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.

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).  

| [[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

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

=== 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.

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.

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.  

* [[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

* [[Diaschismic-tetracot equivalence continuum]]