34edo: Difference between revisions
Jump to navigation
Jump to search
→Intervals: Removed 7mus column |
updated ups/down notation, added color names to the comma list, general cleanup |
||
| Line 1: | Line 1: | ||
= Theory = | |||
34edo divides the octave into 34 equal steps of approximately 35.29412 [[cent|cent]]s. 34edo contains two [[17edo|17edo]]'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than [[31edo|31edo]], but with a sharp fifth rather than a flat one, and 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|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. | 34edo divides the octave into 34 equal steps of approximately 35.29412 [[cent|cent]]s. 34edo contains two [[17edo|17edo]]'s and the half-octave tritone of 600 cents. It excels as a 5-limit system, with tuning even more accurate than [[31edo|31edo]], but with a sharp fifth rather than a flat one, and 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|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. | ||
=Intervals= | =Intervals= | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| Line 153: | Line 43: | ||
| style="text-align:center;" | ^1, vm2 | | style="text-align:center;" | ^1, vm2 | ||
| style="text-align:center;" | up unison, downminor 2nd | | style="text-align:center;" | up unison, downminor 2nd | ||
| style="text-align:center;" | D | | style="text-align:center;" | ^D, vEb | ||
|- | |- | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
| Line 176: | Line 66: | ||
| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| style="text-align:center;" | upminor 2nd | | style="text-align:center;" | upminor 2nd | ||
| style="text-align:center;" | Eb | | style="text-align:center;" | ^Eb | ||
|- | |- | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| Line 187: | Line 77: | ||
| style="text-align:center;" | ~2 | | style="text-align:center;" | ~2 | ||
| style="text-align:center;" | mid 2nd | | style="text-align:center;" | mid 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | vvE | ||
|- | |- | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| Line 199: | Line 89: | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| style="text-align:center;" | downmajor 2nd | | style="text-align:center;" | downmajor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | vE | ||
|- | |- | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| Line 222: | Line 112: | ||
| style="text-align:center;" | ^M2, vm3 | | style="text-align:center;" | ^M2, vm3 | ||
| style="text-align:center;" | upmajor 2nd, downminor 3rd | | style="text-align:center;" | upmajor 2nd, downminor 3rd | ||
| style="text-align:center;" | E | | style="text-align:center;" | ^E, vF | ||
|- | |- | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| Line 245: | Line 135: | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| style="text-align:center;" | upminor 3rd | | style="text-align:center;" | upminor 3rd | ||
| style="text-align:center;" | F | | style="text-align:center;" | ^F | ||
|- | |- | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| Line 256: | Line 146: | ||
| style="text-align:center;" | ~3 | | style="text-align:center;" | ~3 | ||
| style="text-align:center;" | mid 3rd | | style="text-align:center;" | mid 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^F | ||
|- | |- | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| Line 268: | Line 158: | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| style="text-align:center;" | downmajor 3rd | | style="text-align:center;" | downmajor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | vF# | ||
|- | |- | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| Line 291: | Line 181: | ||
| style="text-align:center;" | ^M3, v4 | | style="text-align:center;" | ^M3, v4 | ||
| style="text-align:center;" | upmajor 3rd,down 4th | | style="text-align:center;" | upmajor 3rd,down 4th | ||
| style="text-align:center;" | F# | | style="text-align:center;" | ^F#, vG | ||
|- | |- | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| Line 314: | Line 204: | ||
| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| style="text-align:center;" | up 4th | | style="text-align:center;" | up 4th | ||
| style="text-align:center;" | G | | style="text-align:center;" | ^G | ||
|- | |- | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| Line 323: | Line 213: | ||
| style="text-align:center;" | 36/25, 18/13 | | style="text-align:center;" | 36/25, 18/13 | ||
| style="text-align:center;" | 11/8 | | style="text-align:center;" | 11/8 | ||
| style="text-align:center;" | | | style="text-align:center;" | ~4, d5 | ||
| style="text-align:center;" | | | style="text-align:center;" | mid 4th, dim 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^G, Ab | ||
|- | |- | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| Line 337: | Line 227: | ||
| style="text-align:center;" | vA4, ^d5 | | style="text-align:center;" | vA4, ^d5 | ||
| style="text-align:center;" | downaug 4th, updim 5th | | style="text-align:center;" | downaug 4th, updim 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | vG#, ^Ab | ||
|- | |- | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
| Line 346: | Line 236: | ||
| style="text-align:center;" | 25/18, 13/9 | | style="text-align:center;" | 25/18, 13/9 | ||
| style="text-align:center;" | 16/11 | | style="text-align:center;" | 16/11 | ||
| style="text-align:center;" | A4, | | style="text-align:center;" | A4, ~5 | ||
| style="text-align:center;" | aug 4th, | | style="text-align:center;" | aug 4th, mid 5th | ||
| style="text-align:center;" | G#, | | style="text-align:center;" | G#, vvA | ||
|- | |- | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
| Line 360: | Line 250: | ||
| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| style="text-align:center;" | down 5th | | style="text-align:center;" | down 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | vA | ||
|- | |- | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
| Line 383: | Line 273: | ||
| style="text-align:center;" | ^5, vm6 | | style="text-align:center;" | ^5, vm6 | ||
| style="text-align:center;" | up 5th, downminor 6th | | style="text-align:center;" | up 5th, downminor 6th | ||
| style="text-align:center;" | A | | style="text-align:center;" | ^A, vBb | ||
|- | |- | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| Line 406: | Line 296: | ||
| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| style="text-align:center;" | upminor 6th | | style="text-align:center;" | upminor 6th | ||
| style="text-align:center;" | Bb | | style="text-align:center;" | ^Bb | ||
|- | |- | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
| Line 417: | Line 307: | ||
| style="text-align:center;" | ~6 | | style="text-align:center;" | ~6 | ||
| style="text-align:center;" | mid 6th | | style="text-align:center;" | mid 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | vvB | ||
|- | |- | ||
| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
| Line 429: | Line 319: | ||
| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| style="text-align:center;" | downmajor 6th | | style="text-align:center;" | downmajor 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | vB | ||
|- | |- | ||
| style="text-align:center;" | 26 | | style="text-align:center;" | 26 | ||
| Line 452: | Line 342: | ||
| style="text-align:center;" | ^M6, vm7 | | style="text-align:center;" | ^M6, vm7 | ||
| style="text-align:center;" | upmajor 6th, downminor 7th | | style="text-align:center;" | upmajor 6th, downminor 7th | ||
| style="text-align:center;" | B | | style="text-align:center;" | ^B, vC | ||
|- | |- | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| Line 475: | Line 365: | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| style="text-align:center;" | upminor 7th | | style="text-align:center;" | upminor 7th | ||
| style="text-align:center;" | C | | style="text-align:center;" | ^C | ||
|- | |- | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
| Line 486: | Line 376: | ||
| style="text-align:center;" | ~7 | | style="text-align:center;" | ~7 | ||
| style="text-align:center;" | mid 7th | | style="text-align:center;" | mid 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^C | ||
|- | |- | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| Line 498: | Line 388: | ||
| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| style="text-align:center;" | downmajor 7th | | style="text-align:center;" | downmajor 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | vC# | ||
|- | |- | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
| Line 521: | Line 411: | ||
| style="text-align:center;" | ^M7, v8 | | style="text-align:center;" | ^M7, v8 | ||
| style="text-align:center;" | upmajor 7th, down 8ve | | style="text-align:center;" | upmajor 7th, down 8ve | ||
| style="text-align:center;" | C# | | style="text-align:center;" | ^C#, vD | ||
|- | |- | ||
| style="text-align:center;" | 34 | | style="text-align:center;" | 34 | ||
| Line 535: | Line 425: | ||
|} | |} | ||
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[ | 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 - Chord names in other EDOs]]. | ||
=Approximations to Just Intonation= | |||
Like [[17edo|17edo]], 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11.* 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, 17/10, 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 5-limit JI. It is not a [[Meantone|meantone ]]system. In layman's terms while no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds, technically will be the same pitch as one somewhere upon the cycle of seventeen fifths. | |||
''Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9.'' ([http://en.wikipedia.org/wiki/34_equal_temperament Wikipedia]) | |||
<ul><li>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 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.</li></ul> | |||
Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. It is an improvement over the yet sharper "dominant seventh" found in jazz - which some listeners are accustomed to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless [[68edo|68edo]] (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just. | |||
==Selected just intervals by error== | ==Selected just intervals by error== | ||
| Line 553: | Line 452: | ||
| style="text-align:center;" | 1.324 | | style="text-align:center;" | 1.324 | ||
|- | |- | ||
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]] | | style="text-align:center;" | [[5/4|'''5/4''']], [[8/5|8/5]] | ||
| style="text-align:center;" | 1.922 | | style="text-align:center;" | '''1.922''' | ||
|- | |- | ||
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]] | | style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]] | ||
| Line 562: | Line 461: | ||
| style="text-align:center;" | 2.604 | | style="text-align:center;" | 2.604 | ||
|- | |- | ||
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]] | | style="text-align:center;" | [[4/3|4/3]], [[3/2|'''3/2''']] | ||
| style="text-align:center;" | 3.927 | | style="text-align:center;" | '''3.927''' | ||
|- | |- | ||
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]] | | style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]] | ||
| Line 580: | Line 479: | ||
| style="text-align:center;" | 6.021 | | style="text-align:center;" | 6.021 | ||
|- | |- | ||
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]] | | style="text-align:center;" | [[16/13|16/13]], [[13/8|'''13/8''']] | ||
| style="text-align:center;" | 6.531 | | style="text-align:center;" | '''6.531''' | ||
|- | |- | ||
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]] | | style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]] | ||
| Line 604: | Line 503: | ||
| style="text-align:center;" | 12.878 | | style="text-align:center;" | 12.878 | ||
|- | |- | ||
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]] | | style="text-align:center;" | [[11/8|'''11/8''']], [[16/11|16/11]] | ||
| style="text-align:center;" | 13.388 | | style="text-align:center;" | '''13.388''' | ||
|- | |- | ||
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]] | | style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]] | ||
| Line 613: | Line 512: | ||
| style="text-align:center;" | 15.482 | | style="text-align:center;" | 15.482 | ||
|- | |- | ||
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]] | | style="text-align:center;" | [[8/7|8/7]], [[7/4|'''7/4''']] | ||
| style="text-align:center;" | 15.885 | | style="text-align:center;" | '''15.885''' | ||
|- | |- | ||
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]] | | style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]] | ||
| Line 698: | Line 597: | ||
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]] | | style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]] | ||
| style="text-align:center;" | 29.273 | | style="text-align:center;" | 29.273 | ||
|} | |||
=34edo and phi= | |||
As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval 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 | -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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo|36edo]]. | |||
=Rank two temperaments= | |||
[[List_of_34edo_rank_two_temperaments_by_badness|List of 34edo rank two temperaments by badness]] | |||
Temperaments sorted by generator | |||
{| class="wikitable" | |||
|- | |||
! | Periods | |||
per octave | |||
! | Generator | |||
! | Cents | |||
! | Temperaments | |||
|- | |||
| | 1 | |||
| | 1\34 | |||
| | 35.294 | |||
| | | |||
|- | |||
| | | |||
| | 3\34 | |||
| | 105.882 | |||
| | | |||
|- | |||
| | | |||
| | 5\34 | |||
| | 176.471 | |||
| | [[Tetracot|Tetracot]]/[[bunya|Bunya]]/[[Monkey|Monkey]] | |||
|- | |||
| | | |||
| | 7\34 | |||
| | 247.059 | |||
| | [[Immunity|Immunity]] | |||
|- | |||
| | | |||
| | 9\34 | |||
| | 317.647 | |||
| | [[Hanson|Hanson]]/[[Keemun|Keemun]] | |||
|- | |||
| | | |||
| | 11\34 | |||
| | 388.235 | |||
| | [[Wuerschmidt|Wuerschmidt]]/[[Worschmidt|Worschmidt]] | |||
|- | |||
| | | |||
| | 13\34 | |||
| | 458.824 | |||
| | | |||
|- | |||
| | | |||
| | 15\34 | |||
| | 529.412 | |||
| | | |||
|- | |||
| | 2 | |||
| | 1\34 | |||
| | 35.294 | |||
| | | |||
|- | |||
| | | |||
| | 2\34 | |||
| | 70.588 | |||
| | [[Vishnu|Vishnu]] | |||
|- | |||
| | | |||
| | 3\34 | |||
| | 105.882 | |||
| | [[Srutal|Srutal]]/[[pajara|Pajara]]/[[Diaschismic|Diaschismic]] | |||
|- | |||
| | | |||
| | 4\34 | |||
| | 141.176 | |||
| | [[Fifive|Fifive]] | |||
|- | |||
| | | |||
| | 5\34 | |||
| | 176.471 | |||
| | | |||
|- | |||
| | | |||
| | 6\34 | |||
| | 211.765 | |||
| | | |||
|- | |||
| | | |||
| | 7\34 | |||
| | 247.059 | |||
| | | |||
|- | |||
| | | |||
| | 8\34 | |||
| | 282.353 | |||
| | | |||
|- | |||
| | 17 | |||
| | 1\34 | |||
| | 35.294 | |||
| | | |||
|} | |} | ||
| Line 708: | Line 710: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | | ! | [[Ratio]] | ||
! | Monzo | ! | [[Monzo]] | ||
! | | ! | [[Cents]] | ||
![[Color notation/Temperament Names|Color Name]] | |||
! | Names | ! | Names | ||
|- | |- | ||
| Line 716: | Line 719: | ||
| |<nowiki> | 27 -17 </nowiki>> | | |<nowiki> | 27 -17 </nowiki>> | ||
| style="text-align:right;" | 66.765 | | style="text-align:right;" | 66.765 | ||
| style="text-align:center;" |Sasawa | |||
| style="text-align:center;" | 17-comma | | style="text-align:center;" | 17-comma | ||
|- | |- | ||
| Line 721: | Line 725: | ||
| |<nowiki> | 5 -9 4 </nowiki>> | | |<nowiki> | 5 -9 4 </nowiki>> | ||
| style="text-align:right;" | 27.660 | | style="text-align:right;" | 27.660 | ||
| style="text-align:center;" |Saquadyo | |||
| style="text-align:center;" | Minimal Diesis, Tetracot Comma | | style="text-align:center;" | Minimal Diesis, Tetracot Comma | ||
|- | |- | ||
| Line 726: | Line 731: | ||
| |<nowiki> | 11 -4 -2 </nowiki>> | | |<nowiki> | 11 -4 -2 </nowiki>> | ||
| style="text-align:right;" | 19.553 | | style="text-align:right;" | 19.553 | ||
| style="text-align:center;" |Sagugu | |||
| style="text-align:center;" | Diaschisma | | style="text-align:center;" | Diaschisma | ||
|- | |- | ||
| Line 731: | Line 737: | ||
| |<nowiki> | 17 1 -8 </nowiki>> | | |<nowiki> | 17 1 -8 </nowiki>> | ||
| style="text-align:right;" | 11.445 | | style="text-align:right;" | 11.445 | ||
| style="text-align:center;" |Saquadbigu | |||
| style="text-align:center;" | Würschmidt comma | | style="text-align:center;" | Würschmidt comma | ||
|- | |- | ||
| Line 736: | Line 743: | ||
| |<nowiki> | -6 -5 6 </nowiki>> | | |<nowiki> | -6 -5 6 </nowiki>> | ||
| style="text-align:right;" | 8.107 | | style="text-align:right;" | 8.107 | ||
| style="text-align:center;" |Tribiyo | |||
| style="text-align:center;" | Kleisma, Semicomma Majeur | | style="text-align:center;" | Kleisma, Semicomma Majeur | ||
|- | |- | ||
| Line 741: | Line 749: | ||
| |<nowiki> | 23 6 -14 </nowiki>> | | |<nowiki> | 23 6 -14 </nowiki>> | ||
| style="text-align:right;" | 3.338 | | style="text-align:right;" | 3.338 | ||
| style="text-align:center;" |Sasepbigu | |||
| style="text-align:center;" | Vishnuzma, Semisuper | | style="text-align:center;" | Vishnuzma, Semisuper | ||
|- | |- | ||
| Line 746: | Line 755: | ||
| |<nowiki> | -3 1 -3 3 </nowiki>> | | |<nowiki> | -3 1 -3 3 </nowiki>> | ||
| style="text-align:right;" | 49.492 | | style="text-align:right;" | 49.492 | ||
| style="text-align:center;" |Trizogu | |||
| style="text-align:center;" | Keega | | style="text-align:center;" | Keega | ||
|- | |- | ||
| Line 751: | Line 761: | ||
| |<nowiki> | 1 0 2 -2 </nowiki>> | | |<nowiki> | 1 0 2 -2 </nowiki>> | ||
| style="text-align:right;" | 34.976 | | style="text-align:right;" | 34.976 | ||
| style="text-align:center;" |Biruyo | |||
| style="text-align:center;" | Jubilisma | | style="text-align:center;" | Jubilisma | ||
|- | |- | ||
| Line 756: | Line 767: | ||
| |<nowiki> | -5 -3 3 1 </nowiki>> | | |<nowiki> | -5 -3 3 1 </nowiki>> | ||
| style="text-align:right;" | 21.902 | | style="text-align:right;" | 21.902 | ||
| style="text-align:center;" |Zotriyo | |||
| style="text-align:center;" | Keema | | style="text-align:center;" | Keema | ||
|- | |- | ||
| Line 761: | Line 773: | ||
| |<nowiki> | 1 2 -3 1 </nowiki>> | | |<nowiki> | 1 2 -3 1 </nowiki>> | ||
| style="text-align:right;" | 13.795 | | style="text-align:right;" | 13.795 | ||
| style="text-align:center;" |Zotrigu | |||
| style="text-align:center;" | Starling comma, Septimal semicomma | | style="text-align:center;" | Starling comma, Septimal semicomma | ||
|- | |- | ||
| Line 766: | Line 779: | ||
| |<nowiki> | 2 -2 2 0 -1</nowiki>> | | |<nowiki> | 2 -2 2 0 -1</nowiki>> | ||
| style="text-align:right;" | 17.399 | | style="text-align:right;" | 17.399 | ||
| style="text-align:center;" |Luyoyo | |||
| style="text-align:center;" | Ptolemisma, Ptolemy's comma | | style="text-align:center;" | Ptolemisma, Ptolemy's comma | ||
|- | |- | ||
| Line 771: | Line 785: | ||
| |<nowiki> | -1 5 0 0 -2 </nowiki>> | | |<nowiki> | -1 5 0 0 -2 </nowiki>> | ||
| style="text-align:right;" | 7.139 | | style="text-align:right;" | 7.139 | ||
| style="text-align:center;" |Lulu | |||
| style="text-align:center;" | Rastma, Neutral third comma | | style="text-align:center;" | Rastma, Neutral third comma | ||
|- | |- | ||
| Line 776: | Line 791: | ||
| |<nowiki> | -7 -1 1 1 1 </nowiki>> | | |<nowiki> | -7 -1 1 1 1 </nowiki>> | ||
| style="text-align:right;" | 4.503 | | style="text-align:right;" | 4.503 | ||
| style="text-align:center;" |Lozoyo | |||
| style="text-align:center;" | Keenanisma | | style="text-align:center;" | Keenanisma | ||
|- | |- | ||
| Line 781: | Line 797: | ||
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | | |<nowiki> | -1 -2 -1 1 0 1 </nowiki>> | ||
| style="text-align:right;" | 19.120 | | style="text-align:right;" | 19.120 | ||
| style="text-align:center;" |Thozogu | |||
| style="text-align:center;" | Superleap | | style="text-align:center;" | Superleap | ||
|} | |} | ||
| Line 789: | Line 806: | ||
=Links= | =Links= | ||
<ul><li>[http://microstick.net/products/34-equal-guitar-by-larry-a-hanson/ 34 Equal Guitar] by [[Larry_Hanson|Larry Hanson]]</li><li>[https://microstick.net http://microstick.net/] websites of Neil Haverstick</li><li>https://myspace.com/microstick</li></ul> | <ul><li>[http://microstick.net/products/34-equal-guitar-by-larry-a-hanson/ 34 Equal Guitar] by [[Larry_Hanson|Larry Hanson]]</li><li>[https://microstick.net http://microstick.net/] websites of Neil Haverstick</li><li>https://myspace.com/microstick</li></ul> | ||
[[Category:34edo]] | [[Category:34edo]] | ||
[[Category:34et]] | [[Category:34et]] | ||
Revision as of 22:12, 14 December 2019
Theory
34edo divides the octave into 34 equal steps of approximately 35.29412 cents. 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 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.
Intervals
| Degree | Solfege | Cents | approx. ratios of
2.3.5.13.17 subgroup |
additional ratios
of 7 and 11 |
ups and downs notation | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| Pure octave | 45ed(7φ+6)\(5φ^2) | 2ed25/24 | 48:49:50-WT | |||||||
| 0 | do | 0.000 | 0 | 0 | 0 | 1/1 | P1 | perfect unison | D | |
| 1 | di | 35.294 | 35.296 | 35.336 | 35.697 | 128/125 (diesis), 51/50 | 50/49, 49/48 | ^1, vm2 | up unison, downminor 2nd | ^D, vEb |
| 2 | rih | 70.588 | 70.592 | 70.672 | 25/24, 648/625 (large diesis) | m2 | minor 2nd | Eb | ||
| 3 | ra | 105.882 | 105.8885 | 106.008 | 106.369 | 17/16, 18/17, 16/15 | 15/14 | ^m2 | upminor 2nd | ^Eb |
| 4 | ru | 141.1765 | 141.185 | 141.345 | 13/12 | 14/13, 12/11 | ~2 | mid 2nd | vvE | |
| 5 | reh | 176.471 | 176.481 | 176.681 | 177.042 | 10/9 | 11/10 | vM2 | downmajor 2nd | vE |
| 6 | re | 211.765 | 211.777 | 212.017 | 9/8, 17/15 | M2 | major 2nd | E | ||
| 7 | raw | 247.059 | 247.073 | 247.3535 | 247.714 | 15/13 | 8/7 | ^M2, vm3 | upmajor 2nd, downminor 3rd | ^E, vF |
| 8 | meh | 282.353 | 282.3695 | 282.69 | 20/17, 75/64 | 7/6, 13/11 | m3 | minor 3rd | F | |
| 9 | me | 317.647 | 317.666 | 318.026 | 318.3865 | 6/5 | 17/14 | ^m3 | upminor 3rd | ^F |
| 10 | mu | 352.941 | 352.962 | 353.362 | 16/13 | 11/9 | ~3 | mid 3rd | ^^F | |
| 11 | mi | 388.235 | 388.258 | 388.698 | 389.059 | 5/4 | vM3 | downmajor 3rd | vF# | |
| 12 | maa | 423.529 | 423.554 | 424.035 | 51/40, 32/25 | 14/11, 9/7 | M3 | major 3rd | F# | |
| 13 | maw | 458.8235 | 458.85 | 459.371 | 459.731 | 13/10, 17/13 | 22/17 | ^M3, v4 | upmajor 3rd,down 4th | ^F#, vG |
| 14 | fa | 494.118 | 494.1465 | 494.707 | 4/3 | P4 | 4th | G | ||
| 15 | fih | 529.412 | 529.443 | 530.043 | 530.40 | 512/375, 34/25 | 15/11 | ^4 | up 4th | ^G |
| 16 | fu | 564.706 | 564.739 | 565.379 | 36/25, 18/13 | 11/8 | ~4, d5 | mid 4th, dim 5th | ^^G, Ab | |
| 17 | fi/se | 600.000 | 600.035 | 600.716 | 601.076 | 17/12, 24/17 | 7/5, 10/7 | vA4, ^d5 | downaug 4th, updim 5th | vG#, ^Ab |
| 18 | su | 635.294 | 635.331 | 636.052 | 25/18, 13/9 | 16/11 | A4, ~5 | aug 4th, mid 5th | G#, vvA | |
| 19 | sih | 670.588 | 670.627 | 671.388 | 671.749 | 375/256, 25/17 | 22/15 | v5 | down 5th | vA |
| 20 | sol | 705.882 | 705.924 | 706.724 | 3/2 | P5 | perfect 5th | A | ||
| 21 | saw | 741.1765 | 741.22 | 742.0605 | 742.421 | 20/13, 26/17 | 17/11 | ^5, vm6 | up 5th, downminor 6th | ^A, vBb |
| 22 | leh | 776.471 | 776.516 | 777.397 | 25/16, 80/51 | 14/9 | m6 | minor 6th | Bb | |
| 23 | le | 811.765 | 811.812 | 812.733 | 813.0935 | 8/5 | ^m6 | upminor 6th | ^Bb | |
| 24 | lu | 847.059 | 847.108 | 848.069 | 13/8 | 18/11 | ~6 | mid 6th | vvB | |
| 25 | la | 882.353 | 882.4045 | 883.405 | 883.766 | 5/3 | 28/17 | vM6 | downmajor 6th | vB |
| 26 | laa | 917.647 | 917.701 | 918.7415 | 17/10 | 12/7, 22/13 | M6 | major 6th | B | |
| 27 | law | 952.941 | 952.997 | 954.078 | 954.438 | 26/15 | 7/4 | ^M6, vm7 | upmajor 6th, downminor 7th | ^B, vC |
| 28 | teh | 988.235 | 988.293 | 989.414 | 16/9, 30/17 | m7 | minor 7th | C | ||
| 29 | te | 1023.529 | 1023.589 | 1024.75 | 1025.111 | 9/5 | 20/11 | ^m7 | upminor 7th | ^C |
| 30 | tu | 1058.8235 | 1058.885 | 1060.086 | 24/13 | 13/7, 11/6 | ~7 | mid 7th | ^^C | |
| 31 | ti | 1094.118 | 1094.182 | 1095.423 | 1095.783 | 32/17, 17/9, 15/8 | 28/15 | vM7 | downmajor 7th | vC# |
| 32 | taa | 1129.412 | 1129.478 | 1130.759 | 48/25, 625/324 | M7 | major 7th | C# | ||
| 33 | da | 1164.706 | 1164.774 | 1166.095 | 1166.456 | 125/64, 100/51 | 49/25, 96/49 | ^M7, v8 | upmajor 7th, down 8ve | ^C#, vD |
| 34 | do | 1200.000 | 1200.07 | 1201.431 | 2/1 | P8 | 8ve | 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.
Approximations to Just Intonation
Like 17edo, 34edo contains good approximations of just intervals involving 13 and 3 -- specifically, 13/8, 13/12, 13/9 and their inversions -- while failing to closely approximate ratios of 7 or 11.* 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, 17/10, 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 5-limit JI. It is not a meantone system. In layman's terms while no number of fifths (frequently ratios of ~3:2) land on major or minor thirds, an even number of major or minor thirds, technically will be the same pitch as one somewhere upon the cycle of seventeen fifths.
Viewed in light of Western diatonic theory, the three extra steps (of 34-et compared to 31-et) in effect widen the intervals between C and D, F and G, and A and B [that is: 6 5 3 6 5 6 3], thus making a distinction between major tones, ratio 9/8 and minor tones, ratio 10/9. (Wikipedia)
- 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 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. It is an improvement over the yet sharper "dominant seventh" found in jazz - which some listeners are accustomed to. The ability to tolerate these errors may depend on subtle natural changes in mood. A few cents either way can bother the hell out of one, but on other days you might spend an hour not knowing of the strings are, or being able to, tuned. Nevertheless 68edo (34 x 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form... nearly just.
Selected just intervals by error
The following table shows how some prominent just intervals are represented in 34edo (ordered by absolute error).
Best direct mapping, even if inconsistent
| Interval, complement | Error (abs., in cents) |
|---|---|
| 15/13, 26/15 | 0.682 |
| 18/13, 13/9 | 1.324 |
| 5/4, 8/5 | 1.922 |
| 6/5, 5/3 | 2.006 |
| 13/12, 24/13 | 2.604 |
| 4/3, 3/2 | 3.927 |
| 13/10, 20/13 | 4.610 |
| 11/9, 18/11 | 5.533 |
| 16/15, 15/8 | 5.849 |
| 10/9, 9/5 | 5.933 |
| 14/11, 11/7 | 6.021 |
| 16/13, 13/8 | 6.531 |
| 13/11, 22/13 | 6.857 |
| 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 | 11.555 |
| 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 | 15.885 |
| 7/5, 10/7 | 17.488 |
Patent val mapping
| Interval, complement | Error (abs., in cents) |
|---|---|
| 15/13, 26/15 | 0.682 |
| 18/13, 13/9 | 1.324 |
| 5/4, 8/5 | 1.922 |
| 6/5, 5/3 | 2.006 |
| 13/12, 24/13 | 2.604 |
| 4/3, 3/2 | 3.927 |
| 13/10, 20/13 | 4.610 |
| 11/9, 18/11 | 5.533 |
| 16/15, 15/8 | 5.849 |
| 10/9, 9/5 | 5.933 |
| 16/13, 13/8 | 6.531 |
| 13/11, 22/13 | 6.857 |
| 15/11, 22/15 | 7.539 |
| 9/8, 16/9 | 7.855 |
| 12/11, 11/6 | 9.461 |
| 11/10, 20/11 | 11.466 |
| 11/8, 16/11 | 13.388 |
| 8/7, 7/4 | 15.885 |
| 7/5, 10/7 | 17.806 |
| 7/6, 12/7 | 19.812 |
| 15/14, 28/15 | 21.734 |
| 14/13, 13/7 | 22.416 |
| 9/7, 14/9 | 23.739 |
| 14/11, 11/7 | 29.273 |
34edo and phi
As a Fibonacci number, 34edo contains a fraction of an octave which is close approximation to the irrational interval 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 | -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. Yes. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and 36edo.
Rank two temperaments
List of 34edo rank two temperaments by badness
Temperaments sorted by generator
| Periods
per octave |
Generator | Cents | Temperaments |
|---|---|---|---|
| 1 | 1\34 | 35.294 | |
| 3\34 | 105.882 | ||
| 5\34 | 176.471 | Tetracot/Bunya/Monkey | |
| 7\34 | 247.059 | Immunity | |
| 9\34 | 317.647 | Hanson/Keemun | |
| 11\34 | 388.235 | Wuerschmidt/Worschmidt | |
| 13\34 | 458.824 | ||
| 15\34 | 529.412 | ||
| 2 | 1\34 | 35.294 | |
| 2\34 | 70.588 | Vishnu | |
| 3\34 | 105.882 | Srutal/Pajara/Diaschismic | |
| 4\34 | 141.176 | Fifive | |
| 5\34 | 176.471 | ||
| 6\34 | 211.765 | ||
| 7\34 | 247.059 | ||
| 8\34 | 282.353 | ||
| 17 | 1\34 | 35.294 |
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. The reader can construct his 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.
Commas
34-EDO tempers out the following commas. (Note: This assumes the val < 34 54 79 95 118 126 |.)
| Ratio | Monzo | Cents | Color Name | Names |
|---|---|---|---|---|
| 134217728/129140163 | | 27 -17 > | 66.765 | Sasawa | 17-comma |
| 20000/19683 | | 5 -9 4 > | 27.660 | Saquadyo | Minimal Diesis, Tetracot Comma |
| 2048/2025 | | 11 -4 -2 > | 19.553 | Sagugu | Diaschisma |
| 393216/390625 | | 17 1 -8 > | 11.445 | Saquadbigu | Würschmidt comma |
| 15625/15552 | | -6 -5 6 > | 8.107 | Tribiyo | Kleisma, Semicomma Majeur |
| 1212717/1210381 | | 23 6 -14 > | 3.338 | Sasepbigu | Vishnuzma, Semisuper |
| 1029/1000 | | -3 1 -3 3 > | 49.492 | Trizogu | Keega |
| 50/49 | | 1 0 2 -2 > | 34.976 | Biruyo | Jubilisma |
| 875/864 | | -5 -3 3 1 > | 21.902 | Zotriyo | Keema |
| 126/125 | | 1 2 -3 1 > | 13.795 | Zotrigu | Starling comma, Septimal semicomma |
| 100/99 | | 2 -2 2 0 -1> | 17.399 | Luyoyo | Ptolemisma, Ptolemy's comma |
| 243/242 | | -1 5 0 0 -2 > | 7.139 | Lulu | Rastma, Neutral third comma |
| 385/384 | | -7 -1 1 1 1 > | 4.503 | Lozoyo | Keenanisma |
| 91/90 | | -1 -2 -1 1 0 1 > | 19.120 | Thozogu | Superleap |
Listen
- Ascension
- Uncomfortable In Crowds (extended) by Robin Perry
Links
- 34 Equal Guitar by Larry Hanson
- http://microstick.net/ websites of Neil Haverstick
- https://myspace.com/microstick