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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! Cents

! Approx. ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.23 [[subgroup]]}}</ref>

! Ratios of 7 Using the 34 Val

! Ratios of 7 Using the 34 Val

! Ratios of 7 Using the 34d 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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| [[15/14]], [[21/20]]

| vA1, ^m2

| downaug 1sn, upminor 2nd

| downaug 1sn, upminor 2nd

| vD#, ^Eb

| fru

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|

| ^M2, vm3

| upmajor 2nd, downminor 3rd

| upmajor 2nd, downminor 3rd

| ^E, vF

| ru/no

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|

| ^M6, vm7

| upmajor 6th, downminor 7th

| upmajor 6th, downminor 7th

| ^B, vC

| lu/tho

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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 ==

=== Ups and downs 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.

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

=== 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]].

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</imagemap>

=== ~~Ups and downs notation~~ ===

=== Kosmorsky's thoughts ===

~~Another way~~ to notate the ~~microtonal notes~~ is to ~~use [[Alternative symbols~~ for ~~ups~~ and ~~downs notation#Sharp-4|ups and downs]]. Here~~, ~~this can~~ be ~~done using sharps~~ and ~~flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:~~

The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away – the double-sharp means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which ''may not be most efficient expressions of some musical purposes.'' Users can construct their own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C … F, instead of seven, might be waste – of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended.

~~{{Sharpness-sharp4}}~~

== Approximation to JI ==

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Like [[17edo]], 34edo contains good approximations of just intervals involving 3, 11, and 13 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7 given its step size. 34edo adds ratios of 5 into the mix – including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions – as well as 17 – including 17/16, 18/17, 17/12, 17/11, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the [[syntonic comma]] of 81/80, from 21.5 cents to 35.3 cents), it is suitable for quasi-5-limit JI but is not a [[meantone]] system. While no number of fifths (3/2) land on major or minor thirds, an even number of major or minor thirds will be the same pitch as a pitch somewhere in the circle of seventeen fifths.

The sharpening of ~13 ~~cents~~ of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.

The sharpening of ~13{{c}} of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.

Likewise the 16~~-cent~~ flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L 3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)

Likewise the 16{{c}} flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L 3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)

=== Interval mappings ===

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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.

~~{| class="wikitable center~~-~~all"~~

|-

~~! colspan="3" | Tuning~~

~~! colspan="3" | Strength~~

~~! colspan="2" | Closest edo~~

~~! colspan="2" | Integer~~ limit

|-

~~! ZPI~~

~~! Steps per octave~~

~~! Step size (cents)~~

~~! Height~~

~~! Integral~~

~~! Gap~~

~~! Edo~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[144zpi]]~~

~~| 34.0448410043159~~

~~| 35.2476312005063~~

~~| 6.685147~~

~~| 1.241437~~

~~| 16.236989~~

~~| 34edo~~

~~| 1198~~.~~41946081721~~

~~| 6~~

|}

== Tuning by ear ==

In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 ~~cents~~, and using a pure 5/4 which is less than 2 ~~cents~~ off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 ~~cents~~, or a relative error of less than 10%.

In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5{{c}}, and using a pure 5/4 which is less than 2{{c}} off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5{{c}}, or a relative error of less than 10%.

== Approximation to irrational intervals ==

As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2 ~~cents~~. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 ~~cents~~, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)

As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2{{c}}. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833{{c}}, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)

=== Counterpoint ===

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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|}

In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[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. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports 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.

In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[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. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports 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 ===

=== Rank-2 temperaments ===

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{| class="wikitable"

|+ Rank-2 temperaments by period and generator

|+ style="font-size: 105%;" | Rank-2 temperaments by period and generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator

! Cents

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| 3\34

| 105.88

| [[11L 1s]] [[11L 12s]]

| [[11L 1s]] [[11L 12s]]

|

|-

| 5\34

| 176.471

| [[6L 1s]] [[7L 6s]] [[7L 13s]] ~~ 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)

|-

| 9\34

| 317.647

| [[4L 3s]] [[4L 7s]] [[4L 11s]] [[15L 4s]]

| [[4L 3s]] [[4L 7s]] [[4L 11s]] [[15L 4s]]

| [[Hanson]], [[keemun]] (34), [[catalan]] (34d), [[catakleismic]] (34d)

|-

| 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)

|-

| 13\34

| 458.824

| [[3L 2s]] [[5L 3s]] [[8L 5s]] [[13L 8s]]

| [[3L 2s]] [[5L 3s]] [[8L 5s]] [[13L 8s]]

| [[Petrtri]]

|-

| 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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| 2\34

| 70.588

| [[16L 2s]]

| [[16L 2s]]

| [[Vishnu]]

|-

| 3\34

| 105.882

| [[2L 6s]] [[2L 8s]] [[10L 2s]] [[12L 10s]]

| [[2L 6s]] [[2L 8s]] [[10L 2s]] [[12L 10s]]

| [[Srutal]] (34d), [[pajara]] (34d), [[diaschismic]] (34)

|-

| 4\34

| 141.176

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

| [[2L 6s]] [[8L 2s]] [[8L 10s]]

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

|-

| 5\34

| 176.471

| [[6L 2s]] [[6L 8s]] [[14L 6s]]

| [[6L 2s]] [[6L 8s]] [[14L 6s]]

| [[Stratosphere]]

|-

| 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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{| class="commatable wikitable center-all left-3 right-4 left-6"

|-

! [[Harmonic limit|Prime limit]]

! [[Harmonic limit|Prime limit]]

! [[Ratio]]<ref group="note">{{rd}}</ref>

! [[Monzo]]

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| Superleap

|}

~~== Notation ==~~

~~=== Kosmorsky's thoughts ===~~

The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away – the double-sharp means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which ''may not be most efficient expressions of some musical purposes.'' Users can construct their own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C … F, instead of seven, might be waste – of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended.

== Scales ==

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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]]~~

* [[Diaschismic-tetracot equivalence continuum]]

* [[Skip fretting system 34 2 9]]

== External links ==

@@ Line 6: / Line 6: @@
 }}
 {{Infobox ET}}
-{{EDO intro|34}}
+{{ED intro}}
 {{Wikipedia| 34 equal temperament }}
 == 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 21: / Line 25: @@
 ! Cents
 ! Approx. ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.23 [[subgroup]]}}</ref>
-! 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 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 65: / Line 70: @@
 | [[15/14]], [[21/20]]
 | vA1, ^m2
-| downaug 1sn, <br>upminor 2nd
+| downaug 1sn,<br />upminor 2nd
 | vD#, ^Eb
 | fru
@@ Line 109: / Line 114: @@
 |
 | ^M2, vm3
-| upmajor 2nd, <br>downminor 3rd
+| upmajor 2nd,<br />downminor 3rd
 | ^E, vF
 | ru/no
@@ Line 329: / Line 334: @@
 |
 | ^M6, vm7
-| upmajor 6th, <br>downminor 7th
+| upmajor 6th,<br />downminor 7th
 | ^B, vC
 | lu/tho
@@ 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 ==
+=== Ups and downs 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.
+{{Sharpness-sharp4a}}
+[[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]].
@@ Line 451: / Line 463: @@
 </imagemap>
-=== Ups and downs notation ===
+=== Kosmorsky's thoughts ===
-Another way to notate the microtonal notes is to use [[Alternative symbols for ups and downs notation#Sharp-4|ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away – the double-sharp means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which ''may not be most efficient expressions of some musical purposes.'' Users can construct their own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C … F, instead of seven, might be waste – of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended.
-{{Sharpness-sharp4}}
 == Approximation to JI ==
@@ Line 460: / Line 470: @@
 Like [[17edo]], 34edo contains good approximations of just intervals involving 3, 11, and 13 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7 given its step size. 34edo adds ratios of 5 into the mix – including 5/4, 6/5, 9/5, 15/8, 13/10, 15/13, and their inversions – as well as 17 – including 17/16, 18/17, 17/12, 17/11, 17/10, 17/13, 17/15 and their inversions. Since it distinguishes between 9/8 and 10/9 (exaggerating the difference between them, the [[syntonic comma]] of 81/80, from 21.5 cents to 35.3 cents), it is suitable for quasi-5-limit JI but is not a [[meantone]] system. While no number of fifths (3/2) land on major or minor thirds, an even number of major or minor thirds will be the same pitch as a pitch somewhere in the circle of seventeen fifths.
-The sharpening of ~13 cents of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.
+The sharpening of ~13{{c}} of 11/8 can fit with the 9/8 and 13/8 which both are about 7 cents sharp. This is the basis of a subtle trick: the guitarist tunes the high 'E' string flat by several cents, enough to be imperceptible in many contexts, but which makes chords/harmonies against those several intervals tuned more justly.
-Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L 3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)
+Likewise the 16{{c}} flat 27\34 approximate 7/4 can be musically useful especially in [[kleismic]] or [[4L&nbsp;3s]] contexts (with generator a 9\34 minor third). On the other hand, the slightly worse and sharper 7/4, 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] contains a copy of 34edo and has the intervals 7/4 and 11/8 tuned nearly just.)
 === Interval mappings ===
@@ Line 468: / 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.
-{| class="wikitable center-all"
-|-
-! colspan="3" | Tuning
-! colspan="3" | Strength
-! colspan="2" | Closest edo
-! colspan="2" | Integer limit
-|-
-! ZPI
-! Steps per octave
-! Step size (cents)
-! Height
-! Integral
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[144zpi]]
-| 34.0448410043159
-| 35.2476312005063
-| 6.685147
-| 1.241437
-| 16.236989
-| 34edo
-| 1198.41946081721
-| 6
-| 6
-|}
 == Tuning by ear ==
-In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5 cents, and using a pure 5/4 which is less than 2 cents off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5 cents, or a relative error of less than 10%.
+In principle, one can approximate 34edo by ear using only 5-limit intervals, using the fact that 17edo is very close to a circle of seventeen [[25/24]] chromatic semitones to within 1.5{{c}}, and using a pure 5/4 which is less than 2{{c}} off for the second chain. The overall tuning error, assuming everything is tuned perfectly, will be less than 3.5{{c}}, or a relative error of less than 10%.
 == Approximation to irrational intervals ==
-As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)
+As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the [[logarithmic phi]] – 21 degrees of 34edo, approximately 741.2{{c}}. Repeated iterations of this interval generates [[moment of symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo| -6 2 6 0 0 -13 }}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833{{c}}, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].)
 === Counterpoint ===
@@ Line 514: / Line 495: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 556: / Line 537: @@
 |}
-In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[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. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports 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.
+In the 5-limit, 34edo [[support]]s [[hanson]], [[srutal]], [[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. By way of the patent val 34 supports [[keemun]] temperament, and 34d is an excellent alternative to [[22edo]] for 7-limit [[pajara]] temperament. In the [[11-limit]], 34de supports 11-limit [[pajaric]], and in fact is quite close to the [[POTE tuning]]; it adds [[4375/4374]] to the commas of 11-limit pajaric. On the other hand, the 34d val supports 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 map|edo=34}}
 === Rank-2 temperaments ===
@@ Line 563: / Line 547: @@
 {| class="wikitable"
-|+ Rank-2 temperaments by period and generator
+|+ style="font-size: 105%;" | Rank-2 temperaments by period and generator
 |-
-! Periods<br>per 8ve
+! Periods<br />per 8ve
 ! Generator
 ! Cents
@@ Line 579: / Line 563: @@
 | 3\34
 | 105.88
-| [[11L 1s]]<br> [[11L 12s]]
+| [[11L&nbsp;1s]]<br />[[11L&nbsp;12s]]
 |
 |-
 | 5\34
 | 176.471
-| [[6L 1s]]<br> [[7L 6s]] <br> [[7L 13s]] <br> 7L 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 4s]] <br> [[5L 9s]] <br> [[5L 14s]] <br> [[5L 19s]] <br>Pathological 5L 24s
+| [[5L&nbsp;4s]]<br />[[5L&nbsp;9s]]<br />[[5L&nbsp;14s]]<br />[[5L&nbsp;19s]]
 | [[Immunity]] (34), [[immunized]] (34d)
 |-
 | 9\34
 | 317.647
-| [[4L 3s]]<br> [[4L 7s]]<br> [[4L 11s]]<br> [[15L 4s]]
+| [[4L&nbsp;3s]]<br />[[4L&nbsp;7s]]<br />[[4L&nbsp;11s]]<br />[[15L&nbsp;4s]]
 | [[Hanson]], [[keemun]] (34), [[catalan]] (34d), [[catakleismic]] (34d)
 |-
 | 11\34
 | 388.235
-| [[3L 7s]]<br> [[3L 10s]]<br> [[3L 13s]]<br> [[3L 16s]]<br> [[3L 19s]] <br>[[3L 22s]]<br> Pathological [[3L 25s]] <br> Pathological 3L 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)
 |-
 | 13\34
 | 458.824
-| [[3L 2s]]<br> [[5L 3s]]<br> [[8L 5s]]<br> [[13L 8s]]
+| [[3L&nbsp;2s]]<br />[[5L&nbsp;3s]]<br />[[8L&nbsp;5s]]<br />[[13L&nbsp;8s]]
 | [[Petrtri]]
 |-
 | 15\34
 | 529.412
-| [[2L 3s]]<br> [[2L 5s]]<br> [[7L 2s]]<br> [[9L 7s]] <br> 9L 16s
+| [[2L&nbsp;3s]]<br />[[2L&nbsp;5s]]<br />[[7L&nbsp;2s]]<br />[[9L&nbsp;7s]]
 | [[Mabila]]
 |-
@@ Line 615: / Line 599: @@
 | 2\34
 | 70.588
-| [[16L 2s]]
+| [[16L&nbsp;2s]]
 | [[Vishnu]]
 |-
 | 3\34
 | 105.882
-| [[2L 6s]]<br>[[2L 8s]]<br> [[10L 2s]]<br> [[12L 10s]]
+| [[2L&nbsp;6s]]<br />[[2L&nbsp;8s]]<br />[[10L&nbsp;2s]]<br />[[12L&nbsp;10s]]
 | [[Srutal]] (34d), [[pajara]] (34d), [[diaschismic]] (34)
 |-
 | 4\34
 | 141.176
-| [[2L 6s]]<br> [[8L 2s]]<br> [[8L 10s]] <br> 8L 16s
+| [[2L&nbsp;6s]]<br />[[8L&nbsp;2s]]<br />[[8L&nbsp;10s]]
 | [[Fifive]], [[crepuscular]] (34d), [[fifives]] (34)
 |-
 | 5\34
 | 176.471
-| [[6L 2s]]<br> [[6L 8s]]<br> [[14L 6s]]
+| [[6L&nbsp;2s]]<br />[[6L&nbsp;8s]]<br />[[14L&nbsp;6s]]
 | [[Stratosphere]]
 |-
 | 6\34
 | 211.765
-| [[4L 2s]] <br> [[6L 4s]] <br> [[6L 10s]] <br> [[6L 16s]] <br> Pathological 6L 22s
+| [[4L&nbsp;2s]]<br />[[6L&nbsp;4s]]<br />[[6L&nbsp;10s]]<br />[[6L&nbsp;16s]]
 | [[Antikythera]]
 |-
 | 7\34
 | 247.059
-| [[4L 2s]] <br> [[4L 6s]] <br> [[10L 4s]] <br> [[10L 14s]]
+| [[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[10L&nbsp;4s]]
 | [[Tobago]]
 |-
 | 8\34
 | 282.353
-| [[2L 2s]] <br> [[4L 2s]] <br> [[4L 6s]] <br> [[4L 10s]] <br> [[4L 14s]] <br> 4L 18s <br> 4L 22s <br> Pathological 4L 26s
+| [[2L&nbsp;2s]]<br />[[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[4L&nbsp;10s]]<br />[[4L&nbsp;14s]]
 | [[Bikleismic]]
 |}
@@ Line 654: / Line 638: @@
 {| class="commatable wikitable center-all left-3 right-4 left-6"
 |-
-! [[Harmonic limit|Prime<br>limit]]
+! [[Harmonic limit|Prime<br />limit]]
 ! [[Ratio]]<ref group="note">{{rd}}</ref>
 ! [[Monzo]]
@@ Line 759: / Line 743: @@
 | Superleap
 |}
-== Notation ==
-=== Kosmorsky's thoughts ===
-The chain of fifths gives you the seven naturals, and their sharps and flats. The sharp or flat of a note is (what is commonly called) a neutral second away – the double-sharp means a minor third away from the natural. This has led certain "complainers", in seeking to notate 17 edo, to create an extra character to raise something a small step of which. To render this symbol philosophically harmonious with 34 tone equal temperament, a symbol indicating an adjustment of 1/34 up or down serves the purpose by using two of it, doubled laterally or vertically as composer. This however emphasizes certain aspects of 34edo which ''may not be most efficient expressions of some musical purposes.'' Users can construct their own notation to the needs of the music and performer. As an example, a system with 15 "nominals" like A, B, C … F, instead of seven, might be waste – of paper, or space, or memory if they aren't used consecutively frequently. The system spelled out here has familiarity as an advantage and disadvantage. The spacing of the nominals and lines is the same. Dense chords of certain types would be very impossible to notate. Finally, the table uses ^ and v for "up" and "down", but these might be reserved for adjustments of 1/68th of an octave, being hollow, and filled in triangles are recommended.
 == Scales ==
@@ Line 771: / 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 794: / 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 818: / 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 846: / Line 842: @@
 == See also ==
-* [[Lumatone mapping for 34edo]]
+* [[Diaschismic-tetracot equivalence continuum]]
-* [[Skip fretting system 34 2 9]]
 == External links ==