Maximal evenness: Difference between revisions
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{{Distinguish|Distributional evenness}} | {{Distinguish|Distributional evenness}} | ||
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{{Wikipedia}} | {{Wikipedia}} | ||
A [[periodic scale|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]]. | A [[periodic scale|periodic]] [[binary scale]] is '''maximally even''' ('''ME''') with respect to an [[equal-step tuning]] if it is the result of rounding a smaller equal tuning to the nearest notes of the parent equal tuning with the same equave. Equivalently, a scale is maximally even 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. An ''m''-note maximally even scale in ''n''-edo is the closest ''n''-edo can get to representing ''m''-edo. | In particular, within every [[edo]], one can specify such a scale for every smaller number of notes. An ''m''-note maximally even scale in ''n''-edo is the closest ''n''-edo can get to representing ''m''-edo. | ||
== | == Mathematics == | ||
Mathematically, if 0 | === Definition === | ||
Mathematically, if {{nowrap|0 < ''n'' < ''m''}}, a ''maximally even (sub)set of size n'' in '''Z'''/''m'''''Z''' is any translate of the set | |||
<math>\operatorname{ME}(n, m) = m\mathbb{Z} + \ | <math>\operatorname{ME}(n, m) = \left\{ m\mathbb{Z} + \ceil{\frac{im}{n}} : i \in \{0, ..., n-1\} \right\} \subseteq \mathbb{Z}/m\mathbb{Z},</math> | ||
where the | where the {{w|ceiling function}} fixes integers and rounds up non-integers to the next higher integer. It can be proven that when ''n'' does not divide ''m'', ME(''n'', ''m'') is a [[MOS scale|MOS subset]] of '''Z'''/''m'''''Z''' where the two step sizes differ by exactly 1, and that the set of degrees where each step size occurs is itself maximally even in '''Z'''/''n'''''Z''', satisfying the informal definition above. ME(''n'', ''m'') is the lexicographically first mode among its rotations, and combined with the fact that it is a MOS, this implies that ME(''n'', ''m'') is the brightest mode in the MOS sense. | ||
It is easy to show that replacing ceil() with round() (rounding half-integers up) gives an equivalent definition; floor() does too, since ME(''n'', ''m'') is a MOS and thus [[chirality|achiral]]. | It is easy to show that replacing ceil() with round() (rounding half-integers up) gives an equivalent definition; floor() does too, since ME(''n'', ''m'') is a MOS and thus [[chirality|achiral]]. | ||
== | === Complement of a maximally even subset is maximally even === | ||
Proof sketch: We may assume that {{nowrap|gcd(''n'', ''m'') {{=}} 1}}; there are two cases. | |||
Case 1: ME(''n'', ''m'') where {{nowrap|''n'' < {{frac|''m''|2}}}}. This is a maximally even subset of Z/mZ with step sizes {{nowrap|L > s > 1}}, which determines the locations of step sizes of 2 in the complement. The rest of the complement's step sizes are 1. The sizes of the chunks of 1 are {{nowrap|L − 2}} and {{nowrap|s − 2}} (0 is a valid chunk size), and the sizes form a maximally even MOS. | |||
Case 2: ME(''n'', ''m'') where {{nowrap|''n'' > {{frac|''m''|2}}}}. This has step sizes 1 and 2. The chunks of 1 (of nonzero size since {{nowrap|''n'' > {{frac|''m''|2}}}}) occupy a maximally even subset of the slots of ME(''n'', ''m'') (*). Now replace each 1 with "|" and each 2 with "$|". | |||
(e.g.) {{nowrap|2112111 → <nowiki>$|||$||||</nowiki>}} | |||
Consider the resulting binary word of "|" and "$". The "|"s form chunks of sizes that differ by 1 and are distributed in a MOS way by (*). The desired complement, occupied by the "$"'s, thus forms a maximally even subset. | |||
== Sound perception == | == Sound perception == | ||
The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31st of an octave instead of one 13th). | The ME scales in 31edo will be closer to equal than those in 13edo, since the two step sizes used to approximate equal will differ by a smaller interval (one 31st of an octave instead of one 13th). | ||
The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's 3 3 3 4 will sound more like 4edo than its 1 1 1 1 1 1 1 1 1 1 1 2 will sound like 12edo. | The parent edo will better represent smaller edos than larger ones. With edos larger than 1/2 of the parent edo, the step sizes will be 2 and 1, which are, proportionally speaking, far from equal. So 13edo's {{nowrap|3 3 3 4}} will sound more like 4edo than its {{nowrap|1 1 1 1 1 1 1 1 1 1 1 2}} will sound like 12edo. | ||
Maximally even sets tend to be familiar and musically relevant scale collections. Examples: | Maximally even sets tend to be familiar and musically relevant scale collections. Examples: | ||
* The maximally even heptatonic set of [[19edo]] is, like the one in [[12edo]], a [[5L 2s|diatonic scale]]. | * The maximally even heptatonic set of [[19edo]] is, like the one in [[12edo]], a [[5L 2s|diatonic scale]]. | ||
* The maximally even heptatonic sets of [[ | * The maximally even heptatonic sets of [[17edo]] and [[24edo]], in contrary, are [[4L 3s|mosh scales]] (Neutrominant[7]). | ||
* The maximally even heptatonic set of [[ | * The maximally even heptatonic set of [[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 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. | * 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. | ||
[[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. | |||
=== Quasi-equal scales === | |||
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. | 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. | ||
[[ | Examples of quasi-equal scales include [[equipentatonic]] and [[equiheptatonic]] scales among others. | ||
== Discovery of temperaments with a given generator == | == 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 | Maximum evenness scales' generator and amount of notes follow the formula {{nowrap|''LU'' (mod ''N'') {{=}} 1}}, where ''L'' is the note amount per period, ''U'' is the generator, and ''N'' is the EDO's cardinality. Note that ''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. | 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 === | === 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. | Generator of 12edo's diatonic is 7\12, as is the amount of notes. As such, we simply carry out {{nowrap|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 === | === 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. | 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, which can be interpreted as [[freivald]] or [[emka]]. | 11/8 amounts to 17 steps of 37edo, and the solution to the problem {{nowrap|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 {{nowrap|24 & 37}}, which can be interpreted as [[freivald]] or [[emka]]. | ||
=== Example 3: On-request maximum evenness scales === | === 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. | 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 | We simply merge {{nowrap|52 & 293}} in a selected limit to get our answer. Let's say 17 limit, we get a {{nowrap|52 & 243c}} temperament with a comma list 225/224, 715/714, 2880/2873, 22750/22627 and 60112/60025. | ||
[[Category:Scale]] | [[Category:Scale]] | ||
[[Category:Todo:cleanup]] | [[Category:Todo:cleanup]] | ||