34edo: Difference between revisions

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.

! 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]].

=== 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.

{{Sharpness-sharp4a}}

 

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

{{Sharpness-sharp4}}

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]].

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.)

| steps = 34.0448410043159

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%.

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]].)

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.

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

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

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

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

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

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

| [[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 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

| [[16L 2s]]

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

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

| [[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]]

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

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

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

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

* [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]]

* [[Diaschismic-tetracot equivalence continuum]]