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34edo: Difference between revisions

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| Prime factorization = 2 * 17
| Prime factorization = 2 * 17
| Subgroup = 2.3.5.11.13.17
| Subgroup = 2.3.5.11.13.17
| Step size = 35.294
| Step size = 35.294¢
| Fifth type = leapfrog/archy [[17edo]] 20\34 705.88¢
| Fifth type = 20\34 = 705.88¢ = [[17edo]] leapfrog/archy
| Major 2nd = 6\34 = 212¢
| Minor 2nd = 2\34 = 71¢
| Augmented 1sn = 4\34 = 141¢
| Common uses = blues
| Common uses = blues
| Important MOS =  
| Important MOS =  
Line 12: Line 15:


== Theory ==
== Theory ==
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.
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! 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.  


== Intervals ==
== Intervals ==
Line 42: Line 84:
| [[36/35]]
| [[36/35]]
| ^1, vm2
| ^1, vm2
| up unison, downminor 2nd
| up 1sn, downminor 2nd
| ^D, vEb
| ^D, vEb
|-
|-
Line 51: Line 93:
| [[21/20]], [[36/35]], [[50/49]], [[33/32]]
| [[21/20]], [[36/35]], [[50/49]], [[33/32]]
| [[28/27]], [[49/48]], [[33/32]]
| [[28/27]], [[49/48]], [[33/32]]
| m2
| ^^1, m2
| minor 2nd
| double-up 1sn, minor 2nd
| Eb
| ^^D, Eb
|-
|-
| 3
| 3
Line 61: Line 103:
| [[14/13]]
| [[14/13]]
| [[15/14]], [[21/20]]
| [[15/14]], [[21/20]]
| ^m2
| vA1, ^m2
| upminor 2nd
| downaug 1sn, upminor 2nd
| ^Eb, vD#
| vD#, ^Eb
|-
|-
| 4
| 4
Line 71: Line 113:
| [[15/14]], [[12/11]]
| [[15/14]], [[12/11]]
| [[12/11]], [[14/13]]
| [[12/11]], [[14/13]]
| ~2
| A1, ~2
| mid 2nd
| aug 1sn, mid 2nd
| D#
| D#, vvE
|-
|-
| 5
| 5
Line 83: Line 125:
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
| ^D#, vE
| vE
|-
|-
| 6
| 6
Line 123: Line 165:
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| ^F, vGb
| ^F
|-
|-
| 10
| 10
Line 133: Line 175:
| ~3
| ~3
| mid 3rd
| mid 3rd
| Gb
| ^^F
|-
|-
| 11
| 11
Line 143: Line 185:
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| ^Gb, vF#
| vF#
|-
|-
| 12
| 12
Line 183: Line 225:
| ^4
| ^4
| up 4th
| up 4th
| ^G, vAb
| ^G
|-
|-
| 16
| 16
Line 193: Line 235:
| ~4, d5
| ~4, d5
| mid 4th, dim 5th
| mid 4th, dim 5th
| Ab
| ^^G, Ab
|-
|-
| 17
| 17
Line 213: Line 255:
| A4, ~5
| A4, ~5
| aug 4th, mid 5th
| aug 4th, mid 5th
| G#
| G#, vvA
|-
|-
| 19
| 19
Line 223: Line 265:
| v5
| v5
| down 5th
| down 5th
| ^G#, vA
| vA
|-
|-
| 20
| 20
Line 263: Line 305:
| ^m6
| ^m6
| upminor 6th
| upminor 6th
| ^Bb, #A#
| ^Bb
|-
|-
| 24
| 24
Line 273: Line 315:
| ~6
| ~6
| mid 6th
| mid 6th
| A#
| vvB
|-
|-
| 25
| 25
Line 283: Line 325:
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| ^A#, vB
| vB
|-
|-
| 26
| 26
Line 323: Line 365:
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| ^C, vDb
| ^C
|-
|-
| 30
| 30
Line 333: Line 375:
| ~7
| ~7
| mid 7th
| mid 7th
| Db
| ^^C
|-
|-
| 31
| 31
Line 343: Line 385:
| vM7
| vM7
| downmajor 7th
| downmajor 7th
| ^Db, vC#
| vC#
|-
|-
| 32
| 32
Line 388: Line 430:


=== Selected just intervals by error ===
=== Selected just intervals by error ===
{| class="wikitable center-all"
The following table shows how [[15-odd-limit intervals]] are represented in 34edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
! colspan="2" |
! prime 2
! prime 3
! prime 5
! 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]] (%)
| 0.0
| +11.1
| +5.4
| -45.0
| +37.9
| +18.5
| +2.6
|}
 
The following table shows how [[15-odd-limit intervals]] are represented in 34edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.  
 
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ Direct mapping (even if inconsistent)
|+ Direct mapping (even if inconsistent)
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