MODMOS scale - Revision history

Overthink: /* Ways of Looking at MODMOS Scales */ change some words; improve grammar

2026-05-03T01:33:51Z

Ways of Looking at MODMOS Scales: change some words; improve grammar

← Older revision		Revision as of 01:33, 3 May 2026
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	== Ways of Looking at MODMOS Scales ==		== 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 ==		== Gene's Terminology ==

BudjarnLambeth: /* Introduction */

2024-01-14T06:30:38Z

Introduction

← Older revision		Revision as of 06:30, 14 January 2024
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	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.		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 ~~considered to be~~ [[inflected MOS]] scales, rather than true MODMOS scales.		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.		In the exposition below, we give a formal treatment of MODMOS scales.

BudjarnLambeth at 06:20, 14 January 2024

2024-01-14T06:20:19Z

← Older revision		Revision as of 06:20, 14 January 2024
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	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.		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.		'''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.		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.

BudjarnLambeth: /* Introduction */

2024-01-14T06:14:42Z

Introduction

← Older revision		Revision as of 06:14, 14 January 2024
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	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.		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.		'''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, they can be obtained by applying one chromatic alteration each to one of the [[mode]]s of the diatonic 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.

BudjarnLambeth: Rewrote introduction to make the definition of MODMOS more strict (and more closely match actual usage)

2024-01-14T06:12:31Z

Rewrote introduction to make the definition of MODMOS more strict (and more closely match actual usage)

← Older revision		Revision as of 06:12, 14 January 2024
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	__FORCETOC__		__FORCETOC__
	== Introduction ==		== 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 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 considered to be [[inflected MOS]] scales, rather than true MODMOS scales.

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

	== Definitions ==		== Definitions ==

Mike Battaglia: /* Ways of Looking at MODMOS Scales */

2023-08-02T20:21:40Z

Ways of Looking at MODMOS Scales

← Older revision		Revision as of 20:21, 2 August 2023
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	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 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.		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.

Mike Battaglia: /* Ways of Looking at MODMOS Scales */ better formatting and added basic proof sketch for why there aren't infinitely many monotonic MODMOS's

2023-08-01T00:20:58Z

Ways of Looking at MODMOS Scales: better formatting and added basic proof sketch for why there aren't infinitely many monotonic MODMOS's

← Older revision		Revision as of 00:20, 1 August 2023
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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.		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 '''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.		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.

Inthar: /* Introduction */ Linked to article "interval class".

2023-07-31T21:57:59Z

Introduction: Linked to article "interval class".

← Older revision		Revision as of 21:57, 31 July 2023
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	__FORCETOC__		__FORCETOC__
	== Introduction ==		== 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.		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.		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.

Mike Battaglia: /* Gene's Terminology */

2023-07-31T06:50:13Z

Gene's Terminology

← Older revision		Revision as of 06:50, 31 July 2023
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	== Gene's Terminology ==		== Gene's Terminology ==

	The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ±N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave.		The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ±N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave.

	If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS; these have all steps the same as Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = \|s-c\|; these we may call enharmonic MODMOS. If we were to limit ourselves to note adjustments of one chroma, then in no case can the complexity be more than 2N.		If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS; these have all steps the same as Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = \|s-c\|; these we may call enharmonic MODMOS. If we were to limit ourselves to note adjustments of one chroma, then in no case can the complexity be more than 2N.

			== Melisse Series ==
	One particular way of generating MODMOS is via the Melisse series. For a MOS of size N with octave period, we may put the 1/1 at the base of the generator chain, so that it can be represented by 0, 1, 2 ... N-1. The Melisse series now consists of the N-1 MODMOS 0, 1, 2 ... N-1-k, 2N-k, 2N-k+1 ... 2N-1 for each k 0 < k < N. Since the 1/1 can be any note, we may then pick the desired note and subtract generators so as to make that note correspond to 0, giving the desired mode of the MODMOS.		One particular way of generating MODMOS is via the Melisse series. For a MOS of size N with octave period, we may put the 1/1 at the base of the generator chain, so that it can be represented by 0, 1, 2 ... N-1. The Melisse series now consists of the N-1 MODMOS 0, 1, 2 ... N-1-k, 2N-k, 2N-k+1 ... 2N-1 for each k 0 < k < N. Since the 1/1 can be any note, we may then pick the desired note and subtract generators so as to make that note correspond to 0, giving the desired mode of the MODMOS.

Mike Battaglia: add some notes about useful ways to look at MODMOS's

2023-07-31T06:46:14Z

add some notes about useful ways to look at MODMOS's

← Older revision		Revision as of 06:46, 31 July 2023
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	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.		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.

	~~Numerous~~ options exist for the choice of chromatic alteration, ~~all of which can be obtained by combining and subtracting intervals from within~~ the ~~MOS. The most common~~ is alteration by ''chroma'', where the chroma is the difference between any pair of intervals sharing the same interval class.		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.

	~~This choice~~ of ~~chromatic alteration interval~~ is ~~so common that~~ the ~~term MODMOS has generally come~~ to ~~be associated~~ only ~~with those scales being altered by chroma~~. 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~~.		== Definitions ==
			An MOS scale is constructed by iterating a generator g inside of a period R. That is, for non-negative integers n, we take n*g and reduce it modulo the period R to the range 0 ≤ i < R. We stop only at points when there are exactly two sizes of intervals in the scale, the large interval L and the small interval s. If the period R is 1/P octave for some integer P, we obtain an octave-repeating [[Periodic_scale\|periodic scale]] by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.

	=~~= Definitions ==~~		If we continue by adding another interval, it will either fall short of R, or exceed it, but either way we will now have an interval c = L-s, which has been called the "chroma". A MODMOS, then, is a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS.
	A MOS scale ~~is constructed~~ by ~~iterating~~ a ~~generator g inside of~~ a ~~period R~~. ~~That~~ is, ~~for non-negative integers n~~, ~~we take n*g~~ and ~~reduce~~ it ~~modulo~~ the ~~period R~~ to ~~the range 0 ≤ i < R~~. ~~We stop~~ only ~~at points when there~~ are ~~exactly two sizes of intervals in~~ the scale, the ~~large interval L~~ and the ~~small interval s~~. If the ~~period R~~ is ~~1/P octave~~ for some ~~integer P~~, ~~we obtain an octave-repeating [[Periodic_scale\|periodic~~ scale~~]] by conjoining P copies~~ of the ~~MOS scale inside R so as~~ to ~~produce~~ a MOS ~~scale for~~ the ~~whole octave~~.
			== 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.

			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.

			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.

			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.

			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.

			== Gene's Terminology ==

	~~If we continue by adding another interval, it will either fall short~~ of ~~R, or exceed it, but either way we will now have an interval c = L-s, which has been called~~ the ~~"chroma". A MODMOS, then, is a chromatically altered~~ MOS~~; that~~ is, a ~~MOS altered by chromas. If we take~~ a ~~MOS scale, and adjust one or more its notes up or down~~ by ~~a chroma consistently~~, so that ~~octave periodicity is respected~~, ~~and do not obtain another MOS~~, ~~then we obtain~~ a ~~MODMOS~~. ~~We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that~~ the ~~monotonic ascending ordering~~ of the ~~notes by size be retained~~, ~~or relaxing~~ the ~~condition that notes are adjusted by a single chroma so that they can be adjusted by any~~ number of ~~them~~, ~~but we will take this as~~ the ~~basic definition~~ of ~~a MODMOS~~.		The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ±N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave.

	The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ±N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = \|s-c\|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N ~~if we limit ourselves to note adjustments of one chroma~~.		If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS; these have all steps the same as Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = \|s-c\|; these we may call enharmonic MODMOS. If we were to limit ourselves to note adjustments of one chroma, then in no case can the complexity be more than 2N.

	One particular way of generating MODMOS is via the Melisse series. For a MOS of size N with octave period, we may put the 1/1 at the base of the generator chain, so that it can be represented by 0, 1, 2 ... N-1. The Melisse series now consists of the N-1 MODMOS 0, 1, 2 ... N-1-k, 2N-k, 2N-k+1 ... 2N-1 for each k 0 < k < N. Since the 1/1 can be any note, we may then pick the desired note and subtract generators so as to make that note correspond to 0, giving the desired mode of the MODMOS.		One particular way of generating MODMOS is via the Melisse series. For a MOS of size N with octave period, we may put the 1/1 at the base of the generator chain, so that it can be represented by 0, 1, 2 ... N-1. The Melisse series now consists of the N-1 MODMOS 0, 1, 2 ... N-1-k, 2N-k, 2N-k+1 ... 2N-1 for each k 0 < k < N. Since the 1/1 can be any note, we may then pick the desired note and subtract generators so as to make that note correspond to 0, giving the desired mode of the MODMOS.
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	Of course, MODMOS are not confined to scales of meantone. If we take the [[hobbit scale]] [[prodigy11]] and tune it in a miracle tuning such as [[72edo\|72et]], we obtain a MODMOS of Miracle[11]. In general, if we choose a rank three temperament with an optimal tuning very close to an optimal tuning for a rank two temperament and then tune a hobbit for it in that optimal rank two temperament tuning, we are very likely to construct an interesting MODMOS scale. It is particularly useful in connection with MODMOS of temperaments where the basic MOS doesn't contain a lot of consonant chords, such as Miracle[11].		Of course, MODMOS are not confined to scales of meantone. If we take the [[hobbit scale]] [[prodigy11]] and tune it in a miracle tuning such as [[72edo\|72et]], we obtain a MODMOS of Miracle[11]. In general, if we choose a rank three temperament with an optimal tuning very close to an optimal tuning for a rank two temperament and then tune a hobbit for it in that optimal rank two temperament tuning, we are very likely to construct an interesting MODMOS scale. It is particularly useful in connection with MODMOS of temperaments where the basic MOS doesn't contain a lot of consonant chords, such as Miracle[11].

			Another quite interesting result of this is that if the rank-3 JI 5-limit major scale of (9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1) is tuned to any rank-2 5-limit temperament with a 7-note MOS, this scale will typically be a MODMOS of that MOS in that temperament. So for instance, this major scale will be a MODMOS of porcupine[7] (as Lssssss b4 #7), of tetracot[7] (as LLLLLLs #2 #3 b4 #7), etc.

	[[Category:MODMOS]]		[[Category:MODMOS]]
	[[Category:MOS scale]]		[[Category:MOS scale]]
	[[Category:scales]]		[[Category:scales]]