34edo: Difference between revisions

== Intervals ==

|-

! Cents

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|

|-

|-

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| vD#, ^Eb

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| D#, vvE

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|  

|-

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| 7

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|-

| minor 3rd

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| 9

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|17/14

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| mu

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| mi

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|

|-

| ^M3, v4

| ^F#, vG

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|-

| 15

| ^G

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| ~4, d5

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|

|-

|-

| sih

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| 705.882

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| 741.176

| up 5th, downminor 6th

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| 776.471

|

|-

| 811.765

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| lu

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| la

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| M6

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|

|-

|-

|  

| upminor 7th

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| tu

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| ti

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| 33

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|

| D

|}

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]].

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.

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|34.1|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%.

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]].)

{| class="wikitable center-4 center-5 center-6"

|-

|-

| 2.3.5

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| -2.56

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| -2.82

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| -2.64

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| -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.  

{| class="wikitable"

|-

! Generator

! Temperaments

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|  

|-

|-

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

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

|-

|-

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

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|-

| [[Mabila]]

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| [[16L 2s]]

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

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|-

| [[Antikythera]]

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| [[Bikleismic]]

|}

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

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

|-

! Name(s)

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| 66.765

| Sasawa

| [[17-comma]]

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| Saquadyo

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| Sagugu

| Diaschisma

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| Saquadbigu

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|-

| 3.338

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| Trizogu

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| 35.697

| Zozo

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| 21.902

| Zotriyo

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| Zotrigu

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| Luyoyo

| Ptolemisma, Ptolemy's comma

|-

|-

| 4.503

| Keenanisma

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| 19.120

| Thozogu

|}

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.

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.youtube.com/watch?v=Dcpkrci64Ms look]'' (2023)

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

[[Category:Diaschismic]]