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{{Infobox ET
{{interwiki
| Prime factorization = 2 × 17
| de = 34-EDO
| Step size = 35.294 ¢
| en = 34edo
| Fifth = 20\34 = 705.88¢ (→[[17edo|10\17]])
| es =  
| Major 2nd = 6\34 = 212¢
| ja =  
| Minor 2nd = 2\34 = 71¢
| Augmented 1sn = 4\34 = 141¢
}}
}}
'''34edo''' divides the octave into 34 equal steps of approximately 35.3 [[cent]]s.
{{Infobox ET}}
{{ED intro}}
{{Wikipedia| 34 equal temperament }}


== Theory ==
== Theory ==
{| class="wikitable center-all"
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.
! colspan="2" | <!-- empty cell -->
 
! prime 2
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.
! prime 3
 
! prime 5
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).
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" | Error
! absolute (¢)
| 0.0
| +3.9
| +1.9
| -15.9
| +13.4
| +6.5
| +0.9
|-
! [[Relative error|relative]] (%)
| 0
| +11
| +5
| -45
| +38
| +19
| +3
|-
! colspan="2" | [[nearest edomapping]]
| 34
| 20
| 11
| 27
| 16
| 24
| 3
|}


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.
=== Odd harmonics ===
{{Harmonics in equal|34|intervals=odd|columns=12}}


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-3 left-4 left-5 left-6"
{| class="wikitable center-all right-2 left-3 left-4 left-5"
|-
|-
! Degree
!  
! Solfege
! Cents
! Cents
! Approx. Ratios of<br>2.3.5.11.13.17 [[Subgroup]]
! Approx. ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.23 [[subgroup]]}}</ref>
! Additional Ratios of 7<br>Tending Flat (34 Val)
! Ratios of 7<br />Using the 34 Val
! Additional Ratios of 7<br>Tending Sharp (34d Val)
! Ratios of 7<br />Using the 34d Val
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^^d2)
! colspan="2" | [[Solfege|Solfeges]]
|-
|-
| 0
| 0
| do
| 0.000
| 0<span style="color: #fff;">.000</span>
| 1/1
| 1/1
|  
|  
|
|  
| P1
| P1
| perfect unison
| perfect unison
| D
| D
| da
| do
|-
|-
| 1
| 1
| di
| 35.294
| 35.294
| [[81/80]], [[128/125]], [[51/50]]
| [[81/80]], [[128/125]], [[51/50]]
Line 81: Line 50:
| up 1sn, downminor 2nd
| up 1sn, downminor 2nd
| ^D, vEb
| ^D, vEb
| du/fro
| di
|-
|-
| 2
| 2
| rih
| 70.588
| 70.588
| [[25/24]], [[648/625]], [[33/32]]
| [[25/24]], [[26/25]], [[24/23]], [[27/26]], [[23/22]], [[648/625]], [[33/32]]
| [[21/20]], [[36/35]], [[50/49]]
| [[21/20]], [[36/35]], [[50/49]]
| [[28/27]], [[49/48]]
| [[28/27]], [[49/48]]
| ^^1, m2
| ^^1, m2
| double-up 1sn, minor 2nd
| dup 1sn, minor 2nd
| ^^D, Eb
| ^^D, Eb
| fra
| rih
|-
|-
| 3
| 3
| ra
| 105.882
| 105.882
| [[17/16]], [[18/17]], [[16/15]]
| [[17/16]], [[18/17]], [[16/15]]
Line 99: Line 70:
| [[15/14]], [[21/20]]
| [[15/14]], [[21/20]]
| vA1, ^m2
| vA1, ^m2
| downaug 1sn, upminor 2nd
| downaug 1sn,<br />upminor 2nd
| vD#, ^Eb
| vD#, ^Eb
| fru
| ra
|-
|-
| 4
| 4
| ru
| 141.176
| 141.176
| [[13/12]], [[12/11]]
| [[13/12]], [[12/11]], [[25/23]]
| [[15/14]]
| [[15/14]]
| [[14/13]]
| [[14/13]]
Line 111: Line 83:
| aug 1sn, mid 2nd
| aug 1sn, mid 2nd
| D#, vvE
| D#, vvE
| ri
| ru
|-
|-
| 5
| 5
| reh
| 176.471
| 176.471
| [[10/9]], [[11/10]]
| [[10/9]], [[11/10]]
Line 121: Line 94:
| downmajor 2nd
| downmajor 2nd
| vE
| vE
| ro
| reh
|-
|-
| 6
| 6
| re
| 211.765
| 211.765
| [[9/8]], [[17/15]]
| [[9/8]], [[17/15]], [[26/23]]
|  
|  
| [[8/7]]
| [[8/7]]
Line 131: Line 105:
| major 2nd
| major 2nd
| E
| E
| ra
| re
|-
|-
| 7
| 7
| raw
| 247.059
| 247.059
| [[15/13]]
| [[15/13]], [[23/20]]
| [[7/6]], [[8/7]]
| [[7/6]], [[8/7]]
|
|  
| ^M2, vm3
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| upmajor 2nd,<br />downminor 3rd
| ^E, vF
| ^E, vF
| ru/no
| raw
|-
|-
| 8
| 8
| meh
| 282.353
| 282.353
| [[20/17]], [[75/64]], [[13/11]]
| [[20/17]], [[75/64]], [[27/23]], [[13/11]]
|  
|  
| [[7/6]]
| [[7/6]]
Line 151: Line 127:
| minor 3rd
| minor 3rd
| F
| F
| na
| meh
|-
|-
| 9
| 9
| me
| 317.647
| 317.647
| [[6/5]]
| [[6/5]]
|
|  
|17/14
| 17/14
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| ^F
| ^F
| nu
| me
|-
|-
| 10
| 10
| mu
| 352.941
| 352.941
| [[16/13]], [[11/9]], [[27/22]]
| [[16/13]], [[11/9]], [[27/22]]
Line 171: Line 149:
| mid 3rd
| mid 3rd
| ^^F
| ^^F
| mi
| mu
|-
|-
| 11
| 11
| mi
| 388.235
| 388.235
| [[5/4]]
| [[5/4]]
Line 181: Line 160:
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| mo
| mi
|-
|-
| 12
| 12
| maa
| 423.529
| 423.529
| [[51/40]], [[32/25]]
| [[51/40]], [[32/25]], [[23/18]]
|
|  
| [[9/7]], [[14/11]]
| [[9/7]], [[14/11]]
| M3
| M3
| major 3rd
| major 3rd
| F#
| F#
| ma
| maa
|-
|-
| 13
| 13
| maw
| 458.824
| 458.824
| [[13/10]], [[17/13]], [[22/17]]
| [[13/10]], [[30/23]], [[17/13]], [[22/17]]
| [[9/7]], [[21/16]]
| [[9/7]], [[21/16]]
|  
|  
| ^M3, v4
| ^M3, v4
| upmajor 3rd,down 4th
| upmajor 3rd, down 4th
| ^F#, vG
| ^F#, vG
| mu/fo
| maw
|-
|-
| 14
| 14
| fa
| 494.118
| 494.118
| [[4/3]]
| [[4/3]]
Line 211: Line 193:
| 4th
| 4th
| G
| G
| fa
| fa
|-
|-
| 15
| 15
| fih
| 529.412
| 529.412
| [[27/20]], [[34/25]], [[15/11]]
| [[27/20]], [[34/25]], [[15/11]], [[23/17]]
|  
|  
|  
|  
Line 221: Line 204:
| up 4th
| up 4th
| ^G
| ^G
| fu
| fih
|-
|-
| 16
| 16
| fu
| 564.706
| 564.706
| [[25/18]], [[18/13]], [[11/8]]
| [[25/18]], [[18/13]], [[11/8]], [[32/23]]
| [[7/5]]
| [[7/5]]
|  
|
| ~4, d5
| ~4, d5
| mid 4th, dim 5th
| mid 4th, dim 5th
| ^^G, Ab
| ^^G, Ab
| fi/sha
| fu
|-
|-
| 17
| 17
| fi/se
| 600.000
| 600<span style="color: #fff;">.000</span>
| [[45/32]], [[64/45]], [[17/12]], [[24/17]]
| [[45/32]], [[64/45]], [[17/12]], [[24/17]]
|
|
Line 241: Line 226:
| downaug 4th, updim 5th
| downaug 4th, updim 5th
| vG#, ^Ab
| vG#, ^Ab
| po/shu
| fi/se
|-
|-
| 18
| 18
| su
| 635.294
| 635.294
| [[36/25]], [[13/9]], [[16/11]]
| [[36/25]], [[13/9]], [[16/11]], [[23/16]]
| [[10/7]]
| [[10/7]]
|  
|
| A4, ~5
| A4, ~5
| aug 4th, mid 5th
| aug 4th, mid 5th
| G#, vvA
| G#, vvA
| pa/si
| su
|-
|-
| 19
| 19
| sih
| 670.588
| 670.588
| [[40/27]], [[25/17]], [[22/15]]
| [[40/27]], [[25/17]], [[22/15]], [[34/23]]
|  
|  
|  
|  
Line 261: Line 248:
| down 5th
| down 5th
| vA
| vA
| so
| sih
|-
|-
| 20
| 20
| sol
| 705.882
| 705.882
| [[3/2]]
| [[3/2]]
|  
|
| [[32/21]]
| [[32/21]]
| P5
| P5
| perfect 5th
| perfect 5th
| A
| A
| sa
| sol
|-
|-
| 21
| 21
| saw
| 741.176
| 741.176
| [[20/13]], [[26/17]], [[17/11]]
| [[20/13]], [[23/15]], [[26/17]], [[17/11]]
| [[14/9]], [[32/21]]
| [[14/9]], [[32/21]]
|  
|
| ^5, vm6
| ^5, vm6
| up 5th, downminor 6th
| up 5th, downminor 6th
| ^A, vBb
| ^A, vBb
| su/flo
| saw
|-
|-
| 22
| 22
| leh
| 776.471
| 776.471
| [[25/16]], [[80/51]]
| [[25/16]], [[80/51]], [[36/23]]
|
|
| [[14/9]], [[11/7]]
| [[14/9]], [[11/7]]
Line 291: Line 281:
| minor 6th
| minor 6th
| Bb
| Bb
| fla
| leh
|-
|-
| 23
| 23
| le
| 811.765
| 811.765
| [[8/5]]
| [[8/5]]
Line 301: Line 292:
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| flu
| le
|-
|-
| 24
| 24
| lu
| 847.059
| 847.059
| [[13/8]], [[18/11]], [[44/27]]
| [[13/8]], [[18/11]], [[44/27]]
Line 311: Line 303:
| mid 6th
| mid 6th
| vvB
| vvB
| li
| lu
|-
|-
| 25
| 25
| la
| 882.353
| 882.353
| [[5/3]]
| [[5/3]]
|
|  
| [[28/17]]
| [[28/17]]
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| vB
| vB
| lo
| la
|-
|-
| 26
| 26
| laa
| 917.647
| 917.647
| [[17/10]], [[128/75]], [[22/13]]
| [[17/10]], [[128/75]], [[46/27]], [[22/13]]
|  
|  
| [[12/7]]
| [[12/7]]
Line 331: Line 325:
| major 6th
| major 6th
| B
| B
| la
| laa
|-
|-
| 27
| 27
| law
| 952.941
| 952.941
| [[26/15]]
| [[26/15]], [[40/23]]
| [[7/4]], [[12/7]]
| [[7/4]], [[12/7]]
|
|  
| ^M6, vm7
| ^M6, vm7
| upmajor 6th, downminor 7th
| upmajor 6th,<br />downminor 7th
| ^B, vC
| ^B, vC
| lu/tho
| law
|-
|-
| 28
| 28
| teh
| 988.235
| 988.235
| [[16/9]], [[30/17]]
| [[16/9]], [[30/17]], [[23/13]]
|  
|  
| [[7/4]]
| [[7/4]]
Line 351: Line 347:
| minor 7th
| minor 7th
| C
| C
| tha
| teh
|-
|-
| 29
| 29
| te
| 1023.529
| 1023.529
| [[9/5]], [[20/11]]
| [[9/5]], [[20/11]]
Line 361: Line 358:
| upminor 7th
| upminor 7th
| ^C
| ^C
| thu
| te
|-
|-
| 30
| 30
| tu
| 1058.824
| 1058.824
| [[24/13]], [[11/6]]
| [[24/13]], [[11/6]], [[46/25]]
| [[28/15]]
| [[28/15]]
| [[13/7]]
| [[13/7]]
Line 371: Line 369:
| mid 7th
| mid 7th
| ^^C
| ^^C
| ti
| tu
|-
|-
| 31
| 31
| ti
| 1094.118
| 1094.118
| [[32/17]], [[17/9]], [[15/8]]
| [[32/17]], [[17/9]], [[15/8]]
Line 381: Line 380:
| downmajor 7th
| downmajor 7th
| vC#
| vC#
| to
| ti
|-
|-
| 32
| 32
| taa
| 1129.412
| 1129.412
| [[48/25]], [[625/324]],  [[64/33]]
| [[48/25]], [[25/13]], [[23/12]], [[625/324]],  [[64/33]]
| [[40/21]], [[35/18]], [[49/25]]
| [[40/21]], [[35/18]], [[49/25]]
| [[27/14]], [[96/49]]
| [[27/14]], [[96/49]]
Line 391: Line 391:
| major 7th
| major 7th
| C#
| C#
| ta
| taa
|-
|-
| 33
| 33
| da
| 1164.706
| 1164.706
| [[160/81]], [[125/64]], [[100/51]]
| [[160/81]], [[125/64]], [[100/51]]
Line 400: Line 401:
| ^M7, v8
| ^M7, v8
| upmajor 7th, down 8ve
| upmajor 7th, down 8ve
| ^C#, vD
| vD
| tu/do
| da
|-
|-
| 34
| 34
| do
| 1200.000
| 1200<span style="color: #fff;">.000</span>
| [[2/1]]
| [[2/1]]
|  
|  
|
|  
| P8
| P8
| 8ve
| 8ve
| D
| D
| da
| do
|}
|}


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]].
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 ==
== Notation ==
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 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.
=== 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}}


''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:34_equal_temperament|Wikipedia]])
[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}


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.
=== Sagittal notation ===
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]].


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 × 2) preserves the structure and has these intervals 7/8 and 11/8 in more perfect form… nearly just.
==== Evo flavor ====
<imagemap>
File:34-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 431 0 591 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:34-EDO_Evo_Sagittal.svg]]
</imagemap>


=== Selected just intervals by error ===
==== Revo flavor ====
The following table shows how [[15-odd-limit intervals]] are represented in 34edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
<imagemap>
{| class="wikitable center-all"
File:34-EDO_Revo_Sagittal.svg
|+ Direct mapping (even if inconsistent)
desc none
|-
rect 80 0 300 50 [[Sagittal_notation]]
! Interval, complement
rect 423 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
! Error (abs, [[Cent|¢]])
rect 20 80 120 106 [[81/80]]
|-
rect 120 80 240 106 [[33/32]]
| [[15/13]], [[26/15]]
default [[File:34-EDO_Revo_Sagittal.svg]]
| 0.682
</imagemap>
|-
 
| [[18/13]], [[13/9]]
==== Evo-SZ flavor ====
| 1.324
<imagemap>
|-
File:34-EDO_Evo-SZ_Sagittal.svg
| '''[[5/4]], [[8/5]]'''
desc none
| '''1.922'''
rect 80 0 300 50 [[Sagittal_notation]]
|-
rect 423 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
| [[6/5]], [[5/3]]
rect 20 80 120 106 [[81/80]]
| 2.006
rect 120 80 240 106 [[33/32]]
|-
default [[File:34-EDO_Evo-SZ_Sagittal.svg]]
| [[13/12]], [[24/13]]
</imagemap>
| 2.604
 
|-
=== Kosmorsky's thoughts ===
| '''[[4/3]], [[3/2]]'''
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.
| '''3.927'''
 
|-
== Approximation to JI ==
| [[13/10]], [[20/13]]
[[File:34ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 34edo]]
| 4.610
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.
|-
 
| [[11/9]], [[18/11]]
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.
| 5.533
 
|-
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.)
| [[16/15]], [[15/8]]
 
| 5.849
=== Interval mappings ===
|-
{{Q-odd-limit intervals|34}}
| [[10/9]], [[9/5]]
{{Q-odd-limit intervals|34.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 34d val mapping}}
| 5.933
 
|-
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.
| ''[[14/11]], [[11/7]]''
 
| ''6.021''
== Tuning by ear ==
|-
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%.
| '''[[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''
|}


{| class="wikitable center-all"
== Approximation to irrational intervals ==
|+ Patent val mapping
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]].)
|-
! Interval, complement
! Error (abs, [[Cent|¢]])
|-
| [[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''
|}


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


Note: the 34d val is used for lower error.
== Regular temperament properties ==
{| class="wikitable center-all"
{| class="wikitable center-4 center-5 center-6"
! colspan="2" |
|-
! 3-limit
! rowspan="2" | [[Subgroup]]
! 5-limit
! rowspan="2" | [[Comma list]]
! 7-limit
! rowspan="2" | [[Mapping]]
! 11-limit
! rowspan="2" | Optimal<br />8ve stretch (¢)
! 13-limit
! colspan="2" | Tuning error
! 17-limit
! 2.3.5.13.17
! 2.3.5.11.13.17
|-
|-
! colspan="2" |Octave stretch (¢)
! [[TE error|Absolute]] (¢)
| -1.24
! [[TE simple badness|Relative]] (%)
| -1.10
| -2.56
| -2.82
| -2.64
| -2.30
| -1.06
| -1.53
|-
|-
! rowspan="2" |Error
| 2.3.5
! [[TE error|absolute]] (¢)
| 2048/2025, 15625/15552
| 1.24
| {{mapping| 34 54 79 }}
| −1.10
| 1.03
| 1.03
| 2.92
|-
| 2.3.5.7
| 50/49, 64/63, 4375/4374
| {{mapping| 34 54 79 96 }} (34d)
| −2.56
| 2.66
| 2.66
| 7.57
|-
| 2.3.5.7.11
| 50/49, 64/63, 99/98, 243/242
| {{mapping| 34 54 79 96 118 }} (34d)
| −2.82
| 2.44
| 2.44
| 6.93
|-
| 2.3.5.7.11.13
| 50/49, 64/63, 78/77, 99/98, 144/143
| {{mapping| 34 54 79 96 118 126 }} (34d)
| −2.64
| 2.26
| 2.26
| 6.42
|-
| 2.3.5.7.11.13.17
| 50/49, 64/63, 78/77, 85/84, 99/98, 144/143
| {{mapping| 34 54 79 96 118 126 139 }} (34d)
| −2.30
| 2.26
| 2.26
| 0.94
| 1.35
|-
! [[TE simple badness|relative]] (%)
| 3.51
| 2.92
| 7.57
| 6.93
| 6.42
| 6.41
| 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.  
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.  
* 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.


== Tuning by ear ==
=== Uniform maps ===
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%.
{{Uniform map|edo=34}}
 
== 34edo and 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. 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 ==
=== Rank-2 temperaments ===
* [[List of 34edo rank two temperaments by badness]]
* [[List of 34edo rank two temperaments by badness]]
* [[List of edo-distinct 34d rank two temperaments]]
* [[List of edo-distinct 34d rank two temperaments]]


{| class="wikitable"
{| class="wikitable"
|+ Temperaments sorted by generator
|+ style="font-size: 105%;" | Rank-2 temperaments by period and generator
|-
|-
! Periods<br>per octave
! Periods<br />per 8ve
! Generator
! Generator
! Cents
! Cents
! Mosses
! Temperaments
! Temperaments
|-
|-
| 1
| rowspan="8" style="text-align: center;" | 1
| 1\34
| 1\34
| 35.294
| 35.294
|  
|  
| [[Gammic]]
|-
|-
| "
| 3\34
| 3\34
| 105.882
| 105.88
| [[11L&nbsp;1s]]<br />[[11L&nbsp;12s]]
|  
|  
|-
|-
| "
| 5\34
| 5\34
| 176.471
| 176.471
| [[Tetracot]]/[[Bunya]]/[[Monkey]]
| [[6L&nbsp;1s]]<br />[[7L&nbsp;6s]]<br />[[7L&nbsp;13s]]
| [[Tetracot]], [[bunya]] (34d), [[modus]] (34d), [[monkey]] (34), [[wollemia]] (34)
|-
|-
| "
| 7\34
| 7\34
| 247.059
| 247.059
| [[Immunity]]
| [[5L&nbsp;4s]]<br />[[5L&nbsp;9s]]<br />[[5L&nbsp;14s]]<br />[[5L&nbsp;19s]]
| [[Immunity]] (34), [[immunized]] (34d)
|-
|-
| "
| 9\34
| 9\34
| 317.647
| 317.647
| [[Hanson]]/[[Keemun]]
| [[4L&nbsp;3s]]<br />[[4L&nbsp;7s]]<br />[[4L&nbsp;11s]]<br />[[15L&nbsp;4s]]
| [[Hanson]], [[keemun]] (34), [[catalan]] (34d), [[catakleismic]] (34d)
|-
|-
| "
| 11\34
| 11\34
| 388.235
| 388.235
| [[Wuerschmidt]]/[[Worschmidt]]
| [[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)
|-
|-
| "
| 13\34
| 13\34
| 458.824
| 458.824
| [[Chromatic_pairs#Petrtri|Petrtri]]
| [[3L&nbsp;2s]]<br />[[5L&nbsp;3s]]<br />[[8L&nbsp;5s]]<br />[[13L&nbsp;8s]]
| [[Petrtri]]
|-
|-
| "
| 15\34
| 15\34
| 529.412
| 529.412
| [[Chromatic_pairs#Mabila|Mabila]]
| [[2L&nbsp;3s]]<br />[[2L&nbsp;5s]]<br />[[7L&nbsp;2s]]<br />[[9L&nbsp;7s]]
|-
| [[Mabila]]
| 2
| 1\34
| 35.294
|
|-
|-
| "
| rowspan="8" style="text-align: center;" | 2
| 2\34
| 2\34
| 70.588
| 70.588
| [[16L&nbsp;2s]]
| [[Vishnu]]
| [[Vishnu]]
|-
|-
| "
| 3\34
| 3\34
| 105.882
| 105.882
| [[Srutal]]/[[Pajara]]/[[Diaschismic]]
| [[2L&nbsp;6s]]<br />[[2L&nbsp;8s]]<br />[[10L&nbsp;2s]]<br />[[12L&nbsp;10s]]
| [[Srutal]] (34d), [[pajara]] (34d), [[diaschismic]] (34)
|-
|-
| "
| 4\34
| 4\34
| 141.176
| 141.176
| [[Fifive]]
| [[2L&nbsp;6s]]<br />[[8L&nbsp;2s]]<br />[[8L&nbsp;10s]]
| [[Fifive]], [[crepuscular]] (34d), [[fifives]] (34)
|-
|-
| "
| 5\34
| 5\34
| 176.471
| 176.471
|  
| [[6L&nbsp;2s]]<br />[[6L&nbsp;8s]]<br />[[14L&nbsp;6s]]
| [[Stratosphere]]
|-
|-
| "
| 6\34
| 6\34
| 211.765
| 211.765
|  
| [[4L&nbsp;2s]]<br />[[6L&nbsp;4s]]<br />[[6L&nbsp;10s]]<br />[[6L&nbsp;16s]]
| [[Antikythera]]
|-
|-
| "
| 7\34
| 7\34
| 247.059
| 247.059
|  
| [[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[10L&nbsp;4s]]
| [[Tobago]]
|-
|-
| "
| 8\34
| 8\34
| 282.353
| 282.353
|  
| [[2L&nbsp;2s]]<br />[[4L&nbsp;2s]]<br />[[4L&nbsp;6s]]<br />[[4L&nbsp;10s]]<br />[[4L&nbsp;14s]]
|-
| [[Bikleismic]]
| 17
| 1\34
| 35.294
|
|}
|}


== Notations ==
=== Commas ===
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.
34et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[patent val]] {{val| 34 54 79 95 118 126 }}.
 
== Commas ==
34-EDO [[tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val|34 54 79 95 118 126}}.)


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
|-
! [[Harmonic limit|Prime<br>Limit]]
! [[Harmonic limit|Prime<br />limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Monzo]]
! [[Cents]]
! [[Cents]]
! [[Color name]]
! [[Color name]]
! Name(s)
! Name
|-
|-
| 3
| 3
| [[134217728/129140163|(18 digits)]]
| <abbr title="134217728/129140163">(18 digits)</abbr>
| {{monzo| 27 -17 }}
| {{monzo| 27 -17 }}
| 66.765
| 66.765
Line 765: Line 657:
| 27.660
| 27.660
| Saquadyo
| Saquadyo
| Minimal diesis, tetracot comma
| Tetracot comma
|-
|-
| 5
| 5
Line 775: Line 667:
|-
|-
| 5
| 5
| [[393216/390625|(12 digits)]]
| <abbr title="393216/390625">(12 digits)</abbr>
| {{monzo| 17 1 -8 }}
| {{monzo| 17 1 -8 }}
| 11.445
| 11.445
Line 786: Line 678:
| 8.107
| 8.107
| Tribiyo
| Tribiyo
| Kleisma, semicomma majeur
| Kleisma
|-
|-
| 5
| 5
| [[6115295232/6103515625|(20 digits)]]
| <abbr title="6115295232/6103515625">(20 digits)</abbr>
| {{monzo| 23 6 -14 }}
| {{monzo| 23 6 -14 }}
| 3.338
| 3.338
| Sasepbigu
| Sasepbigu
| [[Vishnuzma]], semisuper comma
| [[Vishnuzma]]
|-
|-
| 7
| 7
Line 807: Line 699:
| 35.697
| 35.697
| Zozo
| Zozo
| Septimal diesis, slendro diesis
| Semaphoresma, slendro diesis
|-
|-
| 7
| 7
Line 821: Line 713:
| 13.795
| 13.795
| Zotrigu
| Zotrigu
| Starling comma, septimal semicomma
| Starling comma
|-
|-
| 11
| 11
Line 828: Line 720:
| 17.399
| 17.399
| Luyoyo
| Luyoyo
| Ptolemisma, Ptolemy's comma
| Ptolemisma
|-
|-
| 11
| 11
Line 835: Line 727:
| 7.139
| 7.139
| Lulu
| Lulu
| Rastma, neutral third comma
| Rastma
|-
|-
| 11
| 11
Line 851: Line 743:
| Superleap
| Superleap
|}
|}
<references/>
 
== Scales ==
=== MOS scales ===
{{main|List of MOS scales in 34edo}}
=== Ternary scales ===
* [[Blackdye]] (5:3:1)
* [[Diachrome]] (5:2:1)
* [[Cthon5m]] (4:2:1)
 
== Instruments ==
=== Lumatone ===
* [[Lumatone mapping for 34edo]]
 
=== Skip fretting ===
* [[Skip fretting system 34 2 9]]
* [[Skip fretting system 34 2 11]]


== Music ==
== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=Mni0bsUVgHk "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – with syntonic-comma adjustment, rendered by Claudi Meneghin (2025)
; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=CwMem5p1R6Y ''Maple Leaf Rag''] (1899) – rendered by Claudi Meneghin (2024)
; {{W|Marco Uccellini}}
* [https://www.youtube.com/watch?v=AOPOHIOgqhQ ''Aria Sopra La Bergamasca''] – arranged for Organ and rendered by Claudi Meneghin (2024)


* [http://www.archive.org/details/Ascension_105 Ascension]
=== 21st century ===
* [https://www.youtube.com/watch?v=FXTM0HeuExk Uncomfortable In Crowds (extended)] by [[Robin Perry]]
; [[Flora Canou]]
* [[:File:A stroll through some retuned maqams.mp3|A stroll through some retuned maqams]] by [[Ray Perlner]]
* [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


== Links ==
; [[E8 Heterotic]]
* [https://youtu.be/8vyiBt-LyR4?si=eCqiwSY6rftnAiFN ''Kythira's Wake''] (2019)
* [https://youtu.be/cp9kxB1N8FI?si=6gQzx0og8PyAao_k ''Septendecimal Samsara''] (2019) – synthwave
* [https://youtu.be/WsC4y7eG8aw?si=cwuySPEyBGJapFXX ''Dodecahedron''] (2019) – contemporary jazz
* [https://youtu.be/SoMEMYpRV4w?si=0COl0PwuPo-eQG8K ''No Threes For You''] (2019)
* [https://youtu.be/_dOIyByMTfU?si=DsWUSQnERSKi0zbu "Elements - Water"] from ''Elements'' (2019–2020)
 
; [[Francium]]
* "Travel To Stay" from ''Mysteries'' (2023) – [https://open.spotify.com/track/2S2UlMKNNL5yB3280KpgvK Spotify] | [https://francium223.bandcamp.com/track/travel-to-stay Bandcamp] | [https://www.youtube.com/watch?v=fDkW1SnMcdw YouTube]
* "Locksmiths" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/2Hzun107B8bxcZaMOClN6T Spotify] | [https://francium223.bandcamp.com/track/locksmiths Bandcamp] | [https://www.youtube.com/watch?v=pQLbtF0Obhc YouTube]
* [https://www.youtube.com/watch?v=HG0kJBHjuZ4 ''Plane Sonatina No. 2''] (2025)
* [https://www.youtube.com/watch?v=ZrjbxQbdVw4 ''cucumber service''] (2025)
 
; [[Adam Freese]]
* [https://www.youtube.com/watch?v=lfdHBBPHvLc ''Austice''] (2023)
 
; [[Hideya]]
* [https://www.youtube.com/watch?v=wT6v8dlUFWo ''Like refracted light''] (2023)
 
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/Ascension_105 ''Ascension''] (2010)
 
; [[luphoria]]
* ''[https://www.youtube.com/watch?v=Dcpkrci64Ms look]'' (2023)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=O_7w4KoaOY4 ''Semitone-Canon on The Mother's Malison Theme, for Cor Anglais & Violin''] (2022)
 
; [[Ray Perlner]]
* [[:File:A stroll through some retuned maqams.mp3|''A stroll through some retuned maqams'']] (2020)
 
; [[Robin Perry]]
* [https://www.youtube.com/watch?v=FXTM0HeuExk ''Uncomfortable In Crowds'' (extended)] (2013)
 
; [[Tapeworm Saga]]
* [https://www.youtube.com/watch?v=BhgxwP9_cSw ''A 3/4 piece in 34edo on 12/31/23''] (2023)
 
; [[Sintel]]
* [https://www.youtube.com/watch?v=hM7p_VVyeQ0 ''Diversion in 34edo''] (2021) – [https://www.youtube.com/watch?v=yTG0z4Znimw transcription by Stephen Weigel]
 
; [[Cam Taylor]]
* [https://www.youtube.com/watch?v=fbrXu7ls5tI ''34-equal Luma: a little sentimental''] (2023)
* [https://www.youtube.com/watch?v=9ORQlBmu_60 ''34 equal: classic triads''] (2023)
* [https://www.youtube.com/watch?v=zojGuuJqGQk&t=2s ''Diaschismatic/Srutal<nowiki>[12]</nowiki> in 34-equal on the harpsichord''] (2024)
 
; [[Userminusone]]
* [https://www.youtube.com/watch?v=5M_Kh94_t1Q ''Perspective''] (2021)
 
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=o2FVaCPJk1k 傘がなくても嬉しい ''(The Puddle Song)''] (2021)
 
; [[Xotla]]
* from ''Lesser Groove'' (2020)
** "Apparatus" – [https://open.spotify.com/track/4pqelsaCnLZrjniGVt3jbN Spotify] | [https://xotla.bandcamp.com/track/apparatus-34edo Bandcamp] | [https://www.youtube.com/watch?v=A6bkzYtsLas YouTube] – sci-fi electro
** "Electrostat" – [https://open.spotify.com/track/5LIPr8n6uQySeLUfM11U2W Spotify] | [https://xotla.bandcamp.com/track/electrostat-tetracot-13 Bandcamp] | [https://www.youtube.com/watch?v=5SAuoyDwpgc YouTube] – ambient electro, tetracot[13] in 34edo tuning
** "Dreams Of Ambience" – [https://open.spotify.com/track/2J5wCauw4aY96DsIEuaDZh Spotify] | [https://xotla.bandcamp.com/track/dreams-of-ambience-34edo Bandcamp] | [https://www.youtube.com/watch?v=jkDA6lhqOYM YouTube] – ambient electro
* from ''Xotla's Microtonal Funk & Blues Vol. 2'' (2020)
** "Chaparral" – [https://open.spotify.com/track/6aOBS6z1Yadj8jUiFcl8l7 Spotify] | [https://xotla.bandcamp.com/track/chaparral Bandcamp] | [https://www.youtube.com/watch?v=k73ktCHq-ac YouTube] – space rock
** "Lull" – [https://open.spotify.com/track/7fDmfyAGNYJ79rkqcnxERm Spotify] | [https://xotla.bandcamp.com/track/lull Bandcamp] | [https://www.youtube.com/watch?v=9-RG2TDESK0 YouTube] – ambient
** "Vortices" – [https://open.spotify.com/track/3MJLNBVL6lMjUxnHeqQc6B Spotify] | [https://xotla.bandcamp.com/track/vortices Bandcamp] | [https://www.youtube.com/watch?v=xGPASo79vZc YouTube] – electro-rock
** "Escapade" – [https://open.spotify.com/track/5An9IJRQSG43gKtLnAUpOY Spotify] | [https://xotla.bandcamp.com/track/escapade Bandcamp] | [https://www.youtube.com/watch?v=3dRQy9hvWaU YouTube] – electronic rock
* [https://www.youtube.com/watch?v=56HuAL1jC8E ''Between Space''] (2022) – ambient sci-fi
 
; [[Zhea Erose]]
* [https://www.youtube.com/watch?v=xYZwye9PWSo ''Modal Studies in Tetracot''] (2021)
 
== See also ==
* [[Diaschismic-tetracot equivalence continuum]]
 
== External links ==
* [http://microstick.net/products/34-equal-guitar-by-larry-a-hanson/ 34 Equal Guitar] by [[Larry Hanson]] {{dead link}}
* [http://microstick.net/products/34-equal-guitar-by-larry-a-hanson/ 34 Equal Guitar] by [[Larry Hanson]] {{dead link}}
* [https://microstick.net http://microstick.net/] websites of Neil Haverstick
* [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)
* [https://myspace.com/microstick] – somehow broken (if you scroll to right, you'll find the songs, playing them, you can't hear anything)
 
== Notes ==
<references group="note" />


[[Category:34edo]]
[[Category:Diaschismic]]
[[Category:34et]]
[[Category:Keemun]]
[[Category:diaschismic]]
[[Category:Kleismic]]
[[Category:Equal divisions of the octave]]
[[Category:Pajara]]
[[Category:keemun]]
[[Category:Oneirotonic]]
[[Category:kleismic]]
[[Category:Listen]]
[[Category:listen]]
[[Category:Würschmidt]]
[[Category:pajara]]
[[Category:Tetracot]]
[[Category:selenium]]
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