34edo: Difference between revisions

{{Infobox ET

| Prime factorization = 2 × 17

| Fifth = 20\34 = 705.88¢ (→[[17edo|10\17]])

| Major 2nd = 6\34 = 212¢

| Minor 2nd = 2\34 = 71¢

| Augmented 1sn = 4\34 = 141¢

'''34edo''' divides the octave into 34 equal steps of approximately 35.3 [[cent]]s.

== Introduction ==

{{Primes in edo|34|columns=11}}

34edo contains two [[17edo]]'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than [[31edo]], but with a sharp fifth rather than a flat one, and ''not'' tempering out [[81/80]] unlike 31edo.

{| class="wikitable center-all right-3 left-4 left-5 left-6"

! Degree

! Approx. Ratios of<br>2.3.5.11.13.17 [[Subgroup]]

! Additional Ratios of 7<br>Tending Flat (34 Val)

! Additional Ratios of 7<br>Tending Sharp (34d Val)

! colspan="3" | [[Ups and Downs Notation]]

|

| [[25/24]], [[648/625]], [[33/32]]

| double-up 1sn, minor 2nd

| downaug 1sn, upminor 2nd

| [[13/12]], [[12/11]]

| [[9/8]], [[17/15]]

| [[15/13]]

|

| upmajor 2nd, downminor 3rd

| [[20/17]], [[75/64]], [[13/11]]

|

|17/14

| [[51/40]], [[32/25]]

|

| [[13/10]], [[17/13]], [[22/17]]

| upmajor 3rd,down 4th

| [[27/20]], [[34/25]], [[15/11]]

| [[25/18]], [[18/13]], [[11/8]]

|  

| [[36/25]], [[13/9]], [[16/11]]

|  

| [[40/27]], [[25/17]], [[22/15]]

|  

| [[20/13]], [[26/17]], [[17/11]]

|  

| [[25/16]], [[80/51]]

|

| [[17/10]], [[128/75]], [[22/13]]

| [[26/15]]

|

| upmajor 6th, downminor 7th

| [[16/9]], [[30/17]]

| [[24/13]], [[11/6]]

| [[48/25]], [[625/324]],  [[64/33]]

| ^C#, 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]].

== Approximations to Just Intonation ==

Like [[17edo]], 34edo contains good approximations of just intervals involving 13, 11, and 3 – specifically, 13/8, 13/12, 13/11, 13/9, 11/9 and their inversions – while failing to closely approximate ratios of 7. 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 (frequently ratios of ~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.)

=== Selected just intervals by error ===

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

! Error (abs, [[Cent|¢]])

| [[15/13]], [[26/15]]

| [[18/13]], [[13/9]]

| '''[[5/4]], [[8/5]]'''

| '''1.922'''

| [[6/5]], [[5/3]]

| 2.006

|-

| [[13/12]], [[24/13]]

| 2.604

|-

| '''[[4/3]], [[3/2]]'''

| [[13/10]], [[20/13]]

| [[11/9]], [[18/11]]

| 5.533

| [[16/15]], [[15/8]]

| [[10/9]], [[9/5]]

| ''[[14/11]], [[11/7]]''

| '''[[16/13]], [[13/8]]'''

| [[13/11]], [[22/13]]

| [[15/11]], [[22/15]]

| 7.539

| [[9/8]], [[16/9]]

| 7.855

|-

| [[12/11]], [[11/6]]

| 9.461

| [[11/10]], [[20/11]]

| 11.466

| ''[[9/7]], [[14/9]]''

| ''[[14/13]], [[13/7]]''

| ''12.878''

| '''[[11/8]], [[16/11]]'''

| '''13.388'''

| ''[[15/14]], [[28/15]]''

| ''13.560''

| ''[[7/6]], [[12/7]]''

| ''15.482''

| '''[[8/7]], [[7/4]]'''

|-

| ''[[7/5]], [[10/7]]''

| ''17.488''

|}

{| class="wikitable center-all"

| [[15/13]], [[26/15]]

|-

|-

| '''[[5/4]], [[8/5]]'''

| [[9/8]], [[16/9]]

| ''[[14/11]], [[11/7]]''

| ''29.273''

=== Temperament measures ===

The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 34et.  

{| class="wikitable center-all"

! colspan="2" |

! colspan="2" |Octave stretch (¢)

! rowspan="2" |Error

! [[TE error|absolute]] (¢)

| 1.24

| 2.66

! [[TE simple badness|relative]] (%)

| 3.51

| 2.92

| 7.57

| 6.93

| 6.42

| 6.41

| 2.65

| 3.83

* 34et has a lower relative error than any previous ETs in the 5-limit. The next ET that does better in this subgroup is 53.  

* 34et is most prominent in the 2.3.5.13.17 and 2.3.5.11.13.17 subgroups. The next ET that does better in these subgroups is 217 and 87, respectively.

== MOS scales ==

{| class="wikitable"

! Periods<br>per octave

! Generator

! Cents

! MOSes

| 1

|  

| "

| 3\34

| 105.88

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

| "

| 5\34

| 176.471

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

| "

| 7\34

| 247.059

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

| "

| 9\34

| 317.647

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

| "

| 11\34

| 388.235

| 13\34

| 458.824

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

| [[2L 3s]]<br/> [[2L 5s]]<br/> [[7L 2s]]<br/> [[9L 7s]] <br/> 9L 16s

| 2

| 2\34

| 70.588

| [[16L 2s]]

| 105.882

| 6\34

| 211.765

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%.

== 34edo and 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 [[MOSScales|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 as a regular temperament ==

In the 5-limit, 34edo supports [[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 two temperaments ===

|+ Temperaments sorted by generator

! Periods<br>per octave

| 1

| 105.882

| [[Tetracot]]/[[Bunya]]/[[Monkey]]

| [[Immunity]]

| [[Hanson]]/[[Keemun]]

| [[Wuerschmidt]]/[[Worschmidt]]

| [[Chromatic_pairs#Petrtri|Petrtri]]

| [[Chromatic_pairs#Mabila|Mabila]]

| "

| [[Srutal]]/[[Pajara]]/[[Diaschismic]]

| [[Fifive]]

|  

|  

|  

|  

|-

34-EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[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>

! Name(s)

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

| Minimal diesis, tetracot comma

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

| Kleisma, semicomma majeur

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

| [[Vishnuzma]], semisuper comma

| Septimal diesis, slendro diesis

| Starling comma, septimal semicomma

| Ptolemisma, Ptolemy's comma

| Rastma, neutral third comma

== Notations ==

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.

* [http://www.archive.org/details/Ascension_105 Ascension]

* [https://www.youtube.com/watch?v=FXTM0HeuExk Uncomfortable In Crowds (extended)] by [[Robin Perry]]

* [[:File:A stroll through some retuned maqams.mp3|A stroll through some retuned maqams]] by [[Ray Perlner]]

* [https://www.youtube.com/watch?v=5M_Kh94_t1Q Perspective] by [[User:Userminusone|Userminusone]]

== Links ==

* [https://microstick.net http://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)

[[Category:34edo]]

[[Category:34et]]

[[Category:diaschismic]]

[[Category:Equal divisions of the octave]]

[[Category:selenium]]