34edo: Difference between revisions

{{Infobox ET

| Prime factorization = 2 × 17

| Fifth = 20\34 (705.9¢) (→[[17edo|10\17]])

| Semitones = 4:2 (141.2¢ : 70.6¢)

| Consistency = 5

'''34edo''' divides the octave into 34 equal steps of approximately 35.3 [[cent]]s.

== Introduction ==

{{Odd harmonics in edo|34|}}

34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than [[31edo]], but with a sharp fifth rather than a flat one, and ''not'' tempering out [[81/80]] unlike 31edo.

{| class="wikitable center-all right-3 left-4 left-5 left-6"

! Degree

! Approx. Ratios of<br>2.3.5.11.13.17.23 [[subgroup]]

! Additional Ratios of 7<br>Tending Flat (34 Val)

! Additional Ratios of 7<br>Tending Sharp (34d Val)

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

|

| [[81/80]], [[128/125]], [[51/50]]

| [[25/24]], [[26/25]], [[24/23]], [[27/26]], [[23/22]], [[648/625]], [[33/32]]

|

|

|17/14

|

| upmajor 3rd,down 4th

|  

|  

|  

|  

|

|

| upmajor 6th, downminor 7th

| [[48/25]], [[25/13]], [[23/12]], [[625/324]],  [[64/33]]

| ^C#, vD

|

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]].

== JI approximation ==

Like [[17edo]], 34edo contains good approximations of just intervals involving 13, 11, and 3 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7. 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 (frequently ratios of ~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.

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.)

=== Selected just intervals by error ===

{| class="wikitable center-all"

|-

! Error (abs, [[Cent|¢]])

|-

| [[15/13]], [[26/15]]

| [[18/13]], [[13/9]]

|-

| '''1.922'''

| [[6/5]], [[5/3]]

| 2.006

|-

| [[13/12]], [[24/13]]

| '''[[4/3]], [[3/2]]'''

| '''3.927'''

| [[13/10]], [[20/13]]

| [[11/9]], [[18/11]]

| 5.533

|-

| [[16/15]], [[15/8]]

| 5.849

|-

| [[10/9]], [[9/5]]

| ''[[14/11]], [[11/7]]''

| '''[[16/13]], [[13/8]]'''

| [[13/11]], [[22/13]]

| [[15/11]], [[22/15]]

| 7.539

| [[9/8]], [[16/9]]

| [[12/11]], [[11/6]]

| [[11/10]], [[20/11]]

| 11.466

| ''[[9/7]], [[14/9]]''

| ''11.555''

|-

| ''[[14/13]], [[13/7]]''

| ''12.878''

| '''[[11/8]], [[16/11]]'''

| '''13.388'''

| ''[[15/14]], [[28/15]]''

| ''13.560''

| ''[[7/6]], [[12/7]]''

| ''15.482''

| '''[[8/7]], [[7/4]]'''

|-

| ''[[7/5]], [[10/7]]''

| ''17.488''

|}

{| class="wikitable center-all"

| [[15/13]], [[26/15]]

|-

|-

| '''[[5/4]], [[8/5]]'''

| [[9/8]], [[16/9]]

| ''[[14/11]], [[11/7]]''

| ''29.273''

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%.

== 34edo and logarithmic phi ==

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 [[MOSScales|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]].)

! rowspan="2" | Subgroup

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

| [{{val| 34 54 79 }}]

| -1.10

| [{{val| 34 54 79 96 }}] (34d)

| -2.56

| [{{val| 34 54 79 96 118 }}] (34d)

| -2.82

| [{{val| 34 54 79 96 118 126 }}] (34d)

| -2.64

| [{{val| 34 54 79 96 118 126 139 }}] (34d)

| -2.30

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.  

|+ Rank-2 temperaments by period and generator

! Periods<br>per octave

! MOSes

| 1

| [[11L 1s]]<br/> [[11L 12s]]

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

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

| [[5L 4s]] <br/> [[5L 9s]] <br/> [[5L 14s]] <br/> [[5L 19s]] <br/>Pathological 5L 24s

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

| [[4L 3s]]<br/> [[4L 7s]]<br/> [[4L 11s]]<br/> [[15L 4s]]

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

| [[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

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

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

| [[2L 3s]]<br/> [[2L 5s]]<br/> [[7L 2s]]<br/> [[9L 7s]] <br/> 9L 16s

| 2

| [[16L 2s]]

| [[2L 6s]]<br/>[[2L 8s]]<br/> [[10L 2s]]<br/> [[12L 10s]]

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

| [[2L 6s]]<br/> [[8L 2s]]<br/> [[8L 10s]] <br/> 8L 16s

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

| [[6L 2s]]<br/> [[6L 8s]]<br/> [[14L 6s]]

| [[4L 2s]] <br/> [[6L 4s]] <br/> [[6L 10s]] <br/> [[6L 16s]] <br/> Pathological 6L 22s

| [[4L 2s]] <br/> [[4L 6s]] <br/> [[10L 4s]] <br/> [[10L 14s]]

| [[Chromatic_pairs#Tobago|Tobago]]

| [[2L 2s]] <br/> [[4L 2s]] <br/> [[4L 6s]] <br/> [[4L 10s]] <br/> [[4L 14s]] <br/> 4L 18s <br/> 4L 22s <br/> Pathological 4L 26s

|  

34-EDO [[tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 34 54 79 95 118 126 }}.

! [[Harmonic limit|Prime<br>Limit]]

! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>

! Name(s)

| [[134217728/129140163|(18 digits)]]

| Minimal diesis, tetracot comma

| [[393216/390625|(12 digits)]]

| Kleisma, semicomma majeur

| [[6115295232/6103515625|(20 digits)]]

| [[Vishnuzma]], semisuper comma

| Septimal diesis, slendro diesis

| Starling comma, septimal semicomma

| Ptolemisma, Ptolemy's comma

| Rastma, neutral third comma

<references/>

== Notations ==

=== Kosmorsky's thoughts ===

== 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 [[strict contrary motion]] in 34edo.

* [https://www.archive.org/details/Ascension_105 Ascension]

* [https://www.youtube.com/watch?v=FXTM0HeuExk Uncomfortable In Crowds (extended)] by [[Robin Perry]]

* [[:File:A stroll through some retuned maqams.mp3|A stroll through some retuned maqams]] by [[Ray Perlner]]

* [https://www.youtube.com/watch?v=5M_Kh94_t1Q Perspective] by [[User:Userminusone|Userminusone]]

* [https://www.youtube.com/watch?v=xYZwye9PWSo Modal Studies in Tetracot] by [[User:Harmonic_witch|Zhea Erose]]

* [[Lumatone mapping for 34edo]]

* [https://microstick.net http://microstick.net/] websites of Neil Haverstick

* https://myspace.com/microstick -- somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything)