; [[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]]		; [[Claudi Meneghin]]

FloraC: /* Music */ cleanup

2026-05-07T14:40:38Z

Music: cleanup

← Older revision		Revision as of 14:40, 7 May 2026
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	=== Modern renderings ===		=== Modern renderings ===
	; {{W\|Frederick Chopin}}		; {{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 ===		=== 21st century ===
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	; [[Budjarn Lambeth]]		; [[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]]		; [[Chuckles McGee]]
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	; [[Claudi Meneghin]]		; [[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/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)		* [https://www.youtube.com/shorts/c9rCrQwF1HI ''NEOBAROQUE CANON, 3-in-1 without Bass in 35-edo for Baroque Consort: Oboe, Recorder, Violin''] (2025)

Overthink: /* Caveats of dual-fifth */ add an important note

2026-05-01T00:07:03Z

Caveats of dual-fifth: add an important note

← Older revision		Revision as of 00:07, 1 May 2026
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	==== Caveats of dual-fifth ====		==== 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.		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.

Overthink: /* Caveats of Dual-fifth */ chain-of-fifths notation; caps

2026-04-17T19:42:25Z

Caveats of Dual-fifth: chain-of-fifths notation; caps

← Older revision		Revision as of 19:42, 17 April 2026
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	Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15\|1/(6:5:4)]], [[6:7:9]], and [[14:18:21\|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28\|1/(8:7:6)]], [[9:10:12]], and [[15:18:20\|1/(12:10:9)]], are also tuned fairly close to DR.		Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15\|1/(6:5:4)]], [[6:7:9]], and [[14:18:21\|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28\|1/(8:7:6)]], [[9:10:12]], and [[15:18:20\|1/(12:10:9)]], are also tuned fairly close to DR.

	==== Caveats of ~~Dual~~-fifth ====		==== 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. 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.

	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. 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.		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.

	35edo is only one of many dual-fifth systems, with others including [[18edo]], [[23edo]], [[25edo]], [[28edo]], [[30edo]], [[37edo]], and [[40edo]], each with their own unique properties.		35edo is only one of many dual-fifth systems, with others including [[18edo]], [[23edo]], [[25edo]], [[28edo]], [[30edo]], [[37edo]], and [[40edo]], each with their own unique properties.

Overthink: /* Theory */ mention dual-fifth earlier; remove confusing statement (not all mos scales supported by 35edo also exist in 22edo)

2026-04-17T00:00:28Z

Theory: mention dual-fifth earlier; remove confusing statement (not all mos scales supported by 35edo also exist in 22edo)

← Older revision		Revision as of 00:00, 17 April 2026
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	== Theory ==		== Theory ==
	As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{Cent}} and 5edo's wide fifth of 720{{C}}. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{C}} of error.		As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{Cent}} and 5edo's wide fifth of 720{{C}}. Since it has two approximations of the perfect fifth which are close to equally off, 35edo is a classic example of a [[dual-fifth]] system. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{C}} of error.

	35edo can also represent the 2.3.5.7.11.17 [[subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of ~~all other subgroup generators~~ (7/5 and 17/11 stand out, having less than 1 cent error). Therefore among [[whitewood]] tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings)~~, and if you ignore [[22edo]]'s more in-tune versions of 35edo MOS's and consistent representation of both subgroups~~.		35edo can also represent the 2.3.5.7.11.17 [[subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of the higher primes ([[7/5]] and [[17/11]] stand out, having less than 1 cent error). Therefore among [[whitewood]] tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings).

	35edo has the optimal [[patent val]] for [[greenwood]] and [[secund]] temperaments, as well as 11-limit [[muggles]], and the 35f val is an excellent tuning for 13-limit muggles. 35edo is the largest edo with a lack of a [[diatonic scale]] (unless 7edo is considered a diatonic scale).		35edo has the optimal [[patent val]] for [[greenwood]] and [[secund]] temperaments, as well as 11-limit [[muggles]], and the 35f val is an excellent tuning for 13-limit muggles. 35edo is the largest edo with a lack of a [[diatonic scale]] (unless 7edo is considered a diatonic scale).

Overthink: /* Dual-fifth harmony */ add more

2026-04-16T23:54:34Z

Dual-fifth harmony: add more

← Older revision		Revision as of 23:54, 16 April 2026
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	Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24\|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120\|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).		Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24\|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120\|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).

	Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. ~~Even though there~~ are two mappings of the perfect fifth, ~~it all fits well in~~ the ~~end~~, ~~as we will soon see~~.		Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. There are two mappings of the perfect fifth, with some chords preferring the flat fifth, while other chords prefer the sharp fifth.

	For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21\|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18\|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. This leads to an approximation of the [[6:7:9:10]] harmonic sixth chord (sometimes known as the ''subminor tetrad'') at 0–8–21–26 steps (0–274–720–891{{C}}), and an approximation of the [[70:90:105:126\|1/(9:7:6:5)]] subharmonic seventh chord (sometimes called the ''supermajor tetrad'') at 0–13–21–30 steps (0–446–720–1029{{C}}).		For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21\|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18\|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. This leads to an approximation of the [[6:7:9:10]] harmonic sixth chord (sometimes known as the ''subminor tetrad'') at 0–8–21–26 steps (0–274–720–891{{C}}), and an approximation of the [[70:90:105:126\|1/(9:7:6:5)]] subharmonic seventh chord (sometimes called the ''supermajor tetrad'') at 0–13–21–30 steps (0–446–720–1029{{C}}).

	The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315\|1/(9:7:6:5:4)]] as 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune.		The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo that is a subset of the 9-odd-limit otonal or utonal pentad (up to octave equivalence) can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315\|1/(9:7:6:5:4)]], that being 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune.

	Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15\|1/(6:5:4)]], [[6:7:9]], and [[14:18:21\|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28\|1/(8:7:6)]], [[9:10:12]], and [[15:18:20\|1/(12:10:9)]], are also tuned fairly close to DR.		Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15\|1/(6:5:4)]], [[6:7:9]], and [[14:18:21\|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28\|1/(8:7:6)]], [[9:10:12]], and [[15:18:20\|1/(12:10:9)]], are also tuned fairly close to DR.

			==== 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.

			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. 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.

			35edo is only one of many dual-fifth systems, with others including [[18edo]], [[23edo]], [[25edo]], [[28edo]], [[30edo]], [[37edo]], and [[40edo]], each with their own unique properties.

	=== Subsets and supersets ===		=== Subsets and supersets ===

Overthink: /* Intervals */ remove nonstandard link to commas

2026-03-25T02:25:39Z

Intervals: remove nonstandard link to commas

← Older revision		Revision as of 02:25, 25 March 2026
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	\| 0.000		\| 0.000
	\| '''1/1'''		\| '''1/1'''
	\| ~~(see [[#Commas]])~~		\|
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	\|		\|

Overthink: /* Dual-fifth harmony */ not too sure actually

2026-03-25T02:25:02Z

Dual-fifth harmony: not too sure actually

← Older revision		Revision as of 02:25, 25 March 2026
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	The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315\|1/(9:7:6:5:4)]] as 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune.		The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315\|1/(9:7:6:5:4)]] as 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune.

	Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15\|1/(6:5:4)]], [[6:7:9]], and [[14:18:21\|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28\|1/(8:7:6)]], [[9:10:12]], and [[15:18:20\|1/(12:10:9)]], are also tuned fairly close to DR. The wide voicings that span the twelfth, those being 4:7:12, 1/(12:7:4), 3:5:9, and 1/(9:5:3) are even closer to DR than the voicings that span the fourth, though not as close as the triads spanning the fifth.		Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15\|1/(6:5:4)]], [[6:7:9]], and [[14:18:21\|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28\|1/(8:7:6)]], [[9:10:12]], and [[15:18:20\|1/(12:10:9)]], are also tuned fairly close to DR.

	=== Subsets and supersets ===		=== Subsets and supersets ===

Overthink: /* Dual-fifth harmony */ mention subminor and supermajor tetrads; move DR to new paragraph

2026-03-24T22:00:15Z

Dual-fifth harmony: mention subminor and supermajor tetrads; move DR to new paragraph

← Older revision		Revision as of 22:00, 24 March 2026
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	=== Dual-fifth harmony ===		=== Dual-fifth harmony ===
	35edo has two viable mappings of the [[3/2\|perfect fifth]], one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15\|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned~~. Interestingly, the approximations of 4:5:6 and 10:12:15 are fairly close to [[DR]] tunings as well~~.		35edo has two viable mappings of the [[3/2\|perfect fifth]], one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15\|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned.

	Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24\|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. The 6:7:8 and 4:7:12 voicings of 4:6:7, which split the fourth and the twelfth respectively, are tuned fairly close to DR. The 1/(8:7:6) and 1/(12:7:4) inversions of 1/(12:8:7) are also tuned close to DR, though their delta signatures are significantly more complex. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120\|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).		Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24\|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120\|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).

	Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. Even though there are two mappings of the perfect fifth, it all fits well in the end, as we will soon see.		Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. Even though there are two mappings of the perfect fifth, it all fits well in the end, as we will soon see.

	For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21\|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth~~. Similarly to 4:5:6 and 10:12:15, these are also fairly close to DR tunings~~. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18\|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. ~~The~~ 9:10~~:12 voicing~~ and ~~especially~~ the 3:5:9 ~~voicing, as well as their inverses, are tuned fairly close to DR~~.		For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21\|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18\|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. This leads to an approximation of the [[6:7:9:10]] harmonic sixth chord (sometimes known as the ''subminor tetrad'') at 0–8–21–26 steps (0–274–720–891{{C}}), and an approximation of the [[70:90:105:126\|1/(9:7:6:5)]] subharmonic seventh chord (sometimes called the ''supermajor tetrad'') at 0–13–21–30 steps (0–446–720–1029{{C}}).

	The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315\|1/(9:7:6:5:4)]] as 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune. Additionally, many triads are tuned very close to DR, which may make them sound less out of tune as well.		The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315\|1/(9:7:6:5:4)]] as 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune.

			Additionally, many triads are tuned very close to [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15\|1/(6:5:4)]], [[6:7:9]], and [[14:18:21\|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28\|1/(8:7:6)]], [[9:10:12]], and [[15:18:20\|1/(12:10:9)]], are also tuned fairly close to DR. The wide voicings that span the twelfth, those being 4:7:12, 1/(12:7:4), 3:5:9, and 1/(9:5:3) are even closer to DR than the voicings that span the fourth, though not as close as the triads spanning the fifth.

	=== Subsets and supersets ===		=== Subsets and supersets ===