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| == JI approximation == | | == JI approximation == |
| 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. | | 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. |
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| 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. | | 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. |
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| === Selected just intervals by error === | | === 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''. | | 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"
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| |+ Direct mapping (even if inconsistent)
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| |-
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| ! Interval, complement
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| ! Error (abs, [[Cent|¢]])
| |
| |-
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| | [[15/13]], [[26/15]]
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| | 0.682
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| |-
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| | [[18/13]], [[13/9]]
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| | 1.324
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| |-
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| | '''[[5/4]], [[8/5]]'''
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| | '''1.922'''
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| |-
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| | [[6/5]], [[5/3]]
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| | 2.006
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| |-
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| | [[13/12]], [[24/13]]
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| | 2.604
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| |-
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| | '''[[4/3]], [[3/2]]'''
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| | '''3.927'''
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| |-
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| | [[13/10]], [[20/13]]
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| | 4.610
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| |-
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| | [[11/9]], [[18/11]]
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| | 5.533
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| |-
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| | [[16/15]], [[15/8]]
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| | 5.849
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| |-
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| | [[10/9]], [[9/5]]
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| | 5.933
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| |-
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| | ''[[14/11]], [[11/7]]''
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| | ''6.021''
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| |-
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| | '''[[16/13]], [[13/8]]'''
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| | '''6.531'''
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| |-
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| | [[13/11]], [[22/13]]
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| | 6.857
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| |-
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| | [[15/11]], [[22/15]]
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| | 7.539
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| |-
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| | [[9/8]], [[16/9]]
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| | 7.855
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| |-
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| | [[12/11]], [[11/6]]
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| | 9.461
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| |-
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| | [[11/10]], [[20/11]]
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| | 11.466
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| |-
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| | ''[[9/7]], [[14/9]]''
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| | ''11.555''
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| |-
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| | ''[[14/13]], [[13/7]]''
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| | ''12.878''
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| |-
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| | '''[[11/8]], [[16/11]]'''
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| | '''13.388'''
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| |-
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| | ''[[15/14]], [[28/15]]''
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| | ''13.560''
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| |-
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| | ''[[7/6]], [[12/7]]''
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| | ''15.482''
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| |-
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| | '''[[8/7]], [[7/4]]'''
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| | '''15.885'''
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| |-
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| | ''[[7/5]], [[10/7]]''
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| | ''17.488''
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| |}
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| {| class="wikitable center-all" | | {{15-odd-limit|34}} |
| |+ Patent val mapping
| | {{15-odd-limit|34.1|title=15-odd-limit intervals by 34d val mapping}} |
| |-
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| ! Interval, complement
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| ! Error (abs, [[Cent|¢]])
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| |-
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| | [[15/13]], [[26/15]]
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| | 0.682
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| |-
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| | [[18/13]], [[13/9]]
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| | 1.324
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| |-
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| | '''[[5/4]], [[8/5]]'''
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| | '''1.922'''
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| |-
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| | [[6/5]], [[5/3]]
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| | 2.006
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| |-
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| | [[13/12]], [[24/13]]
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| | 2.604
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| |-
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| | '''[[4/3]], [[3/2]]'''
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| | '''3.927'''
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| |-
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| | [[13/10]], [[20/13]]
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| | 4.610
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| |-
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| | [[11/9]], [[18/11]]
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| | 5.533
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| |- | |
| | [[16/15]], [[15/8]]
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| | 5.849
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| |-
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| | [[10/9]], [[9/5]]
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| | 5.933
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| |-
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| | '''[[16/13]], [[13/8]]''' | |
| | '''6.531'''
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| |- | |
| | [[13/11]], [[22/13]]
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| | 6.857
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| |-
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| | [[15/11]], [[22/15]]
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| | 7.539
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| |-
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| | [[9/8]], [[16/9]]
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| | 7.855
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| |-
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| | [[12/11]], [[11/6]]
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| | 9.461
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| |-
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| | [[11/10]], [[20/11]]
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| | 11.466
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| |-
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| | '''[[11/8]], [[16/11]]'''
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| | '''13.388'''
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| |-
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| | '''[[8/7]], [[7/4]]'''
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| | '''15.885'''
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| |-
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| | ''[[7/5]], [[10/7]]''
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| | ''17.806''
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| |-
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| | ''[[7/6]], [[12/7]]''
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| | ''19.812''
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| |-
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| | ''[[15/14]], [[28/15]]''
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| | ''21.734''
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| |-
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| | ''[[14/13]], [[13/7]]''
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| | ''22.416''
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| |-
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| | ''[[9/7]], [[14/9]]''
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| | ''23.739''
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| |-
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| | ''[[14/11]], [[11/7]]''
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| | ''29.273''
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| |}
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| == Tuning by ear == | | == Tuning by ear == |