MODMOS scale
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=Near-MOS (NMOS) Scales=
=**Basic Approach**=
The modern jazz view of music theory is predominantly modal. Jazz musicians are often taught to think of a complex chord as implying one or more "background" modes that fill in the cracks between the notes in the chord. This is used as a tool to aid the musician in melodic improvisation, as well as in finding the most "similar" sounding harmonic extensions for a certain chord. When more than one mode fits a particular chord or melody, the choice is often left up to the improvisational caprice of the soloist. One good reference for more information on the specifics of modal harmony as used in jazz is Ron Miller's [[@http://www.amazon.com/Modal-Jazz-Composition-Harmony-1/dp/B000MMQ2HG|Modal Jazz Composition and Harmony.]]
Much of this paradigm was originally derived from the techniques used by composers during the impressionistic era, and most likely emerged from the attempts of jazz musicians such as Bill Evans to find "proper" scale closures for some of the novel harmonic concepts that were being employed by
Many of the modes commonly used are modes of the melodic minor, harmonic minor, and harmonic major scales. These scales are all obtained by making a single chromatic alteration to the diatonic scale; they are **Near-MOS's** in which only some interval classes fall into two sizes. Furthermore, all of the scales most often used in this fashion are proper. Propriety is so commonly seen that if a chromatic alteration produces a near-MOS that is improper, but is a subset of some other proper scale, the encompassing proper scale will be used. For instance, if one starts with lydian and flattens the 7 to lydian b7, C D E F# G A Bb C is produced (commonly called "lydian dominant"). If one desires to raise the 2 to a #2, the resultant improper scale is produced - C D# E F# G A Bb C, sometimes called the "Hungarian Major" scale. In this case, musicians will commonly reframe this scale as a 7-note subset of the octatonic scale, C Db Eb E F# G A Bb C, which is proper.
A system of indexing exists for these near-MOS's - they have been given names which are often used in common parlance. The modes of melodic minor are generally indexed as mixolydian #4, lydian #5, phrygian #6, dorian #7, etc, or alternatively lydian b7, phrygian b1, dorian b2, ionian b3, which are equivalent. So in a sense, much of the modern jazz approach to modal harmony is already a theory of near-MOS; musicians are often taught this comprehensive system of indexing so as to learn how one scale can chromatically transform into another to aid in the fluid navigation of the 12-tet landscape in live improvisation.
We will see that it is possible to extend this paradigm to other MOS's than just 5L2s. Furthermore, we will see that a lot of the existing terminology, most notably the sharp (#) and flat (b) signs, can also fit into the extended structure in a mathematically rigorous way. Furthermore, the extensions to this paradigm have no need for recourse to ratios or Fokker Periodicity Blocks; the near-MOS's of an MOS can be viewed as inharmonic scale abstractions for purely melodic purposes.
It may prove particularly useful in exploring the near-MOS's of temperaments where the basic MOS doesn't contain a lot of consonant chords; miracle[10] may be a good example of this.
=**Outline**=
# The chromatic vector for an MOS should assume that the MOS is the [[Chromatic Pairs|albitonic]] scale of a [[chromatic pairs|chromatic pair]].
# The chromatic vector doesn't have to be defined in terms of ratios, mappings, or [[Fokker Blocks|periodicity blocks]]. **In general, the chromatic vector c = L-s**, regardless of what mapping you use and regardless of whether or not the scale is proper or improper.
# To apply this systematically to an MOS, we need to define a which mode of the MOS we're making alterations on.
# Sharpening one of the notes in an MOS by this vector can be denoted by an accidental. To keep with tradition, we will use the # sign and the scale degree of the base mode that is being altered, where the first note is scale degree 1. Flattening one of the notes can be denoted by another accidental, in this case the b sign.
# So, to demonstrate over the LLsLLLs mode of 5L2s (Ionian)
## The melodic minor NMOS parent scale is reached by Ionian b3 or Ionian #4.
## The harmonic minor NMOS parent scale is reached by Ionian #4 or Ionian b3, b6.
## The harmonic major NMOS parent scale is reached by Ionian b6 or Ionian b3
## The locrian major NMOS parent scale is reached by Ionian b2,b3 or Ionian #1,#2.
### Other NMOS's exist, but they may be wildly improper; above I stick only to the proper NMOS's that exist in 12-tet.
### One can also arrive at these same NMOS's by making different alterations.
### There are also NMOS's that can be arrived at by three alterations; I chose to not explore that far.
### Sometimes a chromatic alteration simply gives you another mode of the same scale.
# The same procedure can be applied to porcupine. In 22-tet, c = L-s = 3\22 - 2\22 = 1\22. So the chromatic vector here is about 55 cents.
# Let's say we're performing manipulations on the Lssssss mode ("porcupine major"). In 22-tet, this is 4 3 3 3 3 3 3. Some interesting near-MOS's are
## P-major b3 or P-major #7 - 4 2 4 3 3 3 3 - this is a P-major scale where the 5/4 has been replaced by 6/5; for a different mode replace the 11/6 with 15/8.
## P-major b4 - 4 3 2 4 3 3 3 - this is a P-major scale where the 11/8 has been replaced by 4/3; this gives it more of a "fractured" and less of a "wind chimes"y sound.
## P-major b5 - 4 3 3 2 4 3 3 - this is a P-major scale where the 3/2 has been replaced by an approximate 16/11; this ~650 cent interval can function in certain circumstances as a very flat "false fifth"
## P-major b6 - 4 3 3 3 2 4 3 - this is a P-major scale where the 5/3 has been flattened to 8/5. Very gothic sound.
## P-major b7 - 4 3 3 3 3 2 4 - this is a P-major scale where the 11/6 has been flattened to an approximate 7/4. Very "otonal" sounding, as an 8:9:10:11:12:14 hexad exists in this scale.
## P-major #3 - 4 4 2 3 3 3 3 - this is a P-major scale where the 5/4 has been sharpened to a 9/7. Very "bright and brassy" sounding.
## There are many more.
# If the chromatic interval is a generalized version of the "sharp" accidental, then generalized versions of the "half-sharp" accidental also exist.
## If you go from the albitonic scale up to the chromatic scale, a chromatic vector c is implied. If you go up one more level to the hyperchromatic MOS, the large step in the chromatic MOS is split into two new intervals. If the albitonic scale was strictly proper, then its s > c, so s is what gets split. Otherwise, c is what gets split. If the scale is proper, no further shades of chromaticism exist.
## Regardless of which gets split, the size of the new interval, which we will denote c2, is |c-s|.
## Depending on the propriety of the scale you're working with, c2 may or may not be smaller than c, so the "half-sharp" moniker may not always be appropriate.
## For meantone, in 31-tet, this interval is the diesis, which I will notate by "^" and "v" for upward and downward alteration, respectively. This leads to such near-near-MOS's as
### C D Ev F G A B C - Ionian with a neutral third
### C D Ebv F G A B C - In 31-tet, Ebv maps to 7/6, so this may well be thought of as a septimal Dorian scale
### C D E F^ G A B C - Ionian with 4/3 replaced with 11/8
### C D E F^ G A Bbv C - This is Ionian with 4/3 replaced with 11/8 and 9/5 replaced with 7/4
### C D E F^ G Av Bbv C - This is Ionian with 4/3 replaced with 11/8, 9/5 replaced with 7/4, and 5/3 replaced with ~13/8.
### As you can see, the more alterations we make, the less this scale starts to resemble the actual meantone MOS that it originated from.
# One can theoretically alter a scale as many times as one wants.
## However, it is suggested by Rothenberg that the near-MOS's that will be most useful are those that are proper. The question of how to deal with near-MOS's that are derived from scales which are themselves improper, as in superpyth[7], is left up to future research.
## It is also suggested, that, as a problem of managing the complexity of the sheer number of these resulting scales, that if more than two alterations are made, the resultant scale may best be viewed as a new scale in its own right and not a near-MOS of the original scale.
=Outline for General Algorithm=
# Start with the albitonic MOS that you want to modify.
# Compute the chromatic step = L-s.
# Find all of the resultant scales that lie at most N chromatic alteration away from the original MOS, where N is the near-MOS maximum alteration complexity that you want to search for.
# If any of these scales end up being permutations of one another, prune the duplicates.
# If so desired, prune the results to eliminate improper scales.Original HTML content:
<html><head><title>MODMOS Scales</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Near-MOS (NMOS) Scales"></a><!-- ws:end:WikiTextHeadingRule:0 -->Near-MOS (NMOS) Scales</h1> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Basic Approach"></a><!-- ws:end:WikiTextHeadingRule:2 --><strong>Basic Approach</strong></h1> The modern jazz view of music theory is predominantly modal. Jazz musicians are often taught to think of a complex chord as implying one or more "background" modes that fill in the cracks between the notes in the chord. This is used as a tool to aid the musician in melodic improvisation, as well as in finding the most "similar" sounding harmonic extensions for a certain chord. When more than one mode fits a particular chord or melody, the choice is often left up to the improvisational caprice of the soloist. One good reference for more information on the specifics of modal harmony as used in jazz is Ron Miller's <a class="wiki_link_ext" href="http://www.amazon.com/Modal-Jazz-Composition-Harmony-1/dp/B000MMQ2HG" rel="nofollow" target="_blank">Modal Jazz Composition and Harmony.</a><br /> <br /> Much of this paradigm was originally derived from the techniques used by composers during the impressionistic era, and most likely emerged from the attempts of jazz musicians such as Bill Evans to find "proper" scale closures for some of the novel harmonic concepts that were being employed by<br /> <br /> Many of the modes commonly used are modes of the melodic minor, harmonic minor, and harmonic major scales. These scales are all obtained by making a single chromatic alteration to the diatonic scale; they are <strong>Near-MOS's</strong> in which only some interval classes fall into two sizes. Furthermore, all of the scales most often used in this fashion are proper. Propriety is so commonly seen that if a chromatic alteration produces a near-MOS that is improper, but is a subset of some other proper scale, the encompassing proper scale will be used. For instance, if one starts with lydian and flattens the 7 to lydian b7, C D E F# G A Bb C is produced (commonly called "lydian dominant"). If one desires to raise the 2 to a #2, the resultant improper scale is produced - C D# E F# G A Bb C, sometimes called the "Hungarian Major" scale. In this case, musicians will commonly reframe this scale as a 7-note subset of the octatonic scale, C Db Eb E F# G A Bb C, which is proper.<br /> <br /> A system of indexing exists for these near-MOS's - they have been given names which are often used in common parlance. The modes of melodic minor are generally indexed as mixolydian #4, lydian #5, phrygian #6, dorian #7, etc, or alternatively lydian b7, phrygian b1, dorian b2, ionian b3, which are equivalent. So in a sense, much of the modern jazz approach to modal harmony is already a theory of near-MOS; musicians are often taught this comprehensive system of indexing so as to learn how one scale can chromatically transform into another to aid in the fluid navigation of the 12-tet landscape in live improvisation.<br /> <br /> We will see that it is possible to extend this paradigm to other MOS's than just 5L2s. Furthermore, we will see that a lot of the existing terminology, most notably the sharp (#) and flat (b) signs, can also fit into the extended structure in a mathematically rigorous way. Furthermore, the extensions to this paradigm have no need for recourse to ratios or Fokker Periodicity Blocks; the near-MOS's of an MOS can be viewed as inharmonic scale abstractions for purely melodic purposes.<br /> <br /> It may prove particularly useful in exploring the near-MOS's of temperaments where the basic MOS doesn't contain a lot of consonant chords; miracle[10] may be a good example of this.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Outline"></a><!-- ws:end:WikiTextHeadingRule:4 --><strong>Outline</strong></h1> <ol><li>The chromatic vector for an MOS should assume that the MOS is the <a class="wiki_link" href="/Chromatic%20Pairs">albitonic</a> scale of a <a class="wiki_link" href="/chromatic%20pairs">chromatic pair</a>.</li><li>The chromatic vector doesn't have to be defined in terms of ratios, mappings, or <a class="wiki_link" href="/Fokker%20Blocks">periodicity blocks</a>. <strong>In general, the chromatic vector c = L-s</strong>, regardless of what mapping you use and regardless of whether or not the scale is proper or improper.</li><li>To apply this systematically to an MOS, we need to define a which mode of the MOS we're making alterations on.</li><li>Sharpening one of the notes in an MOS by this vector can be denoted by an accidental. To keep with tradition, we will use the # sign and the scale degree of the base mode that is being altered, where the first note is scale degree 1. Flattening one of the notes can be denoted by another accidental, in this case the b sign.</li><li>So, to demonstrate over the LLsLLLs mode of 5L2s (Ionian)<ol><li>The melodic minor NMOS parent scale is reached by Ionian b3 or Ionian #4.</li><li>The harmonic minor NMOS parent scale is reached by Ionian #4 or Ionian b3, b6.</li><li>The harmonic major NMOS parent scale is reached by Ionian b6 or Ionian b3</li><li>The locrian major NMOS parent scale is reached by Ionian b2,b3 or Ionian #1,#2.<ol><li>Other NMOS's exist, but they may be wildly improper; above I stick only to the proper NMOS's that exist in 12-tet.</li><li>One can also arrive at these same NMOS's by making different alterations.</li><li>There are also NMOS's that can be arrived at by three alterations; I chose to not explore that far.</li><li>Sometimes a chromatic alteration simply gives you another mode of the same scale.</li></ol></li></ol></li><li>The same procedure can be applied to porcupine. In 22-tet, c = L-s = 3\22 - 2\22 = 1\22. So the chromatic vector here is about 55 cents.</li><li>Let's say we're performing manipulations on the Lssssss mode ("porcupine major"). In 22-tet, this is 4 3 3 3 3 3 3. Some interesting near-MOS's are<ol><li>P-major b3 or P-major #7 - 4 2 4 3 3 3 3 - this is a P-major scale where the 5/4 has been replaced by 6/5; for a different mode replace the 11/6 with 15/8.</li><li>P-major b4 - 4 3 2 4 3 3 3 - this is a P-major scale where the 11/8 has been replaced by 4/3; this gives it more of a "fractured" and less of a "wind chimes"y sound.</li><li>P-major b5 - 4 3 3 2 4 3 3 - this is a P-major scale where the 3/2 has been replaced by an approximate 16/11; this ~650 cent interval can function in certain circumstances as a very flat "false fifth"</li><li>P-major b6 - 4 3 3 3 2 4 3 - this is a P-major scale where the 5/3 has been flattened to 8/5. Very gothic sound.</li><li>P-major b7 - 4 3 3 3 3 2 4 - this is a P-major scale where the 11/6 has been flattened to an approximate 7/4. Very "otonal" sounding, as an 8:9:10:11:12:14 hexad exists in this scale.</li><li>P-major #3 - 4 4 2 3 3 3 3 - this is a P-major scale where the 5/4 has been sharpened to a 9/7. Very "bright and brassy" sounding.</li><li>There are many more.</li></ol></li><li>If the chromatic interval is a generalized version of the "sharp" accidental, then generalized versions of the "half-sharp" accidental also exist.<ol><li>If you go from the albitonic scale up to the chromatic scale, a chromatic vector c is implied. If you go up one more level to the hyperchromatic MOS, the large step in the chromatic MOS is split into two new intervals. If the albitonic scale was strictly proper, then its s > c, so s is what gets split. Otherwise, c is what gets split. If the scale is proper, no further shades of chromaticism exist.</li><li>Regardless of which gets split, the size of the new interval, which we will denote c2, is |c-s|.</li><li>Depending on the propriety of the scale you're working with, c2 may or may not be smaller than c, so the "half-sharp" moniker may not always be appropriate.</li><li>For meantone, in 31-tet, this interval is the diesis, which I will notate by "^" and "v" for upward and downward alteration, respectively. This leads to such near-near-MOS's as<ol><li>C D Ev F G A B C - Ionian with a neutral third</li><li>C D Ebv F G A B C - In 31-tet, Ebv maps to 7/6, so this may well be thought of as a septimal Dorian scale</li><li>C D E F^ G A B C - Ionian with 4/3 replaced with 11/8</li><li>C D E F^ G A Bbv C - This is Ionian with 4/3 replaced with 11/8 and 9/5 replaced with 7/4</li><li>C D E F^ G Av Bbv C - This is Ionian with 4/3 replaced with 11/8, 9/5 replaced with 7/4, and 5/3 replaced with ~13/8.</li><li>As you can see, the more alterations we make, the less this scale starts to resemble the actual meantone MOS that it originated from.</li></ol></li></ol></li><li>One can theoretically alter a scale as many times as one wants.<ol><li>However, it is suggested by Rothenberg that the near-MOS's that will be most useful are those that are proper. The question of how to deal with near-MOS's that are derived from scales which are themselves improper, as in superpyth[7], is left up to future research.</li><li>It is also suggested, that, as a problem of managing the complexity of the sheer number of these resulting scales, that if more than two alterations are made, the resultant scale may best be viewed as a new scale in its own right and not a near-MOS of the original scale.</li></ol></li></ol><!-- ws:start:WikiTextHeadingRule:6:<h1> --><h1 id="toc3"><a name="Outline for General Algorithm"></a><!-- ws:end:WikiTextHeadingRule:6 -->Outline for General Algorithm</h1> <ol><li>Start with the albitonic MOS that you want to modify.</li><li>Compute the chromatic step = L-s.</li><li>Find all of the resultant scales that lie at most N chromatic alteration away from the original MOS, where N is the near-MOS maximum alteration complexity that you want to search for.</li><li>If any of these scales end up being permutations of one another, prune the duplicates.</li><li>If so desired, prune the results to eliminate improper scales.</li></ol></body></html>