34edo: Difference between revisions

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== Theory ==

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. Its primes 7 and 11 are less accurate, but still usable (with the 34d val for prime 7) with a sharp tendency, in fact mapping all [[15-odd-limit]] intervals consistently except for 7/4 and 8/7 in the 34d val.

34edo contains two [[17edo]]'s and the half-octave tritone of 600{{c}}. 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. Its primes 7 and 11 are less accurate, but still usable (with the 34d val for prime 7) with a sharp tendency, in fact mapping all [[15-odd-limit]] intervals consistently except for 7/4 and 8/7 in the 34d val.

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.

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{{c}}, and a neutral third of 353{{c}}. For [[extraclassical tonality]], a tendo third of 459{{c}} and an arto third of 247{{c}} are also available, approximating 13/10 and 15/13 respectively.

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).

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).

=== Odd harmonics ===

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== Approximation to JI ==

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.

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{{c}} to 35.3{{c}}), 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.

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.

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 == Theory ==
-edo 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. Its primes 7 and 11 are less accurate, but still usable (with the 34d val for prime 7) with a sharp tendency, in fact mapping all [[15-odd-limit]] intervals consistently except for 7/4 and 8/7 in the 34d val.
+edo contains two [[17edo]]'s and the half-octave tritone of 600{{c}}. 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. Its primes 7 and 11 are less accurate, but still usable (with the 34d val for prime 7) with a sharp tendency, in fact mapping all [[15-odd-limit]] intervals consistently except for 7/4 and 8/7 in the 34d val.
-edo'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.
+edo'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{{c}}, and a neutral third of 353{{c}}. For [[extraclassical tonality]], a tendo third of 459{{c}} and an arto third of 247{{c}} are also available, approximating 13/10 and 15/13 respectively.
-edo 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).
+edo supports the [[diatonic scale]], both the simpler 5L&nbsp;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&nbsp;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).
 === Odd harmonics ===
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 == Approximation to JI ==
 [[File:34ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 34edo]]
-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.
+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{{c}} to 35.3{{c}}), 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.
 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.