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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|} | |} | ||
In the 5-limit, 34edo [[support]]s [[hanson]], [[ | In the 5-limit, 34edo [[support]]s [[hanson]], [[diaschismic]], [[tetracot]], [[würschmidt]], and [[vishnu]] temperaments. It does less well in the [[7-limit]], with two mappings possible for [[7/4]]: a flat one from the [[patent val]], and a sharp one from the 34d val (which is the same as [[17edo]]'s 7/4). By way of the patent val 34 supports [[keemun]] temperament. The two different 7-limit vals allow for different extensions of diaschismic: the patent val supports [[septimal diaschismic]], while 34d supports [[pajara]], being an excellent alternative to [[22edo]] for 7-limit pajara temperament. In the [[11-limit]], the 34d val supports undecimal pajara, vishnu and würschmidt, adding 4375/4374 to the commas of pajara. Among subgroup temperaments, the patent val supports [[semaphore]] on the 2.3.7 subgroup. | ||
=== Uniform maps === | === Uniform maps === | ||
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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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=== 21st century === | === 21st century === | ||
; [[Amano Hideya]] | |||
* [https://www.youtube.com/watch?v=wT6v8dlUFWo ''Like refracted light''] (2023) | |||
; [[bili_33093783396]] | ; [[bili_33093783396]] | ||
* [https://www.bilibili.com/video/BV1CggPztEEi/ ''A Show of Tetracot Modulation''] (2025) | * [https://www.bilibili.com/video/BV1CggPztEEi/ ''A Show of Tetracot Modulation''] (2025) | ||
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* [https://www.youtube.com/shorts/PDANHoJhs3I ''34edo groove''] (2026) | * [https://www.youtube.com/shorts/PDANHoJhs3I ''34edo groove''] (2026) | ||
* [https://www.youtube.com/watch?v=CY4IlT1UEFs ''groove 34edo''] (2026) | * [https://www.youtube.com/watch?v=CY4IlT1UEFs ''groove 34edo''] (2026) | ||
* [https://www.youtube.com/watch?v=JKblQe375pw ''34edo bossa''] (2026) | |||
; [[E8 Heterotic]] | ; [[E8 Heterotic]] | ||
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* [https://youtu.be/SoMEMYpRV4w?si=0COl0PwuPo-eQG8K ''No Threes For You''] (2019) | * [https://youtu.be/SoMEMYpRV4w?si=0COl0PwuPo-eQG8K ''No Threes For You''] (2019) | ||
* [https://youtu.be/_dOIyByMTfU?si=DsWUSQnERSKi0zbu "Elements - Water"] from ''Elements'' (2019–2020) | * [https://youtu.be/_dOIyByMTfU?si=DsWUSQnERSKi0zbu "Elements - Water"] from ''Elements'' (2019–2020) | ||
; [[Zhea Erose]] | |||
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021) | |||
; [[Francium]] | ; [[Francium]] | ||
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; [[Adam Freese]] | ; [[Adam Freese]] | ||
* [https://www.youtube.com/watch?v=lfdHBBPHvLc ''Austice''] (2023) | * [https://www.youtube.com/watch?v=lfdHBBPHvLc ''Austice''] (2023) | ||
; [[Peter Kosmorsky]] | ; [[Peter Kosmorsky]] | ||
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; [[Claudi Meneghin]] | ; [[Claudi Meneghin]] | ||
* [https://www.youtube.com/watch?v=O_7w4KoaOY4 ''Semitone-Canon on The Mother's Malison Theme, for Cor Anglais & Violin''] (2022) | * [https://www.youtube.com/watch?v=O_7w4KoaOY4 ''Semitone-Canon on The Mother's Malison Theme, for Cor Anglais & Violin''] (2022) | ||
; [[Ocean Stegosaurus Tardigrade]] | |||
* [https://www.youtube.com/watch?v=-D1ZjT3H5xM ''4 string instrument test (34ed2)''] (2026) | |||
; [[Ray Perlner]] | ; [[Ray Perlner]] | ||
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** "Escapade" – [https://open.spotify.com/track/5An9IJRQSG43gKtLnAUpOY Spotify] | [https://xotla.bandcamp.com/track/escapade Bandcamp] | [https://www.youtube.com/watch?v=3dRQy9hvWaU YouTube] – electronic rock | ** "Escapade" – [https://open.spotify.com/track/5An9IJRQSG43gKtLnAUpOY Spotify] | [https://xotla.bandcamp.com/track/escapade Bandcamp] | [https://www.youtube.com/watch?v=3dRQy9hvWaU YouTube] – electronic rock | ||
* [https://www.youtube.com/watch?v=56HuAL1jC8E ''Between Space''] (2022) – ambient sci-fi | * [https://www.youtube.com/watch?v=56HuAL1jC8E ''Between Space''] (2022) – ambient sci-fi | ||
== See also == | == See also == | ||
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== External links == | == External links == | ||
* [ | * [https://microstick.net/a-34-equal-guitar-by-larry-a-hanson/ 34 Equal Guitar] by [[Larry Hanson]] | ||
* [https://microstick.net Websites of Neil Haverstick] | * [https://microstick.net Websites of Neil Haverstick] | ||
* [https://myspace.com/microstick] – somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything) | * [https://myspace.com/microstick] – somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything) | ||