36edo: Difference between revisions

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36edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).

36edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount.

{{Harmonics in equal|36|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 36edo (continued)}}

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=== Additional properties ===

36edo offers a good approximation to the [[acoustic phi|acoustic golden ratio]], as 25\36. [[Heinz Bohlen]] proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.

36edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount.

Thanks to its sevenths, 36edo is an ideal tuning for its size for [[metallic harmony]].

@@ Line 12: / Line 12: @@
 edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).
+edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount.
 {{Harmonics in equal|36|intervals=odd|prec=2|columns=14}}
 {{Harmonics in equal|36|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 36edo (continued)}}
@@ Line 22: / Line 24: @@
 === Additional properties ===
 edo offers a good approximation to the [[acoustic phi|acoustic golden ratio]], as 25\36. [[Heinz Bohlen]] proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.
-edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount.
 Thanks to its sevenths, 36edo is an ideal tuning for its size for [[metallic harmony]].