34edo: Difference between revisions

{{EDO intro|34}}

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.

{{Harmonics in equal|34|intervals=odd}}

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

! Ratios of 7<br>Using the 34 Val

! Ratios of 7<br>Using the 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]]

| downaug 1sn, <br>upminor 2nd

| upmajor 2nd, <br>downminor 3rd

| upmajor 6th, <br>downminor 7th

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

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 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.

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.)

The following table shows how [[15-odd-limit intervals]] are represented in 34edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.

{{15-odd-limit|34}}

{{15-odd-limit|34.1|title=15-odd-limit intervals by 34d val mapping}}

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%.

== Approximation to 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 [[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" | [[Comma list|Comma List]]

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

! colspan="2" | Tuning Error

| -1.10

| -2.56

| -2.82

| -2.64

| -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 8ve

| 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]]

| [[Petrtri]]

| [[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

34edo [[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>

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

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

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

| Slendro diesis

<references />

== Notation ==

=== 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://soundcloud.com/floracanou/october-dieting-plan?in=floracanou/sets/totmc-suite-vol-1 "October Dieting Plan"] from [https://soundcloud.com/floracanou/sets/totmc-suite-vol-1 ''TOTMC Suite Vol. 1''] (2023) – [[modus]] in 34edo tuning

* [[Lumatone mapping for 34edo]]

* [[Skip fretting system 34 2 9]]