34edo: Difference between revisions

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== Approximations to Just Intonation ==

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.

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.

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.

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.

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.

Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. On the other hand, the "sharper 7/4", 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] (34 × 2) preserves the 34edo structure and has the intervals 7/8 and 11/8 tuned nearly just.)

=== Selected just intervals by error ===

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 == Approximations to Just Intonation ==
-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.
+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.
-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.
+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.
-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.
+Likewise the 16-cent flat 27\34 approximate 7/4 can be musically useful. On the other hand, the "sharper 7/4", 28\34, sounds more like the "dominant seventh" found in blues and jazz – which some listeners are accustomed to. ([[68edo]] (34 × 2) preserves the 34edo structure and has the intervals 7/8 and 11/8 tuned nearly just.)
 === Selected just intervals by error ===