Maximal evenness: Difference between revisions

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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 ~~Maqamic~~[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.

<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.

</li></ul>

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.

[[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 ==

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 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 Maqamic[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.
+<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.
 </li></ul>
 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.
 [[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 ==