MODMOS scale: Difference between revisions

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__FORCETOC__

== Introduction ==

A scale is considered to be a ''[[MOS scale]]'' if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. '''MODMOS''' scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of "chromatic alterations" to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.

~~In theory~~, although ~~numerous options exist for~~ the ~~choice~~ of chromatic alteration, the ~~standard is alteration~~ by the MOS's ''chroma'', where the chroma is the difference between any pair of intervals sharing the same interval class. ~~This choice~~ of ~~chromatic alteration~~ interval is ~~so fundamental to~~ the ~~structure~~ of ~~these~~ scales ~~that the term MODMOS~~, ~~in its main sense~~, ~~is generally interpreted as referring to only those~~ scales ~~being altered by this interval in particular~~. In the exposition below, we give a formal treatment of MODMOS~~'s that looks only at chroma-altered~~ scales~~. These scales are distinguished by the sense that they are [[Periodic scale|epimorphic]], and hence of special musical interest. However, alterations by other intervals may also be useful~~.

A scale is considered to be a [[MOS scale]] if every generic [[interval class]] comes in two specific [[interval]] sizes. For example, the familiar [[diatonic scale]] is an MOS.

'''MODMOS scales''', also known as '''altered MOS scales''', generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of "chromatic alterations" to an MOS.

The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS, they can be obtained by applying one chromatic alteration each to one of the [[mode]]s of the diatonic MOS.

A chromatic alteration means changing the size of an interval by increments of the MOS's [[chroma]], where the chroma is the difference between any pair of intervals sharing the same interval class.

Alteration by increments of some other interval is possible, but they lack the useful properties of MODMOS scales, most importantly [[epimorphism]], so they are [[inflected MOS]] scales, rather than true MODMOS scales.

In the exposition below, we give a formal treatment of MODMOS scales.

== Definitions ==

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Any number of alterations are permitted; it is up to the judgment of the composer which of the resulting scales are most musically useful. However, clearly, some MODMOS's will be more useful than others, and it is good to talk about some of the ways this can be the case.

For starters, certain alterations will cause the notes of the scale to no longer be "monotonic" (in ascending order). Typically we are most interested in those MODMOS's which are. In fact, for any MOS, only finitely many MODMOS's will be monotonic in this way.

For starters, certain alterations will cause the notes of the scale to no longer be "'''monotonic'''" (in ascending order). Typically we are most interested in those MODMOS's which are. In fact, for any MOS, only finitely many MODMOS's will be monotonic in this way (up to transpositional equivalence). To see this, note that there is a smallest possible type of step any MODMOS of the original MOS can have, which has been chroma-flattened as much as possible; thus there is a '''flattest MODMOS''' which is made up of N-1 of these minimal seconds in a row, followed by one huge "maximal second" to make up the difference with the octave. Similarly, there will be a '''sharpest MODMOS''' which starts with one huge second, and then the N-1 minimal seconds. Every monotonic MODMOS will be intermediate to these two, formed from various intermediate seconds (of which there are only finitely many type).

Another important note is that the more alterations are made, the less the resulting scale will resemble the original MOS. Thus, it can be very useful, when trying to "organize" the universe of MODMOS's generated by an MOS, to sort them by the total number of alterations that have been made. Thus one can look at ~~singly~~-~~altered~~ MODMOS's, ~~doubly~~-~~altered~~ MODMOS's, and so on, each of which gets further from the character of the core MOS. Similarly, one can look at the maximum number of chroma-alterations that has been made to any particular note at a time: are all notes formed by one chroma alteration, or do we have any notes which have been "doubly adjusted?" Or triply adjusted? etc.

Another important note is that the more alterations are made, the less the resulting scale will resemble the original MOS. Thus, it can be very useful, when trying to "organize" the universe of MODMOS's generated by an MOS, to sort them by the total number of alterations that have been made. Thus one can look at '''single-alteration''' MODMOS's, '''double-alteration''' MODMOS's, and so on, each of which gets further from the character of the core MOS. Similarly, one can look at the maximum number of chroma-alterations that has been made to any particular note at a time: are all notes formed by one chroma alteration, or do we have any notes which have been doubly adjusted? Or triply adjusted? etc.

It is also important to look at, for some MODMOS, how many generators the entire thing will span, which is called the **coverage** of the MODMOS. For instance, the diatonic scale requires 7 contiguous generators, whereas the melodic minor requires 9, the harmonic minor and major scales require 10, and the double harmonic scale requires 11. It can be quite useful to look at the "coverage" of a MODMOS on the generator chain, particularly if one want the MODMOS to fit into a single larger "chromatic" or "enharmonic" sized MOS.

It is also important to look at, for some MODMOS, how many generators the entire thing will span, which is called the '''generator span''' or '''coverage''' of the MODMOS. For instance, the diatonic scale requires 7 contiguous generators, whereas the melodic minor requires 9, the harmonic minor and major scales require 10, and the double harmonic scale requires 11. It can be quite useful to look at the "coverage" of a MODMOS on the generator chain, particularly if one want the MODMOS to fit into a single larger "chromatic" or "enharmonic" sized MOS.

There are doubtless many other useful ways in which one can analyze the MODMOS universe associated to an MOS. As a baseline definition, however, all of these scales are still MODMOS scales.

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 __FORCETOC__
 == Introduction ==
-A scale is considered to be a ''[[MOS scale]]'' if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. '''MODMOS''' scales generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of "chromatic alterations" to an MOS. The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS (the fourths come in three sizes), they can be obtained by applying one chromatic alteration each to one of the modes of the diatonic MOS.
-In theory, although numerous options exist for the choice of chromatic alteration, the standard is alteration by the MOS's ''chroma'', where the chroma is the difference between any pair of intervals sharing the same interval class. This choice of chromatic alteration interval is so fundamental to the structure of these scales that the term MODMOS, in its main sense, is generally interpreted as referring to only those scales being altered by this interval in particular. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished by the sense that they are [[Periodic scale|epimorphic]], and hence of special musical interest. However, alterations by other intervals may also be useful.
+A scale is considered to be a [[MOS scale]] if every generic [[interval class]] comes in two specific [[interval]] sizes. For example, the familiar [[diatonic scale]] is an MOS.
+'''MODMOS scales''', also known as '''altered MOS scales''', generalize the class of scales which are not MOS, but which have been obtained by applying a finite number of "chromatic alterations" to an MOS.
+The familiar melodic and harmonic minor scales are examples of MODMOS's: although these scales are not MOS, they can be obtained by applying one chromatic alteration each to one of the [[mode]]s of the diatonic MOS.
+A chromatic alteration means changing the size of an interval by increments of the MOS's [[chroma]], where the chroma is the difference between any pair of intervals sharing the same interval class.
+Alteration by increments of some other interval is possible, but they lack the useful properties of MODMOS scales, most importantly [[epimorphism]], so they are [[inflected MOS]] scales, rather than true MODMOS scales.
+In the exposition below, we give a formal treatment of MODMOS scales.
 == Definitions ==
@@ Line 13: / Line 22: @@
 Any number of alterations are permitted; it is up to the judgment of the composer which of the resulting scales are most musically useful. However, clearly, some MODMOS's will be more useful than others, and it is good to talk about some of the ways this can be the case.
-For starters, certain alterations will cause the notes of the scale to no longer be "monotonic" (in ascending order). Typically we are most interested in those MODMOS's which are. In fact, for any MOS, only finitely many MODMOS's will be monotonic in this way.
+For starters, certain alterations will cause the notes of the scale to no longer be "'''monotonic'''" (in ascending order). Typically we are most interested in those MODMOS's which are. In fact, for any MOS, only finitely many MODMOS's will be monotonic in this way (up to transpositional equivalence). To see this, note that there is a smallest possible type of step any MODMOS of the original MOS can have, which has been chroma-flattened as much as possible; thus there is a '''flattest MODMOS''' which is made up of N-1 of these minimal seconds in a row, followed by one huge "maximal second" to make up the difference with the octave. Similarly, there will be a '''sharpest MODMOS''' which starts with one huge second, and then the N-1 minimal seconds. Every monotonic MODMOS will be intermediate to these two, formed from various intermediate seconds (of which there are only finitely many type).
-Another important note is that the more alterations are made, the less the resulting scale will resemble the original MOS. Thus, it can be very useful, when trying to "organize" the universe of MODMOS's generated by an MOS, to sort them by the total number of alterations that have been made. Thus one can look at singly-altered MODMOS's, doubly-altered MODMOS's, and so on, each of which gets further from the character of the core MOS. Similarly, one can look at the maximum number of chroma-alterations that has been made to any particular note at a time: are all notes formed by one chroma alteration, or do we have any notes which have been "doubly adjusted?" Or triply adjusted? etc.
+Another important note is that the more alterations are made, the less the resulting scale will resemble the original MOS. Thus, it can be very useful, when trying to "organize" the universe of MODMOS's generated by an MOS, to sort them by the total number of alterations that have been made. Thus one can look at '''single-alteration''' MODMOS's, '''double-alteration''' MODMOS's, and so on, each of which gets further from the character of the core MOS. Similarly, one can look at the maximum number of chroma-alterations that has been made to any particular note at a time: are all notes formed by one chroma alteration, or do we have any notes which have been doubly adjusted? Or triply adjusted? etc.
-It is also important to look at, for some MODMOS, how many generators the entire thing will span, which is called the **coverage** of the MODMOS. For instance, the diatonic scale requires 7 contiguous generators, whereas the melodic minor requires 9, the harmonic minor and major scales require 10, and the double harmonic scale requires 11. It can be quite useful to look at the "coverage" of a MODMOS on the generator chain, particularly if one want the MODMOS to fit into a single larger "chromatic" or "enharmonic" sized MOS.
+It is also important to look at, for some MODMOS, how many generators the entire thing will span, which is called the '''generator span''' or '''coverage''' of the MODMOS. For instance, the diatonic scale requires 7 contiguous generators, whereas the melodic minor requires 9, the harmonic minor and major scales require 10, and the double harmonic scale requires 11. It can be quite useful to look at the "coverage" of a MODMOS on the generator chain, particularly if one want the MODMOS to fit into a single larger "chromatic" or "enharmonic" sized MOS.
 There are doubtless many other useful ways in which one can analyze the MODMOS universe associated to an MOS. As a baseline definition, however, all of these scales are still MODMOS scales.