MODMOS scale: Difference between revisions

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== Ways of Looking at MODMOS Scales ==

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.

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 some MODMOS's will clearly be more useful than others, and it makes sense 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 (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~~).

For starters, certain alterations will cause the notes of the scale to no longer be "'''monotonic'''" (in ascending order), or equivalently cause one of the scale steps to become negative in size. Typically we are most interested in MODMOS's that are monotonic. 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 types).

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.

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 derived from a MOS, to classify them by the total number of alterations from the original MOS. 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 central 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 '''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.

It is also important to look at, for some MODMOS, how many MOS generators scale 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 wants 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.

There are doubtlessly 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.

== Gene's Terminology ==

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 == Ways of Looking at MODMOS Scales ==
-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.
+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 some MODMOS's will clearly be more useful than others, and it makes sense 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 (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).
+For starters, certain alterations will cause the notes of the scale to no longer be "'''monotonic'''" (in ascending order), or equivalently cause one of the scale steps to become negative in size. Typically we are most interested in MODMOS's that are monotonic. 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 types).
-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.
+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 derived from a MOS, to classify them by the total number of alterations from the original MOS. 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 central 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 '''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.
+It is also important to look at, for some MODMOS, how many MOS generators scale 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 wants 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.
+There are doubtlessly 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.
 == Gene's Terminology ==