Maximal evenness: Difference between revisions

Line 19:

[[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic [[5L 2s|diatonic]] scale from [[12edo]], <span style="font-family: monospace;">2 2 1 2 2 2 1</span>, is not only a maximally even scale, but also the Irvian mode of such scale. Such a mode is best shown in odd EDOs, which truly have a "middle" note owing to being odd, and therefore allowing for true symmetric arrangements of notes.

== Discovery of temperaments with a given generator ==

Maximum evenness scales' generator and amount of notes follow the formula LU mod N = 1, where L is the note amount per period, U is the generator, and N is the EDO's cardinality. Note: L and U have to be coprime for the period to be 1 octave.

As such, it's possible to discover a temperament with a given generator in a given EDO simply by [[temperament merging]] the amount of notes with the EDO's cardinality.

=== Example 1: 12edo's diatonic ===

Generator of 12edo's diatonic is 7\12, as is the amount of notes. As such, we simply carry out 7 & 12 to find the desired temperament. In 5-limit, that's meantone, tempering out 81/80, and consistent with world musical practices today.

=== Example 2: 37edo's 11/8 ===

Let's say we want to see what would repeatedly stacking 11th harmonic do well in all of 11-limit, in an EDO that presents it well.

11/8 amounts to 17 steps of 37edo, and the solution to the problem 17*x mod 1 = 37 is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is 24 & 37. The commas that are offered are 2662/2625, 2744/2673, and 3733/3645.

=== Example 3: On-request maximum evenness scales ===

Let's say we want to see what rank two temperament does Sym454 leap rule represent, 62\293 generator with 52/293 note count.

We simply merge 52 & 293 in a selected limit to get our answer. Let's say 17 limit, we get 52 & 243c temperament with a comma list 225/224, 715/714, 2880/2873, 22750/22627 and 60112/60025.

Let's see what temperament does the Tabular Persian or Dee calendar offer (29 & 33). In the 5-limit, we get a contorted Lala-Quinyo (553584375:536870912).

== Real life counterparts ==

@@ Line 19: / Line 19: @@
 [[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic [[5L 2s|diatonic]] scale from [[12edo]], <span style="font-family: monospace;">2 2 1 2 2 2 1</span>, is not only a maximally even scale, but also the Irvian mode of such scale. Such a mode is best shown in odd EDOs, which truly have a "middle" note owing to being odd, and therefore allowing for true symmetric arrangements of notes.
+== Discovery of temperaments with a given generator ==
+Maximum evenness scales' generator and amount of notes follow the formula LU mod N = 1, where L is the note amount per period, U is the generator, and N is the EDO's cardinality. Note: L and U have to be coprime for the period to be 1 octave.
+As such, it's possible to discover a temperament with a given generator in a given EDO simply by [[temperament merging]] the amount of notes with the EDO's cardinality.
+=== Example 1: 12edo's diatonic ===
+Generator of 12edo's diatonic is 7\12, as is the amount of notes. As such, we simply carry out 7 & 12 to find the desired temperament. In 5-limit, that's meantone, tempering out 81/80, and consistent with world musical practices today.
+=== Example 2: 37edo's 11/8 ===
+Let's say we want to see what would repeatedly stacking 11th harmonic do well in all of 11-limit, in an EDO that presents it well.
+/8 amounts to 17 steps of 37edo, and the solution to the problem 17*x mod 1 = 37 is 24, meaning if the generator is 11/8, we are dealing with a 24 tone maximally even scale. As such, the temperament we are looking for is 24 & 37. The commas that are offered are 2662/2625, 2744/2673, and 3733/3645.
+=== Example 3: On-request maximum evenness scales ===
+Let's say we want to see what rank two temperament does Sym454 leap rule represent, 62\293 generator with 52/293 note count.
+We simply merge 52 & 293 in a selected limit to get our answer. Let's say 17 limit, we get 52 & 243c temperament with a comma list 225/224, 715/714, 2880/2873, 22750/22627 and 60112/60025.
+Let's see what temperament does the Tabular Persian or Dee calendar offer (29 & 33). In the 5-limit, we get a contorted Lala-Quinyo (553584375:536870912).
 == Real life counterparts ==