34edo: Difference between revisions

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{{Wikipedia| 34 equal temperament }}

== 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).  

 

{| class="wikitable center-all right-2 left-3 left-4 left-5"

!  

! 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 34d Val

! colspan="3" | [[Ups and downs notation]]

! colspan="2" | [[Solfege|Solfeges]]

|  

| downaug 1sn,<br />upminor 2nd

|  

| upmajor 2nd,<br />downminor 3rd

|  

| 17/14

|  

| upmajor 3rd, down 4th

|

|

|

|

|  

|  

| upmajor 6th,<br />downminor 7th

| 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]].

 

 

 

 

 

 

 

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.

== Approximation to JI ==

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{{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{{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 ===

{{Q-odd-limit intervals|34}}

{{Q-odd-limit intervals|34.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 34d val mapping}}

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.

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{{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]].)

 

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

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

| {{mapping| 34 54 79 }}

| −1.10

| {{mapping| 34 54 79 96 }} (34d)

| −2.56

| {{mapping| 34 54 79 96 118 }} (34d)

| −2.82

| {{mapping| 34 54 79 96 118 126 }} (34d)

| −2.64

| {{mapping| 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.  

 

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

! Periods<br />per 8ve

! Mosses

| rowspan="8" style="text-align: center;" | 1

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

| [[6L&nbsp;1s]]<br />[[7L&nbsp;6s]]<br />[[7L&nbsp;13s]]

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

| [[5L&nbsp;4s]]<br />[[5L&nbsp;9s]]<br />[[5L&nbsp;14s]]<br />[[5L&nbsp;19s]]

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

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

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

| [[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)

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

| [[2L&nbsp;3s]]<br />[[2L&nbsp;5s]]<br />[[7L&nbsp;2s]]<br />[[9L&nbsp;7s]]

| rowspan="8" style="text-align: center;" | 2

| [[16L&nbsp;2s]]

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

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

| [[2L&nbsp;6s]]<br />[[8L&nbsp;2s]]<br />[[8L&nbsp;10s]]

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

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

| [[4L&nbsp;2s]]<br />[[6L&nbsp;4s]]<br />[[6L&nbsp;10s]]<br />[[6L&nbsp;16s]]

| [[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[10L&nbsp;4s]]

| [[Tobago]]

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

| [[Bikleismic]]

34et [[tempering out|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 group="note">{{rd}}</ref>

! Name

| <abbr title="134217728/129140163">(18 digits)</abbr>

| Tetracot comma

| <abbr title="393216/390625">(12 digits)</abbr>

| Kleisma

| <abbr title="6115295232/6103515625">(20 digits)</abbr>

| [[Vishnuzma]]

| Semaphoresma, slendro diesis

| Starling comma

| Ptolemisma

| Rastma

== Scales ==

=== MOS scales ===

{{main|List of MOS scales in 34edo}}

* [[Blackdye]] (5:3:1)

* [[Diachrome]] (5:2:1)

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

== Instruments ==

* [[Lumatone mapping for 34edo]]

* [[Skip fretting system 34 2 9]]

* [[Skip fretting system 34 2 11]]

; [[Zhea Erose]]

* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021)

* [[Diaschismic-tetracot equivalence continuum]]

* [https://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)