34edo: Difference between revisions

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== 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'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 ===

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! Ratios of 7 Using the 34 Val

! Ratios of 7 Using the 34d Val

! colspan="3" | [[Ups and ~~Downs Notation~~]]

! colspan="3" | [[Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: v4A1 and ^^d2)

! 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 ~~Downs Notation~~ #Chord names in other EDOs]].

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 ==

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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}}

~~=== Zeta peak index ===~~

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.

~~{{ZPI~~

~~| zpi = 144~~

~~| steps = 34~~.~~0448410043159~~

~~| step size = 35.2476312005063~~

~~| tempered height = 6.685147~~

~~| pure height = 5.810937~~

~~| integral = 1.241437~~

~~| gap = 16.236989~~

~~| octave = 1198.41946081721~~

~~| consistent = 6~~

~~| distinct = 6~~

}}

== Tuning by ear ==

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=== Uniform maps ===

=== Rank-2 temperaments ===

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| 5\34

| 176.471

| [[6L 1s]] [[7L 6s]] [[7L 13s~~]]<br />[[7L 20s~~]]

| [[6L 1s]] [[7L 6s]] [[7L 13s]]

| [[Tetracot]], [[bunya]] (34d), [[modus]] (34d), [[monkey]] (34), [[wollemia]] (34)

|-

| 7\34

| 247.059

| [[5L 4s]] [[5L 9s]] [[5L 14s]] [[5L 19s]]~~ Pathological 5L 24s~~

| [[5L 4s]] [[5L 9s]] [[5L 14s]] [[5L 19s]]

| [[Immunity]] (34), [[immunized]] (34d)

|-

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| 11\34

| 388.235

| [[3L 7s]] [[3L 10s]] [[3L 13s]] [[3L 16s]] [[3L 19s]] [[3L 22s]] ~~Pathological [[3L 25s]] Pathological 3L 28s~~

| [[3L 7s]] [[3L 10s]] [[3L 13s]] [[3L 16s]] [[3L 19s]] [[3L 22s]]

| [[Würschmidt]] (34d), [[worschmidt]] (34)

|-

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| 15\34

| 529.412

| [[2L 3s]] [[2L 5s]] [[7L 2s]] [[9L 7s]]~~ 9L 16s~~

| [[2L 3s]] [[2L 5s]] [[7L 2s]] [[9L 7s]]

| [[Mabila]]

|-

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| 4\34

| 141.176

| [[2L 6s]] [[8L 2s]] [[8L 10s]]~~ 8L 16s~~

| [[2L 6s]] [[8L 2s]] [[8L 10s]]

| [[Fifive]], [[crepuscular]] (34d), [[fifives]] (34)

|-

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| 6\34

| 211.765

| [[4L 2s]] [[6L 4s]] [[6L 10s]] [[6L 16s]]~~ Pathological 6L 22s~~

| [[4L 2s]] [[6L 4s]] [[6L 10s]] [[6L 16s]]

| [[Antikythera]]

|-

| 7\34

| 247.059

| [[4L 2s]] [[4L 6s]] [[10L 4s~~]] [[10L 14s~~]]

| [[4L 2s]] [[4L 6s]] [[10L 4s]]

| [[Tobago]]

|-

| 8\34

| 282.353

| [[2L 2s]] [[4L 2s]] [[4L 6s]] [[4L 10s]] [[4L 14s]]~~ 4L 18s 4L 22s Pathological 4L 26s~~

| [[2L 2s]] [[4L 2s]] [[4L 6s]] [[4L 10s]] [[4L 14s]]

| [[Bikleismic]]

|}

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* [[Diachrome]] (5:2:1)

* [[Cthon5m]] (4:2:1)

== Instruments ==

=== Lumatone ===

* [[Lumatone mapping for 34edo]]

=== Skip fretting ===

* [[Skip fretting system 34 2 9]]

* [[Skip fretting system 34 2 11]]

== Music ==

=== Modern renderings ===

; {{W|Johann Sebastian Bach}}

* [https://www.youtube.com/watch?v=Mni0bsUVgHk "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)

; {{W|Scott Joplin}}

* [https://www.youtube.com/watch?v=CwMem5p1R6Y ''Maple Leaf Rag''] (1899) – rendered by Claudi Meneghin (2024)

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* "Travel To Stay" from ''Mysteries'' (2023) – [https://open.spotify.com/track/2S2UlMKNNL5yB3280KpgvK Spotify] | [https://francium223.bandcamp.com/track/travel-to-stay Bandcamp] | [https://www.youtube.com/watch?v=fDkW1SnMcdw 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=ZrjbxQbdVw4 ''cucumber service''] (2025)

; [[Adam Freese]]

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; [[Tapeworm Saga]]

* [https://www.youtube.com/watch?v=BhgxwP9_cSw ''A 3/4 piece in 34edo on 12/31/23''] (2023)

; [[Sintel]]

* [https://www.youtube.com/watch?v=hM7p_VVyeQ0 ''Diversion in 34edo''] (2021) – [https://www.youtube.com/watch?v=yTG0z4Znimw transcription by Stephen Weigel]

; [[Cam Taylor]]

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== See also ==

* [[Lumatone mapping for 34edo]]

* [[Skip fretting system 34 2 9]]

* [[Diaschismic-tetracot equivalence continuum]]

@@ Line 11: / Line 11: @@
 == Theory ==
 edo 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.
+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 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.
+edo 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 ===
@@ Line 23: / Line 27: @@
 ! Ratios of 7<br />Using the 34 Val
 ! Ratios of 7<br />Using the 34d Val
-! colspan="3" | [[Ups and Downs Notation]]
+! 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]]
 |-
@@ Line 412: / Line 417: @@
 |}
-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]].
+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 ==
@@ Line 473: / Line 478: @@
 {{Q-odd-limit intervals|34.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 34d val mapping}}
-=== Zeta peak index ===
+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.
-{{ZPI
-| zpi = 144
-| steps = 34.0448410043159
-| step size = 35.2476312005063
-| tempered height = 6.685147
-| pure height = 5.810937
-| integral = 1.241437
-| gap = 16.236989
-| octave = 1198.41946081721
-| consistent = 6
-| distinct = 6
-}}
 == Tuning by ear ==
@@ Line 547: / Line 540: @@
 === Uniform maps ===
-{{Uniform map|13|33.5|34.5}}
+{{Uniform map|edo=34}}
 === Rank-2 temperaments ===
@@ Line 575: / Line 568: @@
 | 5\34
 | 176.471
-| [[6L&nbsp;1s]]<br />[[7L&nbsp;6s]]<br />[[7L&nbsp;13s]]<br&nbsp;/>[[7L&nbsp;20s]]
+| [[6L&nbsp;1s]]<br />[[7L&nbsp;6s]]<br />[[7L&nbsp;13s]]
 | [[Tetracot]], [[bunya]] (34d), [[modus]] (34d), [[monkey]] (34), [[wollemia]] (34)
 |-
 | 7\34
 | 247.059
-| [[5L&nbsp;4s]]<br />[[5L&nbsp;9s]]<br />[[5L&nbsp;14s]]<br />[[5L&nbsp;19s]]<br />Pathological 5L&nbsp;24s
+| [[5L&nbsp;4s]]<br />[[5L&nbsp;9s]]<br />[[5L&nbsp;14s]]<br />[[5L&nbsp;19s]]
 | [[Immunity]] (34), [[immunized]] (34d)
 |-
@@ Line 590: / Line 583: @@
 | 11\34
 | 388.235
-| [[3L&nbsp;7s]]<br />[[3L&nbsp;10s]]<br />[[3L&nbsp;13s]]<br />[[3L&nbsp;16s]]<br />[[3L&nbsp;19s]]<br />[[3L&nbsp;22s]]<br />Pathological [[3L&nbsp;25s]]<br />Pathological 3L&nbsp;28s
+| [[3L&nbsp;7s]]<br />[[3L&nbsp;10s]]<br />[[3L&nbsp;13s]]<br />[[3L&nbsp;16s]]<br />[[3L&nbsp;19s]]<br />[[3L&nbsp;22s]]<br />
 | [[Würschmidt]] (34d), [[worschmidt]] (34)
 |-
@@ Line 600: / Line 593: @@
 | 15\34
 | 529.412
-| [[2L&nbsp;3s]]<br />[[2L&nbsp;5s]]<br />[[7L&nbsp;2s]]<br />[[9L&nbsp;7s]]<br />9L&nbsp;16s
+| [[2L&nbsp;3s]]<br />[[2L&nbsp;5s]]<br />[[7L&nbsp;2s]]<br />[[9L&nbsp;7s]]
 | [[Mabila]]
 |-
@@ Line 616: / Line 609: @@
 | 4\34
 | 141.176
-| [[2L&nbsp;6s]]<br />[[8L&nbsp;2s]]<br />[[8L&nbsp;10s]]<br />8L&nbsp;16s
+| [[2L&nbsp;6s]]<br />[[8L&nbsp;2s]]<br />[[8L&nbsp;10s]]
 | [[Fifive]], [[crepuscular]] (34d), [[fifives]] (34)
 |-
@@ Line 626: / Line 619: @@
 | 6\34
 | 211.765
-| [[4L&nbsp;2s]]<br />[[6L&nbsp;4s]]<br />[[6L&nbsp;10s]]<br />[[6L&nbsp;16s]]<br />Pathological 6L&nbsp;22s
+| [[4L&nbsp;2s]]<br />[[6L&nbsp;4s]]<br />[[6L&nbsp;10s]]<br />[[6L&nbsp;16s]]
 | [[Antikythera]]
 |-
 | 7\34
 | 247.059
-| [[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[10L&nbsp;4s]]<br />[[10L&nbsp;14s]]
+| [[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[10L&nbsp;4s]]
 | [[Tobago]]
 |-
 | 8\34
 | 282.353
-| [[2L&nbsp;2s]]<br />[[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[4L&nbsp;10s]]<br />[[4L&nbsp;14s]]<br />4L&nbsp;18s<br />4L&nbsp;22s<br />Pathological 4L&nbsp;26s
+| [[2L&nbsp;2s]]<br />[[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[4L&nbsp;10s]]<br />[[4L&nbsp;14s]]
 | [[Bikleismic]]
 |}
@@ Line 758: / Line 751: @@
 * [[Diachrome]] (5:2:1)
 * [[Cthon5m]] (4:2:1)
+== Instruments ==
+=== Lumatone ===
+* [[Lumatone mapping for 34edo]]
+=== Skip fretting ===
+* [[Skip fretting system 34 2 9]]
+* [[Skip fretting system 34 2 11]]
 == Music ==
 === Modern renderings ===
+; {{W|Johann Sebastian Bach}}
+* [https://www.youtube.com/watch?v=Mni0bsUVgHk "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)
 ; {{W|Scott Joplin}}
 * [https://www.youtube.com/watch?v=CwMem5p1R6Y ''Maple Leaf Rag''] (1899) – rendered by Claudi Meneghin (2024)
@@ Line 781: / Line 785: @@
 * "Travel To Stay" from ''Mysteries'' (2023) – [https://open.spotify.com/track/2S2UlMKNNL5yB3280KpgvK Spotify] | [https://francium223.bandcamp.com/track/travel-to-stay Bandcamp] | [https://www.youtube.com/watch?v=fDkW1SnMcdw 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=ZrjbxQbdVw4 ''cucumber service''] (2025)
 ; [[Adam Freese]]
@@ Line 805: / Line 811: @@
 ; [[Tapeworm Saga]]
 * [https://www.youtube.com/watch?v=BhgxwP9_cSw ''A 3/4 piece in 34edo on 12/31/23''] (2023)
+; [[Sintel]]
+* [https://www.youtube.com/watch?v=hM7p_VVyeQ0 ''Diversion in 34edo''] (2021) – [https://www.youtube.com/watch?v=yTG0z4Znimw transcription by Stephen Weigel]
 ; [[Cam Taylor]]
@@ Line 833: / Line 842: @@
 == See also ==
-* [[Lumatone mapping for 34edo]]
-* [[Skip fretting system 34 2 9]]
 * [[Diaschismic-tetracot equivalence continuum]]