34edo: Difference between revisions

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== 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.

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. 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:34_equal_temperament|Wikipedia]])

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

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

== Tuning by ear ==

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 == 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.
+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. 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:34_equal_temperament|Wikipedia]])
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 * 34et has a lower relative error than any previous ETs in the 5-limit. The next ET that does better in this subgroup is 53.
 * 34et is most prominent in the 2.3.5.13.17 and 2.3.5.11.13.17 subgroups. The next ET that does better in these subgroups is 217 and 87, respectively.
 == Tuning by ear ==