MODMOS scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 211216318 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 211787514 - 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:genewardsmith|genewardsmith]] and made on <tt>2011-03- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-18 11:59:23 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>211787514</tt>.<br> | ||
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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> | ||
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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 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. | 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 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. 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 c2 = |s-c|; these we may call | 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 c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma. | ||
=Examples= | =Examples= | ||
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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 | ## 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. | ||
## 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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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 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. <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 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. <br /> | ||
<br /> | <br /> | ||
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 c2 = |s-c|; these we may call | 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 c2 = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N, since we have limited ourselves to note adjustments of one chroma.<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> | ||
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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 /> | 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 | <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 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 &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:8:&lt;h1&gt; --><h1 id="toc4"><a name="Outline for General Algorithm"></a><!-- ws:end:WikiTextHeadingRule:8 -->Outline for General Algorithm</h1> | ||
<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> | <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> | ||