Maximal evenness: Difference between revisions

Line 1:

A '''maximally even''' ('''ME''') ~~scale is a [[scale]] inscribed in~~ an [[equal-step tuning]] ~~which contains exactly~~ two step sizes ~~as close in size as possible (differing~~ by exactly one ~~degree~~ of the parent tuning ~~system), and whose steps are distributed as evenly as possible~~. In other words, such a scale satisfies the property of '''maximal evenness'''.

A [[periodic]] [[binary scale]] is '''maximally even''' ('''ME''') with respect to an [[equal-step tuning]] if its two [[step]] sizes are [[Distributional evenness|evenly distributed]] within its [[step pattern]] and differ by exactly one step of the parent tuning. In other words, such a scale satisfies the property of '''maximal evenness'''. The first condition implies that ME scales are [[MOS scale]]s, and the second condition implies that the scale's [[step ratio]] is [[superparticular]].

In particular, within every [[edo]], one can specify such a scale for every smaller number of notes. In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo.

== Mathematics ==

Mathematically, if ''n'' < ''m'', a ME scale of ''n'' notes in ''m''-ed is any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function rounds down to the nearest integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''ed, and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed, satisfying the informal definition above.

In particular, within every [[EDO|edo]] one can specify such a scale for every smaller number of notes.In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo.

~~== Mathematics ==~~

From the MOS theory standpoint, the generator of the scale is a modular multiplicative inverse of it's number of notes and the EDO size. Maximal evenness scale whose generator is equal to it's note amount is called [[concoctic]]. Major and minor scales in standard Western music are such - the generator is a perfect fifth of 7 semitones, as inferred through Pythagorean tuning, and the scale has 7 notes in it.

Line 16:

Line 17:

Maximally even sets tend to be familiar and musically relevant scale collections. Examples:

~~<ul><li>~~The maximally even heptatonic set of [[~~19edo|~~19edo]] is, like the one in 12edo, a diatonic scale.~~</li><li>~~The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are Neutrominant[7].~~</li><li>~~The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], (as well as 13-tonic being an Orwell[13]),while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.~~</li><li>~~The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.~~</li><li>~~The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[~~Circulating~~ temperament~~|circulating temperaments~~]] with the right timbre.

* The maximally even heptatonic set of [[19edo]] is, like the one in [[12edo]], a [[5L 2s|diatonic scale]].

~~</li></ul>~~

* The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are [[4L 3s|mosh scales]] (Neutrominant[7]).

* The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], (as well as 13-tonic being an Orwell[13]),while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.

* The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.

* The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[circulating temperament]]s with the right timbre.

Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[~~Joel_Mandelbaum|~~Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.

Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.

[[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.

[[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic diatonic scale from 12edo, <code>2 2 1 2 2 2 1</code>, 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 ==

Line 43:

Line 47:

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~~ ==

== Sonifications ==

* Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the [[Wikipedia:Hebrew calendar|Hebrew calendar]].

* Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the [[Wikipedia:Iranian calendars|Persian calendar]].

[[Category:Scale]]

[[Category:Todo:cleanup]]

@@ Line 1: / Line 1: @@
 {{Distinguish|Distributional evenness}}
-A '''maximally even''' ('''ME''') scale is a [[scale]] inscribed in an [[equal-step tuning]] which contains exactly two step sizes as close in size as possible (differing by exactly one degree of the parent tuning system), and whose steps are distributed as evenly as possible. In other words, such a scale satisfies the property of '''maximal evenness'''.
+{{Wikipedia}}
+A [[periodic]] [[binary scale]] is '''maximally even''' ('''ME''') with respect to an [[equal-step tuning]] if its two [[step]] sizes are [[Distributional evenness|evenly distributed]] within its [[step pattern]] and differ by exactly one step of the parent tuning. In other words, such a scale satisfies the property of '''maximal evenness'''. The first condition implies that ME scales are [[MOS scale]]s, and the second condition implies that the scale's [[step ratio]] is [[superparticular]].
+In particular, within every [[edo]], one can specify such a scale for every smaller number of notes. In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo.
+== Mathematics ==
 Mathematically, if ''n'' < ''m'', a ME scale of ''n'' notes in ''m''-ed is any [[mode]] of the sequence ME(''n'', ''m'') = [floor(''i''*''m''/''n'') | ''i'' = 1…''n''], where the [[Wikipedia:Floor and ceiling functions|floor]] function rounds down to the nearest integer. It can be proven that ME(''n'', ''m'') is a [[MOS scale]] where the two step sizes differ by exactly 1\''m''ed, and that the indices for the two step sizes are themselves ME when considered as subsets of ''n''-ed, satisfying the informal definition above.
-In particular, within every [[EDO|edo]] one can specify such a scale for every smaller number of notes.In terms of sub-edo representation, a maximally even scale is the closest the parent edo can get to representing the smaller edo.
-== Mathematics ==
 From the MOS theory standpoint, the generator of the scale is a modular multiplicative inverse of it's number of notes and the EDO size. Maximal evenness scale whose generator is equal to it's note amount is called [[concoctic]]. Major and minor scales in standard Western music are such - the generator is a perfect fifth of 7 semitones, as inferred through Pythagorean tuning, and the scale has 7 notes in it.
@@ Line 16: / Line 17: @@
 Maximally even sets tend to be familiar and musically relevant scale collections. Examples:
-<ul><li>The maximally even heptatonic set of [[19edo|19edo]] is, like the one in 12edo, a diatonic scale.</li><li>The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are Neutrominant[7].</li><li>The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], (as well as 13-tonic being an Orwell[13]),while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.</li><li>The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.</li><li>The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[Circulating temperament|circulating temperaments]] with the right timbre.
+* The maximally even heptatonic set of [[19edo]] is, like the one in [[12edo]], a [[5L 2s|diatonic scale]].
-</li></ul>
+* The maximally even heptatonic sets of [[17edo|17edo]] and [[24edo|24edo]], in contrary, are [[4L 3s|mosh scales]] (Neutrominant[7]).
+* The maximally even heptatonic set of [[22edo|22edo]] is Porcupine[7] (the superpythagorean diatonic scale in 22edo is not maximally even), the maximally even octatonic set of 22edo is the octatonic scale of Hedgehog, the maximally even nonatonic set of 22edo is Orwell[9], (as well as 13-tonic being an Orwell[13]),while the maximally even decatonic set of 22edo is the symmetric decatonic scale of Pajara.
+* The maximally even 13-element set in 24edo is Ivan Wyschnegradsky's diatonicized chromatic scale.
+* The maximally even sets in edos 40 and higher have step sizes so close together that they can sound like [[circulating temperament]]s with the right timbre.
-Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel_Mandelbaum|Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.
+Note that "maximally even" is equivalent to "quasi-equal-interval-symmetrical" in [[Joel Mandelbaum]]'s 1961 thesis [http://www.anaphoria.com/mandelbaum.html Multiple Divisions of the Octave and the Tonal Resources of 19-Tone Temperament]. Previous versions of this article have conflated "quasi-equal" with "quasi-equal-interval symmetrical". In fact, "quasi-equal" scales, according to Mandelbaum, meet the first criterion listed above, but not necessarily the second.
-[[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.
+[[Irvian mode]] is a specific mode of the scale, where the notes are also symmetrically arranged. For example, the major mode of the basic diatonic scale from 12edo, <code>2 2 1 2 2 2 1</code>, 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 ==
@@ Line 43: / Line 47: @@
 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 ==
+== Sonifications ==
 * Maximally even heptatonic scale of [[19edo]] is the leap year arrangement of the [[Wikipedia:Hebrew calendar|Hebrew calendar]].
 * Maximally even octatonic scale of [[33edo]] is a leap year arrangement of the Dee calendar and the tabular, evened version of the [[Wikipedia:Iranian calendars|Persian calendar]].
 [[Category:Scale]]
 [[Category:Todo:cleanup]]