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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:mbattaglia1|mbattaglia1]] and made on <tt>2011-11-25 12:20:59 UTC</tt>.<br> | | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-11-25 14:18:41 UTC</tt>.<br> |
| : The original revision id was <tt>279023908</tt>.<br> | | : The original revision id was <tt>279046962</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]] |
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| =Introduction= | | =Introduction= |
| A scale is considered to be an **MOS** 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 to the diatonic scale. | | A scale is considered to be an **MOS** 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. |
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| 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. | | 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. |
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| =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 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. | | Consider the MOS series 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 "c", equal to 193.157 - 117.108 = 78.049 cents. This interval is the chroma for meantone[7], and the adjustment of any note up or down by this interval is represented by the sharp # or flat b accidentals. |
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| | The diatonic scale has steps LLsLLLs, which in the key of C can be written C D E F G A B C'. 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, if we flatten the third, we obtain C D Eb F G A B C', the melodic minor scale, or LsLLLLs. Since this scale contains three types of fourths (C-F, "perfect", Eb-A, "augmented", B-Eb, "diminished"), it is no longer an MOS and is therefore a MODMOS. If we apply a further alteration and flatten the sixth as well, we obtain the harmonic minor scale of C D Eb F G Ab B C', which now has three sizes of second and fourth and is therefore also a MODMOS. However, if we apply one more alteration and flatten the seventh, we're left with the natural minor scale of C D Eb F G Ab Bb C' - this is a mode of the diatonic scale, and hence is a MOS rather than a MODMOS. |
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| If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in [[50edo|50et]], for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called "smithgw_modmos12a.scl" in the [[http://www.huygens-fokker.org/docs/scales.zip|Scala Scale Archive]]. Another MODMOS of Meantone[12] in the archive is wreckpop, "smithgw_wreckpop.scl". This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.) | | If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in [[50edo|50et]], for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called "smithgw_modmos12a.scl" in the [[http://www.huygens-fokker.org/docs/scales.zip|Scala Scale Archive]]. Another MODMOS of Meantone[12] in the archive is wreckpop, "smithgw_wreckpop.scl". This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.) |
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| Of course, MODMOS are not confined to scales of meantone. If we take the [[Hobbits|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 [[Hobbits|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].</pre></div> |
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| =MODMOS in Jazz=
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| 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 [[Chromatic pairs|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.]]
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| 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.
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| 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.
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| 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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| =Proper MODMOS=
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| By confining attention to MODMOS which are [[http://en.wikipedia.org/wiki/Rothenberg_propriety|Rothenberg proper]], one may survey all of the proper MODMOS for a particular proper MOS, which are of special interest.
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| In keeping with tradition, we will may the # sign to denote sharpening by a chroma, b to denote flatening, and numbers to denote scale degrees of the base MOS, where the first note is scale degree 1.
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| # So, to demonstrate over the LLsLLLs mode of 5L2s (Ionian)
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| ## The melodic minor MODMOS parent scale is reached by Ionian b3 or Ionian #4.
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| ## The harmonic minor MODMOS parent scale is reached by Ionian #4 or Ionian b3, b6.
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| ## The harmonic major MODMOS parent scale is reached by Ionian b6 or Ionian b3
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| ## The locrian major MODMOS parent scale is reached by Ionian b2,b3 or Ionian #1,#2.
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| ### Other MODMOS's exist, but they may be wildly improper; above we confine ourselves to MODMOS which are proper in 12et.
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| ### There are also MODMOS that can be arrived at by three alterations, which we do not here list.
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| # The same procedure can be applied to porcupine. In 22et, c = L-s = 3\22 - 2\22 = 1\22. So the chroma here is about 55 cents.
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| # The MOS Lssssss mode ("porcupine major") in 22et is 4333333. Some interesting MODMOS are:
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| ## P-major b3 or P-major #7 - 4243333 - 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.
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| ## P-major b4 - 4324333 - 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.
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| ## P-major b5 - 4332433 - 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"
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| ## P-major b6 - 4333243 - this is a P-major scale where the 5/3 has been flattened to 8/5. Very gothic sound.
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| ## P-major b7 - 4333324 - 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.
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| ## P-major #3 - 4423333 - this is a P-major scale where the 5/4 has been sharpened to a 9/7. Very "bright and brassy" sounding.
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| ## There are many more.
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| # If the chromatic interval is a generalized version of the "sharp" accidental, then generalized versions of the "half-sharp" accidental also exist.
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| ## 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 enharmonic 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.
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| ## Regardless of which gets split, the size of the new interval, which we will denote d, is |c-s|.
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| ## Depending on the propriety of the scale you're working with, d may or may not be smaller than c, so the "half-sharp" moniker may not always be appropriate.
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| ## For meantone, in 31et, this interval is the diesis, which I will notate by "^" and "v" for upward and downward alteration, respectively. This leads to such MODMOS as
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| ### C D Ev F G A B C - Ionian with a neutral third
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| ### C D Ebv F G A B C - In 31et, Ebv maps to 7/6, so this may well be thought of as a septimal Dorian scale
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| ### C D E F^ G A B C - Ionian with 4/3 replaced with 11/8
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| ### 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
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| ### 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.
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| ### As you can see, the more alterations we make, the less this scale starts to resemble the actual meantone MOS that it originated from.
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| # One can theoretically alter a scale as many times as one wants.
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| ## However, it is suggested by Rothenberg that the MODMOS that will be most useful are those that are proper. The question of how to deal with MODMOS that are derived from scales which are themselves improper, as in Superpyth[7], is left up to future research.
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| ## 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 MODMOS of the original scale.
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| =Outline for General Algorithm=
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| # Start with the [[Chromatic pairs|albitonic]] MOS that you want to modify.
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| # Compute the chroma = L-s.
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| # Find all of the resultant scales that lie at most N chromatic alteration away from the original MOS, where N is the MODMOS maximum alteration complexity that you want to search for.
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| # If any of these scales end up being permutations of one another, prune the duplicates.
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| # If so desired, prune the results to eliminate improper scales.</pre></div>
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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:12:&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:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#Definitions">Definitions</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --> | <a href="#MODMOS in Jazz">MODMOS in Jazz</a><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --> | <a href="#Proper MODMOS">Proper MODMOS</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Outline for General Algorithm">Outline for General Algorithm</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | | <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:6:&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:6 --><!-- ws:start:WikiTextTocRule:7: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --> | <a href="#Definitions">Definitions</a><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --> | <a href="#Examples">Examples</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> |
| <!-- ws:end:WikiTextTocRule:19 --><br /> | | <!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1> |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1> | | A scale is considered to be an <strong>MOS</strong> if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. <strong>MODMOS</strong> 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.<br /> |
| A scale is considered to be an <strong>MOS</strong> if every generic interval class comes in two specific interval sizes. For example, the familiar diatonic scale is an MOS. <strong>MODMOS</strong> 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 to the diatonic scale.<br /> | |
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| 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 <strong>chroma</strong>, where the chroma is the difference between any pair of intervals sharing the same interval class.<br /> | | 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 <strong>chroma</strong>, where the chroma is the difference between any pair of intervals sharing the same interval class.<br /> |
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:4 -->Examples</h1> | | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:4 -->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 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.<br /> | | Consider the MOS series 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 &quot;c&quot;, equal to 193.157 - 117.108 = 78.049 cents. This interval is the chroma for meantone[7], and the adjustment of any note up or down by this interval is represented by the sharp # or flat b accidentals.<br /> |
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| | The diatonic scale has steps LLsLLLs, which in the key of C can be written C D E F G A B C'. 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, if we flatten the third, we obtain C D Eb F G A B C', the melodic minor scale, or LsLLLLs. Since this scale contains three types of fourths (C-F, &quot;perfect&quot;, Eb-A, &quot;augmented&quot;, B-Eb, &quot;diminished&quot;), it is no longer an MOS and is therefore a MODMOS. If we apply a further alteration and flatten the sixth as well, we obtain the harmonic minor scale of C D Eb F G Ab B C', which now has three sizes of second and fourth and is therefore also a MODMOS. However, if we apply one more alteration and flatten the seventh, we're left with the natural minor scale of C D Eb F G Ab Bb C' - this is a mode of the diatonic scale, and hence is a MOS rather than a MODMOS.<br /> |
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| If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in <a class="wiki_link" href="/50edo">50et</a>, for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called &quot;smithgw_modmos12a.scl&quot; in the <a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/scales.zip" rel="nofollow">Scala Scale Archive</a>. Another MODMOS of Meantone[12] in the archive is wreckpop, &quot;smithgw_wreckpop.scl&quot;. This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.)<br /> | | If we take the 12 note MOS rather than the seven notes of the diatonic scale, then the chroma is a diesis, which in 1/4 comma meantone is exactly 128/125, or 41.059 cents. However, other meantone tunings will give it a different size, and in <a class="wiki_link" href="/50edo">50et</a>, for example, it is exactly 48 cents in size. Subtracting this adjusts a note twelve fifths up on the chain of fifths, and adding it adjusts it twelve notes down. If we take a chain of eleven 50et fifths up from the unison to create a 12-note MOS, so that we have a generator chain from 0 to 11, we may adjust the 3 up to 15 and the 7 up to 19. This leads to the scale called &quot;smithgw_modmos12a.scl&quot; in the <a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/scales.zip" rel="nofollow">Scala Scale Archive</a>. Another MODMOS of Meantone[12] in the archive is wreckpop, &quot;smithgw_wreckpop.scl&quot;. This takes a gamut of Meantone[12] from -4 to 7, which we may call from Ab to F#, and adjusts the 5 (B) up a 48 cent diesis, and so down to -7 fifths, or Cb, and the -2 (Bb) down a diesis, and so up the chain of fifths to 10 (A#.)<br /> |
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| Of course, MODMOS are not confined to scales of meantone. If we take the <a class="wiki_link" href="/Hobbits">hobbit scale</a> <a class="wiki_link" href="/prodigy11">prodigy11</a> and tune it in a miracle tuning such as <a class="wiki_link" href="/72edo">72et</a>, 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].<br /> | | Of course, MODMOS are not confined to scales of meantone. If we take the <a class="wiki_link" href="/Hobbits">hobbit scale</a> <a class="wiki_link" href="/prodigy11">prodigy11</a> and tune it in a miracle tuning such as <a class="wiki_link" href="/72edo">72et</a>, 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].</body></html></pre></div> |
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="MODMOS in Jazz"></a><!-- ws:end:WikiTextHeadingRule:6 -->MODMOS in Jazz</h1>
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| 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 <a class="wiki_link" href="/Chromatic%20pairs">albitonic</a>-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 />
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| 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 />
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| 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 />
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| 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 />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Proper MODMOS"></a><!-- ws:end:WikiTextHeadingRule:8 -->Proper MODMOS</h1>
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| By confining attention to MODMOS which are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Rothenberg proper</a>, one may survey all of the proper MODMOS for a particular proper MOS, which are of special interest.<br />
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| In keeping with tradition, we will may the # sign to denote sharpening by a chroma, b to denote flatening, and numbers to denote scale degrees of the base MOS, where the first note is scale degree 1.<br />
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| <ol><li>So, to demonstrate over the LLsLLLs mode of 5L2s (Ionian)<ol><li>The melodic minor MODMOS parent scale is reached by Ionian b3 or Ionian #4.</li><li>The harmonic minor MODMOS parent scale is reached by Ionian #4 or Ionian b3, b6.</li><li>The harmonic major MODMOS parent scale is reached by Ionian b6 or Ionian b3</li><li>The locrian major MODMOS parent scale is reached by Ionian b2,b3 or Ionian #1,#2.<ol><li>Other MODMOS's exist, but they may be wildly improper; above we confine ourselves to MODMOS which are proper in 12et.</li><li>There are also MODMOS that can be arrived at by three alterations, which we do not here list.</li></ol></li></ol></li><li>The same procedure can be applied to porcupine. In 22et, c = L-s = 3\22 - 2\22 = 1\22. So the chroma here is about 55 cents.</li><li>The MOS Lssssss mode (&quot;porcupine major&quot;) in 22et is 4333333. Some interesting MODMOS are:<ol><li>P-major b3 or P-major #7 - 4243333 - 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 - 4324333 - this is a P-major scale where the 11/8 has been replaced by 4/3; this gives it more of a &quot;fractured&quot; and less of a &quot;wind chimes&quot;y sound.</li><li>P-major b5 - 4332433 - 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 &quot;false fifth&quot;</li><li>P-major b6 - 4333243 - this is a P-major scale where the 5/3 has been flattened to 8/5. Very gothic sound.</li><li>P-major b7 - 4333324 - this is a P-major scale where the 11/6 has been flattened to an approximate 7/4. Very &quot;otonal&quot; sounding, as an 8:9:10:11:12:14 hexad exists in this scale.</li><li>P-major #3 - 4423333 - this is a P-major scale where the 5/4 has been sharpened to a 9/7. Very &quot;bright and brassy&quot; sounding.</li><li>There are many more.</li></ol></li><li>If the chromatic interval is a generalized version of the &quot;sharp&quot; accidental, then generalized versions of the &quot;half-sharp&quot; accidental also exist.<ol><li>If you go from the <a class="wiki_link" href="/Chromatic%20pairs">albitonic</a> scale up to the chromatic scale, a chroma c is implied. If you go up one more level to the enharmonic MOS, the large step in the chromatic MOS is split into two new intervals. If the albitonic scale was strictly proper, then its s &gt; c, so s is what gets split. Otherwise, c is what gets split.</li><li>Regardless of which gets split, the size of the new interval, which we will denote d, is |c-s|.</li><li>Depending on the propriety of the scale you're working with, d may or may not be smaller than c, so the &quot;half-sharp&quot; moniker may not always be appropriate.</li><li>For meantone, in 31et, this interval is the diesis, which I will notate by &quot;^&quot; and &quot;v&quot; for upward and downward alteration, respectively. This leads to such MODMOS 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 31et, 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 MODMOS that will be most useful are those that are proper. The question of how to deal with MODMOS 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 MODMOS of the original scale.</li></ol></li></ol><!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Outline for General Algorithm"></a><!-- ws:end:WikiTextHeadingRule:10 -->Outline for General Algorithm</h1>
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| <ol><li>Start with the <a class="wiki_link" href="/Chromatic%20pairs">albitonic</a> MOS that you want to modify.</li><li>Compute the chroma = 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 MODMOS 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></pre></div>
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