Step ratio: Difference between revisions

(One intermediate revision by one other user not shown)

Line 5:

|ja =

}}

~~In the context of scales described using whole-number step sizes, a~~ '''step ratio''' is a ratio of a scale'~~s step sizes~~, ~~where~~ the ~~values are listed~~ in ~~decreasing order of size. For a~~ [[~~MOS scale|moment-of-symmetry~~]] ~~scale~~, ~~this is denoted in the general form of L:s. This is also called~~ '''Blackwood's R''', after Easley Blackwood who described it for diatonic ~~mosses~~ and referred to this ratio as R.

The '''step ratio''' of a [[scale]] is the [[Wikipedia:ratio|ratio]] between the [[interval size measure|size]]s of its [[step]]s. The step ratio of a [[binary scale]] is also known as '''hardness''', alluding to the terms ''hard'' and ''soft'' used to name step ratios in [[TAMNAMS]], or as '''Blackwood's R''', after [[Easley Blackwood]] who described it for [[5L 2s|diatonic]] scales in ''The Structure of Recognizable Diatonic Tunings'' and referred to this ratio as ''R''.

~~The melodic sound of a~~ MOS scale is ~~not just affected by~~ the ~~tuning of its intervals~~, ~~but by~~ the sizes of ~~its~~ steps. ~~Step ratios whose~~ large ~~and~~ small step ~~are close to equal to one another may sound smoother, softer, or more mellow~~. In ~~contrast~~, step ~~ratios whose large step~~ is ~~significantly larger than~~ the ~~small step may sound jagged~~, ~~dramatic~~, ~~or sparkly~~.

For binary scales, including [[MOS scale|moment-of-symmetry]] scales, the step ratio is usually written in the form ''x'':''y'', where ''x'' and ''y'' represent the sizes of the large and small steps respectively (''x'' > ''y''). For example, in a scale with a step ratio of 2:1, each large step has twice the size of a small step. In general, for ''n''-ary scales, the step ratio is usually written in the form ''x''1:''x''2:...:''x''''n'', where ''x''1, ''x''2, ..., ''x''''n'' represent the size of each step in decreasing order of size.

At the extremes ~~are step ratios whose~~ large and small steps ~~either~~ equal ~~to one another~~, or ~~where~~ the small step "collapses" to zero. ~~At this point~~, the ~~step pattern of the MOS scale will become increasingly ambiguous; this is as much~~ a ~~feature as a bug - it depends on your intent. The '''~~step ratio~~''' or '''hardness''', the ratio between the sizes of L and s, is thus important~~ to ~~the sound of the scale~~.

The step ratio of a scale is mostly associated with its melodic shape. Scales whose steps are very similar in size may sound melodically smoother, softer, or more mellow. In contrast, scales with steps of very different sizes may sound jagged, dramatic, or sparkly.

A binary scale becomes "softer" as its step ratio approaches 1:1 and "harder" when approaching ∞:1 (or 1:0). At the extremes, either the large and small steps become equal, or the small step "collapses" to zero. In both cases, it [[Wikipedia:degeneracy (mathematics)|degenerate]]s into a unary scale (by omitting the zero-sized intervals if necessary), with a step ratio trivially equal to 1.

== Relative interval sizes ==

{{todo|clarify|inline=1|comment=This section relies on multiple assumptions which are not made clear to the reader (including the knowledge of advanced MOS concepts, which should minimally be linked to, or ideally avoided whenever possible).}}

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.)

Part of this perception stems from the fact that, as these ''x'':''y'' 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 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.

Line 22:

Line 24:

== TAMNAMS naming system for step ratios ==

In [[TAMNAMS]], a step ratio of 2:1 is named ''basic''. Step ratios between 1:1 and 2:1 are named ''soft-of-basic'', with 3:2 being ''soft'', while step ratios between 2:1 and ∞:1 are named ''hard-of-basic'', with 3:1 being ''hard''. Other names are given to several other common ratios, as well as to ranges between these ratios.

It is also proposed to use pairwise naming for step ratios of ternary scales. For example, 3:2:1 would be named ''soft-basic'', because 3:2 is soft and 2:1 is basic.

[[Category:Scale]]

[[Category:MOS scale]]

@@ Line 5: / Line 5: @@
 |ja =
 }}
-In the context of scales described using whole-number step sizes, a '''step ratio''' is a ratio of a scale's step sizes, where the values are listed in decreasing order of size. For a [[MOS scale|moment-of-symmetry]] scale, this is denoted in the general form of L:s. This is also called '''Blackwood's R''', after Easley Blackwood who described it for diatonic mosses and referred to this ratio as R.
+The '''step ratio''' of a [[scale]] is the [[Wikipedia:ratio|ratio]] between the [[interval size measure|size]]s of its [[step]]s. The step ratio of a [[binary scale]] is also known as '''hardness''', alluding to the terms ''hard'' and ''soft'' used to name step ratios in [[TAMNAMS]], or as '''Blackwood's R''', after [[Easley Blackwood]] who described it for [[5L 2s|diatonic]] scales in ''The Structure of Recognizable Diatonic Tunings'' and referred to this ratio as ''R''.
-The melodic sound of a MOS scale is not just affected by the tuning of its intervals, but by the sizes of its steps. Step ratios whose large and small step are close to equal to one another may sound smoother, softer, or more mellow. In contrast, step ratios whose large step is significantly larger than the small step may sound jagged, dramatic, or sparkly.
+For binary scales, including [[MOS scale|moment-of-symmetry]] scales, the step ratio is usually written in the form ''x'':''y'', where ''x'' and ''y'' represent the sizes of the large and small steps respectively (''x'' > ''y''). For example, in a scale with a step ratio of 2:1, each large step has twice the size of a small step. In general, for ''n''-ary scales, the step ratio is usually written in the form ''x''<sub>1</sub>:''x''<sub>2</sub>:...:''x''<sub>''n''</sub>, where ''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub> represent the size of each step in decreasing order of size.
-At the extremes are step ratios whose large and small steps either equal to one another, or where the small step "collapses" to zero. At this point, the step pattern of the MOS scale will become increasingly ambiguous; this is as much a feature as a bug - it depends on your intent. The '''step ratio''' or '''hardness''', the ratio between the sizes of L and s, is thus important to the sound of the scale.
+The step ratio of a scale is mostly associated with its melodic shape. Scales whose steps are very similar in size may sound melodically smoother, softer, or more mellow. In contrast, scales with steps of very different sizes may sound jagged, dramatic, or sparkly.
+A binary scale becomes "softer" as its step ratio approaches 1:1 and "harder" when approaching ∞:1 (or 1:0). At the extremes, either the large and small steps become equal, or the small step "collapses" to zero. In both cases, it [[Wikipedia:degeneracy (mathematics)|degenerate]]s into a unary scale (by omitting the zero-sized intervals if necessary), with a step ratio trivially equal to 1.
 == Relative interval sizes ==
+{{todo|clarify|inline=1|comment=This section relies on multiple assumptions which are not made clear to the reader (including the knowledge of advanced MOS concepts, which should minimally be linked to, or ideally avoided whenever possible).}}
-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.)
+Part of this perception stems from the fact that, as these ''x'':''y'' 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 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.
@@ Line 22: / Line 24: @@
 == TAMNAMS naming system for step ratios ==
-{{main| TAMNAMS #Step ratio spectrum }}
+{{Main| TAMNAMS #Step ratio spectrum }}
+In [[TAMNAMS]], a step ratio of 2:1 is named ''basic''. Step ratios between 1:1 and 2:1 are named ''soft-of-basic'', with 3:2 being ''soft'', while step ratios between 2:1 and ∞:1 are named ''hard-of-basic'', with 3:1 being ''hard''. Other names are given to several other common ratios, as well as to ranges between these ratios.
+It is also proposed to use pairwise naming for step ratios of ternary scales. For example, 3:2:1 would be named ''soft-basic'', because 3:2 is soft and 2:1 is basic.
 [[Category:Scale]]
 [[Category:MOS scale]]