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.

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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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* [[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 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.
@@ Line 466: / 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 474: / 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 520: / 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 562: / 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 569: / 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 585: / 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 621: / 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 660: / 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 773: / 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 796: / 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 820: / 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 848: / Line 842: @@
 == See also ==
-* [[Lumatone mapping for 34edo]]
-* [[Skip fretting system 34 2 9]]
 * [[Diaschismic-tetracot equivalence continuum]]