34edo: Difference between revisions

(9 intermediate revisions by 5 users not shown)

Line 12:

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 ~~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'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).

Line 424:

== Notation ==

=== ~~Ups~~ and downs 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:

=== 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.

~~[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:~~

~~{{Sharpness-sharp4}}~~

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[41edo#Sagittal notation|~~41-EDO~~]], and is a superset of the notation for [[17edo#Sagittal notation|~~17-EDO~~]].

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 ====

Line 491:

Line 492:

=== Counterpoint ===

34edo has such an excellent [[sqrt(25/24)]] that the next edo to have a better one is [[441edo|441]]. Every sequence of intervals available in [[17edo]] ~~are~~ reachable by {{W|Contrary motion|strict contrary motion}} in 34edo.

34edo has such an excellent [[sqrt(25/24)]] that the next edo to have a better one is [[441edo|441]]. Every sequence of intervals available in [[17edo]] is reachable by {{W|Contrary motion|strict contrary motion}} in 34edo.

== Regular temperament properties ==

Line 593:

Line 594:

| 458.824

| [[3L 2s]]<br />[[5L 3s]]<br />[[8L 5s]]<br />[[13L 8s]]

| [[Petrtri]]

| [[Petrtri]], [[Goldis]]

|-

| 15\34

Line 775:

Line 776:

* [[Blackdye]] (5:3:1)

* [[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

* Rotated [[1-3-5-11 hexany]]: 5 4 7 4 5 9

Line 818:

Line 821:

=== 21st century ===

; [[bili_33093783396]]

* [https://www.bilibili.com/video/BV1CggPztEEi/ ''A Show of Tetracot Modulation''] (2025)

; [[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

Line 824:

Line 830:

* [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/PDANHoJhs3I ''34edo groove''] (2026)

* [https://www.youtube.com/watch?v=CY4IlT1UEFs ''groove 34edo''] (2026)

; [[E8 Heterotic]]

Line 864:

Line 872:

; [[Tapeworm Saga]]

* [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]]

@@ Line 12: / Line 12: @@
 edo 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.
-edo'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.
+edo'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.
 edo supports the [[diatonic scale]], both the simpler 5L&nbsp;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&nbsp;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).
@@ Line 424: / Line 424: @@
 == Notation ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo 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 ===
+edo 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}}
-[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
-{{Sharpness-sharp4}}
 === Sagittal notation ===
-This notation uses the same sagittal sequence as [[41edo#Sagittal notation|41-EDO]], and is a superset of the notation for [[17edo#Sagittal notation|17-EDO]].
+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 ====
@@ Line 491: / Line 492: @@
 === Counterpoint ===
-edo has such an excellent [[sqrt(25/24)]] that the next edo to have a better one is [[441edo|441]]. Every sequence of intervals available in [[17edo]] are reachable by {{W|Contrary motion|strict contrary motion}} in 34edo.
+edo has such an excellent [[sqrt(25/24)]] that the next edo to have a better one is [[441edo|441]]. Every sequence of intervals available in [[17edo]] is reachable by {{W|Contrary motion|strict contrary motion}} in 34edo.
 == Regular temperament properties ==
@@ Line 593: / Line 594: @@
 | 458.824
 | [[3L&nbsp;2s]]<br />[[5L&nbsp;3s]]<br />[[8L&nbsp;5s]]<br />[[13L&nbsp;8s]]
-| [[Petrtri]]
+| [[Petrtri]], [[Goldis]]
 |-
 | 15\34
@@ Line 775: / Line 776: @@
 * [[Blackdye]] (5:3:1)
 * [[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
 * Rotated [[1-3-5-11 hexany]]: 5 4 7 4 5 9
@@ Line 818: / Line 821: @@
 === 21st century ===
+; [[bili_33093783396]]
+* [https://www.bilibili.com/video/BV1CggPztEEi/ ''A Show of Tetracot Modulation''] (2025)
 ; [[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
@@ Line 824: / Line 830: @@
 * [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/PDANHoJhs3I ''34edo groove''] (2026)
+* [https://www.youtube.com/watch?v=CY4IlT1UEFs ''groove 34edo''] (2026)
 ; [[E8 Heterotic]]
@@ Line 864: / Line 872: @@
 ; [[Tapeworm Saga]]
 * [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]]