Rank 3 scale: Difference between revisions

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MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.

MOS scales are ''mirror-symmetric'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern LsLLLsL. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode.'' The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the ''even-symmetric mode'' has step pattern sLssLssLss. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.

MOS scales are ''mirror-symmetric'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern LsLLLsL. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of odd cardinality are symmetric about a note, and mirror-symmetric scales of even cardinality are symmetric about an interval. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode.'' The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the even-symmetric mode has step pattern sLssLssLss. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.

MOS scales and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.

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For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

'''Conjecture:''' The only symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

== Trivalent scales ==

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The scale abacaba is the only PWF / PDE / PMOS scale, and the only trivalent scale that is also symmetric.

The scales a…ba…c, and the scale abacaba are the only pairwise-DE scales, and the only MV3 scales that are also symmetric.

The scales a…ba…c, and the scale abacaba are the only pairwise-DE scales, and the only MV3 scales that are also mirror-symmetric.

There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales.

@@ Line 12: / Line 12: @@
 MOS scales can be generated by stacking a single generator modulo a period. Not all generated scales are MOS.
-MOS scales are ''mirror-symmetric'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern LsLLLsL. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode.'' The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the ''even-symmetric mode'' has step pattern sLssLssLss. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.
+MOS scales are ''mirror-symmetric'', wherein the scale is symmetric about a point. In mirror-symmetric scales of odd cardinality, the axis of symmetric lies on a note of the scale, and so the scale has a ''symmetric mode'', wherein the inverse of each interval (about the period) also exists in the mode. The ''step arrangement'' of the scale in such a mode is a palindrome - e.g., the diatonic scale in Dorian mode has step pattern LsLLLsL. For mirror-symmetric scales of even cardinality, the axis of symmetric lies exactly half-way between two notes of the scale, and no such mode exists. Mirror-symmetric scales of odd cardinality are symmetric about a note, and mirror-symmetric scales of even cardinality are symmetric about an interval. Mirror-symmetric scales of even cardinality can be written in a mode for which the inverse of every interval in the scale about the largest interval of the scale bar the period also exists in the mode. We will call such a mode the ''even-symmetric mode.'' The step pattern of such a mode is a palindrome, followed by a single step size. For example, Magic[10] in the even-symmetric mode has step pattern sLssLssLss. Mirror-symmetric scales may alternatively be defined as scales for which the inverse of every mode is also a mode of the scale. Clearly the symmetric mode is an inverse of itself.
 MOS scales and can be uniquely defined by their ''MOS signature'', i.e. the diatonic scale by 5L 2s.
@@ Line 28: / Line 28: @@
 For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.
-'''Conjecture:''' The only symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
+'''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
 == Trivalent scales ==
@@ Line 68: / Line 68: @@
 The scale abacaba is the only PWF / PDE / PMOS scale, and the only trivalent scale that is also symmetric.
-The scales a…ba…c, and the scale abacaba are the only pairwise-DE scales, and the only MV3 scales that are also symmetric.
+The scales a…ba…c, and the scale abacaba are the only pairwise-DE scales, and the only MV3 scales that are also mirror-symmetric.
 There is only one way to arrange the steps of these scales such that they are pairwise-DE. This means that they can be uniquely described by a signature, like MOS scales.