MODMOS scale: Difference between revisions
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Wikispaces>mbattaglia1 **Imported revision 210413818 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-03-14 16:45:00 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>210413818</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]] | ||
=Definitions= | |||
=Definitions= | |||
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]] by conjoining P copies of the MOS scale inside R so produce a MOS scale for the whole octave. | 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]] by conjoining P copies of the MOS scale inside R so produce a MOS scale for the whole octave. | ||
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 is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust 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 require that the monotonic ascending ordering of the notes by size be retained, or relax 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. | 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 is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust 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 require that the monotonic ascending ordering of the notes by size be retained, or relax 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. | ||
=Examples= | =Examples= | ||
Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and | Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and | ||
do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS. | do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS. | ||
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=MODMOS in Jazz= | =MODMOS in Jazz= | ||
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.]] | 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; all of these scales commonly used are Rothenberg proper scales and can be thought of as albitonic-sized proper scale "closures" of 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 | 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 a theory to explain some of the novel harmonic concepts that were being employed by composers such as Debussy and Ravel. | ||
Many of the | Many of the scales 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 MODMOS 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 MODMOS 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 MODMOS - 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 MODMOS; 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. | A system of indexing exists for these MODMOS - 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 MODMOS; 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. | ||
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## There are many more. | ## 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 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 [[Chromatic pairs|albitonic]] scale up to the chromatic scale, a chroma 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 you go from the [[Chromatic pairs|albitonic]] scale up to the chromatic scale, a chroma 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. | ||
## Regardless of which gets split, the size of the new interval, which we will denote c2, is |c-s|. | ## 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. | ## 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. | ||
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<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>MODMOS Scales</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definitions">Definitions</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#MODMOS in Jazz">MODMOS in Jazz</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Proper MODMOS">Proper MODMOS</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Outline for General Algorithm">Outline for General Algorithm</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>MODMOS Scales</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definitions">Definitions</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#MODMOS in Jazz">MODMOS in Jazz</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Proper MODMOS">Proper MODMOS</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Outline for General Algorithm">Outline for General Algorithm</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | ||
<!-- ws:end:WikiTextTocRule:16 --> | <!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definitions"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definitions</h1> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definitions"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definitions</h1> | 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 &lt;= i &lt; 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 <a class="wiki_link" href="/periodic%20scale">periodic scale</a> by conjoining P copies of the MOS scale inside R so produce a MOS scale for the whole octave.<br /> | ||
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 &lt;= i &lt; 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 <a class="wiki_link" href="/periodic%20scale">periodic scale</a> by conjoining P copies of the MOS scale inside R so produce a MOS scale for the whole octave.<br /> | |||
<br /> | <br /> | ||
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 &quot;chroma&quot;. A MODMOS is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust 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 require that the monotonic ascending ordering of the notes by size be retained, or relax 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.<br /> | 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 &quot;chroma&quot;. A MODMOS is in essence, a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust 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 require that the monotonic ascending ordering of the notes by size be retained, or relax 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.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> | ||
Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and | Consider the MOS of 1/4-comma meantone, where the generator g is log2(5)/4 = 0.58048 octaves, or 696.578 cents. The seven note MOS is the diatonic scale tuned in 1/4 comma tuning, with large step L a whole tone of 193.157 cents and small step s a diatonic semitone of 117.108 cents. L-s is the chromatic semitone, 78.049 cents. This is the interval whose adjustment of a note up or down is represented by a sharp # or flat b symbol. The diatonic scale has steps LLsLLLs, which in the key of C can be written CDEFGABC'. From the definition of a MODMOS, if we add sharps and flats to this, and <br /> | ||
do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.<br /> | do not get another diatonic scale, then we have a MODMOS. For example LsLLLLs, which is CDEbFGABC', is the melodic minor scale, which is therefore a MODMOS. The harmonic minor scale is CDEbFGAbBC', and is therefore also a MODMOS. However, the natural minor, CDEbFGAbBbC' is a mode of the diatonic scale, and a MOS rather than a MODMOS.<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="MODMOS in Jazz"></a><!-- ws:end:WikiTextHeadingRule:4 -->MODMOS in Jazz</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="MODMOS in Jazz"></a><!-- ws:end:WikiTextHeadingRule:4 -->MODMOS in Jazz</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 &quot;background&quot; 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 /> | 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 &quot;background&quot; modes that fill in the cracks between the notes in the chord; all of these scales commonly used are Rothenberg proper scales and can be thought of as albitonic-sized proper scale &quot;closures&quot; of 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 /> | <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 | 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 a theory to explain some of the novel harmonic concepts that were being employed by composers such as Debussy and Ravel.<br /> | ||
<br /> | <br /> | ||
Many of the | Many of the scales 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 MODMOS 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 MODMOS 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 &quot;Lydian Dominant&quot;). 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 &quot;Hungarian Major&quot; 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 /> | <br /> | ||
A system of indexing exists for these MODMOS - 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 MODMOS; 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 /> | A system of indexing exists for these MODMOS - 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 MODMOS; 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 /> | ||