34edo: Difference between revisions
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34edo contains two [[17edo]]'s and the half-octave tritone of 600{{c}}. It excels in approximating harmonics 3, 5, 13, 17, and 23 (2.3.5.13.17.23 [[subgroup]] a.k.a. the no-7's no-11's no-19's 23-limit), with tuning even more accurate than [[31edo]] in the 5-limit, but with a sharp tendency and fifth rather than a flat one, and ''not'' tempering out [[81/80]] unlike 31edo. Its primes 7 and 11 are less accurate, but still usable (with the 34d val for prime 7) with a sharp tendency, in fact mapping all [[15-odd-limit]] intervals consistently except for 7/4 and 8/7 in the 34d val. | 34edo contains two [[17edo]]'s and the half-octave tritone of 600{{c}}. It excels in approximating harmonics 3, 5, 13, 17, and 23 (2.3.5.13.17.23 [[subgroup]] a.k.a. the no-7's no-11's no-19's 23-limit), with tuning even more accurate than [[31edo]] in the 5-limit, but with a sharp tendency and fifth rather than a flat one, and ''not'' tempering out [[81/80]] unlike 31edo. Its primes 7 and 11 are less accurate, but still usable (with the 34d val for prime 7) with a sharp tendency, in fact mapping all [[15-odd-limit]] intervals consistently except for 7/4 and 8/7 in the 34d val. | ||
34edo's significance in regards to JI approximation comes from making many simple and natural equivalences between JI intervals. For example, a key characteristic of 34edo is that it splits the standard whole tone of [[9/8]] into six parts, such that three chromatic semitones of [[25/24]] or two diatonic semitones of [[16/15]] result in 9/8. Additionally, if you stack a five-step [[10/9]] interval four times, you reach the perfect fifth [[3/2]], supporting [[tetracot]]. This also means that the perfect fifth is mapped to 20 steps. Given that and the fact that the major third [[5/4]] is mapped to 11 steps, one can see that 34edo takes advantage of a natural logarithmic approximation of 5/4 as a portion of 3/2, or equivalently [[6/5]] as a portion of 5/4, resulting in [[gammic | 34edo's significance in regards to JI approximation comes from making many simple and natural equivalences between JI intervals. For example, a key characteristic of 34edo is that it splits the standard whole tone of [[9/8]] into six parts, such that three chromatic semitones of [[25/24]] or two diatonic semitones of [[16/15]] result in 9/8. Additionally, if you stack a five-step [[10/9]] interval four times, you reach the perfect fifth [[3/2]], supporting [[tetracot]]. This also means that the perfect fifth is mapped to 20 steps. Given that and the fact that the major third [[5/4]] is mapped to 11 steps, one can see that 34edo takes advantage of a natural logarithmic approximation of 5/4 as a portion of 3/2, or equivalently [[6/5]] as a portion of 5/4, resulting in [[gammic]] temperament. It also has the thirds from 17edo: "neogothic" minor and major thirds of about 282 and 424{{c}}, and a neutral third of 353{{c}}. For [[extraclassical tonality]], a tendo third of 459{{c}} and an arto third of 247{{c}} are also available, approximating 13/10 and 15/13 respectively. | ||
34edo supports the [[diatonic scale]], both the simpler 5L 2s [[Moment-of-symmetry scale|moment-of-symmetry]] form and a more complex [[nicetone]] scale representing the [[zarlino]] diatonic. This can be extended into a 12-note chromatic scale of [[10L 2s]] by stacking the two different varieties of semitones, with an intuitive non-MOS form appearing at LLsLLLLLLsLL (created by first subdividing 34edo into the standard [[pentic]] scale and then splitting that into further smaller steps). | 34edo supports the [[diatonic scale]], both the simpler 5L 2s [[Moment-of-symmetry scale|moment-of-symmetry]] form and a more complex [[nicetone]] scale representing the [[zarlino]] diatonic. This can be extended into a 12-note chromatic scale of [[10L 2s]] by stacking the two different varieties of semitones, with an intuitive non-MOS form appearing at LLsLLLLLLsLL (created by first subdividing 34edo into the standard [[pentic]] scale and then splitting that into further smaller steps). | ||
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== Notation == | == Notation == | ||
=== | === Stein–Zimmermann–Gould notation === | ||
34edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat. | [[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows: | ||
{{Sharpness-sharp4-szg}} | |||
=== Kite's ups and downs notation === | |||
34edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat. | |||
{{Ups and downs sharpness}} | {{Ups and downs sharpness}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as [[41edo#Sagittal notation| | This notation uses the same sagittal sequence as [[41edo #Sagittal notation|41edo]], and is a superset of the notation for [[17edo #Sagittal notation|17edo]]. | ||
==== Evo flavor ==== | ==== Evo flavor ==== | ||
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| 458.824 | | 458.824 | ||
| [[3L 2s]]<br />[[5L 3s]]<br />[[8L 5s]]<br />[[13L 8s]] | | [[3L 2s]]<br />[[5L 3s]]<br />[[8L 5s]]<br />[[13L 8s]] | ||
| [[Petrtri]] | | [[Petrtri]], [[Goldis]] | ||
|- | |- | ||
| 15\34 | | 15\34 | ||
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* [[Blackdye]] (5:3:1) | * [[Blackdye]] (5:3:1) | ||
* [[Diachrome]] (5:2:1) | * [[Diachrome]] (5:2:1) | ||
* [[Cthon5m]] (4:2:1) === Combination product sets === | * [[Cthon5m]] (4:2:1) | ||
=== Combination product sets === | |||
* [[1-3-5-9 hexany]]: 6 5 9 5 6 3 | * [[1-3-5-9 hexany]]: 6 5 9 5 6 3 | ||
* Rotated [[1-3-5-11 hexany]]: 5 4 7 4 5 9 | * Rotated [[1-3-5-11 hexany]]: 5 4 7 4 5 9 | ||
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=== 21st century === | === 21st century === | ||
; [[bili_33093783396]] | |||
* [https://www.bilibili.com/video/BV1CggPztEEi/ ''A Show of Tetracot Modulation''] (2025) | |||
; [[Flora Canou]] | ; [[Flora Canou]] | ||
* [https://soundcloud.com/floracanou/october-dieting-plan?in=floracanou/sets/totmc-suite "October Dieting Plan"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in [[modus]], 34edo tuning | * [https://soundcloud.com/floracanou/october-dieting-plan?in=floracanou/sets/totmc-suite "October Dieting Plan"] from [https://soundcloud.com/floracanou/sets/totmc-suite ''TOTMC Suite''] (2023–2025) – in [[modus]], 34edo tuning | ||
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* [https://www.youtube.com/shorts/uVZ6tJ1y6ak ''34edo improv''] (2025) | * [https://www.youtube.com/shorts/uVZ6tJ1y6ak ''34edo improv''] (2025) | ||
* [https://www.youtube.com/shorts/Azk7a2bAwOo ''In My Room - Julia Wolf (microtonal cover in 34edo)''] (2026) | * [https://www.youtube.com/shorts/Azk7a2bAwOo ''In My Room - Julia Wolf (microtonal cover in 34edo)''] (2026) | ||
* [https://www.youtube.com/shorts/PDANHoJhs3I ''34edo groove''] (2026) | |||
* [https://www.youtube.com/watch?v=CY4IlT1UEFs ''groove 34edo''] (2026) | |||
; [[E8 Heterotic]] | ; [[E8 Heterotic]] | ||
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; [[Tapeworm Saga]] | ; [[Tapeworm Saga]] | ||
* [https://www.youtube.com/watch?v=BhgxwP9_cSw ''A 3/4 piece in 34edo on 12/31/23''] (2023) | * [https://www.youtube.com/watch?v=BhgxwP9_cSw ''A 3/4 piece in 34edo on 12/31/23''] (2023) | ||
; [[Shanyuan Baihe-Yuri]] (杉原百合-Yuri) | |||
* [https://www.bilibili.com/video/BV1CK411b72L/ ''Lost Memories -1#''] (2023) | |||
* [https://www.bilibili.com/video/BV1Dw411h7Af/ ''Hold a Memorial Ceremony for Myself''] (2023) | |||
; [[Sintel]] | ; [[Sintel]] | ||