17edo: Difference between revisions

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== Theory ==

17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].

17edo's [[3/2|perfect fifth]] is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].

Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.

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{{Harmonics in equal|17|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 17edo (continued)}}

=== Octave stretch ===

17edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[27edt]] or [[44ed6]].

=== Subsets and supersets ===

17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

17edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]], so it does not contain any nontrivial subset edos, though it contains [[17ed4]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.

== Intervals ==

@@ Line 10: / Line 10: @@
 == Theory ==
-edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].
+edo's [[3/2|perfect fifth]] is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to a [[5L 2s|diatonic]] scale with more constrasting large and small steps. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]].
 Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating.
@@ Line 23: / Line 23: @@
 {{Harmonics in equal|17|intervals=odd|columns=11}}
 {{Harmonics in equal|17|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 17edo (continued)}}
+=== Octave stretch ===
+edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[27edt]] or [[44ed6]].
 === Subsets and supersets ===
-edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.
+edo is the seventh [[prime edo]], following [[13edo]] and coming before [[19edo]], so it does not contain any nontrivial subset edos, though it contains [[17ed4]]. [[34edo]], which doubles it, provides a good correction to harmonics 5 and 17; while [[68edo]], which quadruples it, provides additionally the primes 7, 19, and 31.
 == Intervals ==