8edo: Difference between revisions

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8edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1187 cents gives the tuning called [[29ed12]].

~~Pure~~-~~octaves~~ 8edo ~~approximates~~ only 2 ~~of the first 11 prime harmonics~~ within 15 cents~~. 29ed12 approximates 5 of them~~.

Of all prime [[harmonic]]s up to 31, pure-octave 8edo only manages to approximate 2/1 and 19/1 within 15 [[cents]], completely missing all the others.

~~Pure-octaves 8edo~~ approximates ~~only 6 of the first 27 integer harmonics~~ within 20 cents~~. 29ed12 approximates 10 of them~~.

By contrast, 29ed12 approximates 2/1, 11/1, 13/1, 17/1 and 31/1 all within 15 cents.

~~In this way~~, ~~29ed12 (compressed-octaves 8edo) is able to provide a wider and stronger palette of [[consonance]]s compared to~~ pure-~~octaves~~ 8edo.

Of all integer harmonics up to 30, pure-octave 8edo approximates the following within 20 cents:

* 2, 4, 8, 16, 19, 27.

~~There are~~ [[zeta peak index]] tunings ~~nearby~~ 8edo ~~as well, however they damage the octave by over~~ 20 cents, ~~rendering~~ them unrecognisable as stretched or compressed 8edo ~~and~~ more like entirely new scales in their own right.

Of all integer harmonics up to 30, 29ed12 approximates the following within 20 cents:

* 2, 6, 11, 12, 13, 17, 20, 22, 25, 26.

This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.

The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.

=== Subsets and supersets ===

@@ Line 30: / Line 30: @@
 edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1187 cents gives the tuning called [[29ed12]].
-Pure-octaves 8edo approximates only 2 of the first 11 prime harmonics within 15 cents. 29ed12 approximates 5 of them.
+Of all prime [[harmonic]]s up to 31, pure-octave 8edo only manages to approximate 2/1 and 19/1 within 15 [[cents]], completely missing all the others.
-Pure-octaves 8edo approximates only 6 of the first 27 integer harmonics within 20 cents. 29ed12 approximates 10 of them.
+By contrast, 29ed12 approximates 2/1, 11/1, 13/1, 17/1 and 31/1 all within 15 cents.
-In this way, 29ed12 (compressed-octaves 8edo) is able to provide a wider and stronger palette of [[consonance]]s compared to pure-octaves 8edo.
+Of all integer harmonics up to 30, pure-octave 8edo approximates the following within 20 cents:
+* 2, 4, 8, 16, 19, 27.
-There are [[zeta peak index]] tunings nearby 8edo as well, however they damage the octave by over 20 cents, rendering them unrecognisable as stretched or compressed 8edo and more like entirely new scales in their own right.
+Of all integer harmonics up to 30, 29ed12 approximates the following within 20 cents:
+* 2, 6, 11, 12, 13, 17, 20, 22, 25, 26.
+This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.
+The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.
 === Subsets and supersets ===