34edo: Difference between revisions
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== Theory == | == Theory == | ||
34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. 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. | 34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. 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. | ||
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 cents, and a neutral third of 353 cents. For [[extraclassical tonality]], a tendo third of 459 cents and an arto third of 247 cents 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). | |||
=== Odd harmonics === | === Odd harmonics === | ||
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! Ratios of 7<br />Using the 34 Val | ! Ratios of 7<br />Using the 34 Val | ||
! Ratios of 7<br />Using the 34d Val | ! Ratios of 7<br />Using the 34d Val | ||
! colspan="3" | [[Ups and | ! colspan="3" | [[Ups and downs notation]] | ||
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^^d2) | |||
! colspan="2" | [[Solfege|Solfeges]] | ! colspan="2" | [[Solfege|Solfeges]] | ||
|- | |- | ||
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|} | |} | ||
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and | Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chord names in other EDOs]]. | ||
== Notation == | == Notation == | ||
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{{Q-odd-limit intervals|34.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 34d val mapping}} | {{Q-odd-limit intervals|34.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 34d val mapping}} | ||
Of particular interest is the fact that the 34d val allows all 15-odd-limit intervals to be mapped consistently except for 7/4 and 8/7. | |||
== Tuning by ear == | == Tuning by ear == | ||
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* "Locksmiths" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/2Hzun107B8bxcZaMOClN6T Spotify] | [https://francium223.bandcamp.com/track/locksmiths Bandcamp] | [https://www.youtube.com/watch?v=pQLbtF0Obhc YouTube] | * "Locksmiths" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/2Hzun107B8bxcZaMOClN6T Spotify] | [https://francium223.bandcamp.com/track/locksmiths Bandcamp] | [https://www.youtube.com/watch?v=pQLbtF0Obhc YouTube] | ||
* [https://www.youtube.com/watch?v=HG0kJBHjuZ4 ''Plane Sonatina No. 2''] (2025) | * [https://www.youtube.com/watch?v=HG0kJBHjuZ4 ''Plane Sonatina No. 2''] (2025) | ||
* [https://www.youtube.com/watch?v=ZrjbxQbdVw4 ''cucumber service''] (2025) | |||
; [[Adam Freese]] | ; [[Adam Freese]] | ||