Step ratio: Difference between revisions

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== Relative interval sizes ==

Part of this perception stems from the fact that, as these L:s ratios change and pass certain critical rational values, the *next* MOS in the sequence changes structure entirely. For instance, when we have L:s > 2, the next MOS changes from "~~xLys~~" to "~~yLxs~~." As an example, with the "~~5L2s~~" diatonic MOS, if we have L/s < 2, the next MOS will be "~~7L5s~~," and if we have L/s > 2, the next MOS will be "~~5L7s~~." (At the point L/s = 2, we have that the next MOS is an equal temperament.)

Part of this perception stems from the fact that, as these L:s ratios change and pass certain critical rational values, the ''next'' MOS in the sequence changes structure entirely. For instance, when we have L:s > 2, the next MOS changes from "''x''L ''y''s" to "''y''L ''x''s". As an example, with the "5L 2s" diatonic MOS, if we have L/s < 2, the next MOS will be "7L 5s", and if we have L/s > 2, the next MOS will be "5L 7s". (At the point L/s = 2, we have that the next MOS is an equal temperament.)

Similar things happen with *all* of these rational points. As the L:s ratio decreases and passes 3/2, for instance, the MOS that is *two* steps after the current one changes. Again, as an example, with the familiar ~~5L2s~~ diatonic MOS sequence, if we have 3:2 < L:s < 2:1, the next two MOS's have 19 and 31 notes, whereas if we have L:s < 3:2, the next two MOS's have 19 and 26 notes.

Similar things happen with ''all'' of these rational points. As the L:s ratio decreases and passes 3/2, for instance, the MOS that is ''two'' steps after the current one changes. Again, as an example, with the familiar 5L 2s diatonic MOS sequence, if we have 3:2 < L:s < 2:1, the next two MOS's have 19 and 31 notes, whereas if we have L:s < 3:2, the next two MOS's have 19 and 26 notes.

Another way to look at this is using [[Rothenberg propriety]]: it so happens that, with one small exception, if a MOS has L:s < 2:1, it is "strictly proper", if it has L:s > 2:1, it is "improper", and if it has L:s = 2:1, it is "proper," all using Rothenberg's definition. The one exception is if the MOS has a single small step (e.g. it is of the form ~~xL1s~~), at which point it is always "strictly proper." Similarly we pass the L:s 3:2 boundary, the *next* MOS changes from strictly proper to improper, and so on.

Another way to look at this is using [[Rothenberg propriety]]: it so happens that, with one small exception, if a MOS has L:s < 2:1, it is "strictly proper", if it has L:s > 2:1, it is "improper", and if it has L:s = 2:1, it is "proper", all using Rothenberg's definition. The one exception is if the MOS has a single small step (e.g. it is of the form ''x''L 1s), at which point it is always "strictly proper". Similarly we pass the L:s = 3:2 boundary, the ''next'' MOS changes from strictly proper to improper, and so on.

The special ratio L:s = ~~phi~~ is unique in that it is the only ratio in which the MOS is strictly proper, and all of the following MOS's are also strictly proper.

The special ratio L:s = φ is unique in that it is the only ratio in which the MOS is strictly proper, and all of the following MOS's are also strictly proper.

== TAMNAMS naming system for step ratios ==

:{{main|TAMNAMS#Step ratio spectrum}}

[[Category:Scale theory]]

[[Category:MOS scales]]

@@ Line 9: / Line 9: @@
 == Relative interval sizes ==
-Part of this perception stems from the fact that, as these L:s ratios change and pass certain critical rational values, the *next* MOS in the sequence changes structure entirely. For instance, when we have L:s > 2, the next MOS changes from "xLys" to "yLxs." As an example, with the "5L2s" diatonic MOS, if we have L/s < 2, the next MOS will be "7L5s," and if we have L/s > 2, the next MOS will be "5L7s." (At the point L/s = 2, we have that the next MOS is an equal temperament.)
+Part of this perception stems from the fact that, as these L:s ratios change and pass certain critical rational values, the ''next'' MOS in the sequence changes structure entirely. For instance, when we have L:s > 2, the next MOS changes from "''x''L ''y''s" to "''y''L ''x''s". As an example, with the "5L 2s" diatonic MOS, if we have L/s < 2, the next MOS will be "7L 5s", and if we have L/s > 2, the next MOS will be "5L 7s". (At the point L/s = 2, we have that the next MOS is an equal temperament.)
-Similar things happen with *all* of these rational points. As the L:s ratio decreases and passes 3/2, for instance, the MOS that is *two* steps after the current one changes. Again, as an example, with the familiar 5L2s diatonic MOS sequence, if we have 3:2 < L:s < 2:1, the next two MOS's have 19 and 31 notes, whereas if we have L:s < 3:2, the next two MOS's have 19 and 26 notes.
+Similar things happen with ''all'' of these rational points. As the L:s ratio decreases and passes 3/2, for instance, the MOS that is ''two'' steps after the current one changes. Again, as an example, with the familiar 5L 2s diatonic MOS sequence, if we have 3:2 < L:s < 2:1, the next two MOS's have 19 and 31 notes, whereas if we have L:s < 3:2, the next two MOS's have 19 and 26 notes.
-Another way to look at this is using [[Rothenberg propriety]]: it so happens that, with one small exception, if a MOS has L:s < 2:1, it is "strictly proper", if it has L:s > 2:1, it is "improper", and if it has L:s = 2:1, it is "proper," all using Rothenberg's definition. The one exception is if the MOS has a single small step (e.g. it is of the form xL1s), at which point it is always "strictly proper." Similarly we pass the L:s 3:2 boundary, the *next* MOS changes from strictly proper to improper, and so on.
+Another way to look at this is using [[Rothenberg propriety]]: it so happens that, with one small exception, if a MOS has L:s < 2:1, it is "strictly proper", if it has L:s > 2:1, it is "improper", and if it has L:s = 2:1, it is "proper", all using Rothenberg's definition. The one exception is if the MOS has a single small step (e.g. it is of the form ''x''L 1s), at which point it is always "strictly proper". Similarly we pass the L:s = 3:2 boundary, the ''next'' MOS changes from strictly proper to improper, and so on.
-The special ratio L:s = phi is unique in that it is the only ratio in which the MOS is strictly proper, and all of the following MOS's are also strictly proper.
+The special ratio L:s = φ is unique in that it is the only ratio in which the MOS is strictly proper, and all of the following MOS's are also strictly proper.
 == TAMNAMS naming system for step ratios ==
-:{{main|TAMNAMS#Step ratio spectrum}}
+{{main| TAMNAMS #Step ratio spectrum }}
+[[Category:Scale theory]]
+[[Category:MOS scales]]