MODMOS scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>~~2013~~-09-~~26 13~~:14:56 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-06-20 19:19:19 UTC</tt>.

: The original revision id was <tt>~~454433324~~</tt>.

: The original revision id was <tt>514557760</tt>.

: The revision comment was: <tt></tt>

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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, then, is a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS.

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

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

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

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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, then, is a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS.

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

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

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

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2013-09-26 13:14:56 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-06-20 19:19:19 UTC</tt>.<br>
-: The original revision id was <tt>454433324</tt>.<br>
+: The original revision id was <tt>514557760</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>
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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, then, is a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS.
-The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ∓N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N if we limit ourselves to note adjustments of one chroma.
+The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ±N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N if we limit ourselves to note adjustments of one chroma.
 One particular way of generating MODMOS is via the Melisse series. For a MOS of size N with octave period, we may put the 1/1 at the base of the generator chain, so that it can be represented by 0, 1, 2 ... N-1. The Melisse series now consists of the N-1 MODMOS 0, 1, 2 ... N-1-k, 2N-k, 2N-k+1 ... 2N-1 for each k 0 &lt; k &lt; N. Since the 1/1 can be any note, we may then pick the desired note and subtract generators so as to make that note correspond to 0, giving the desired mode of the MODMOS.
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 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 &amp;quot;chroma&amp;quot;. A MODMOS, then, is a chromatically altered MOS; that is, a MOS altered by chromas. If we take a MOS scale, and adjust one or more its notes up or down by a chroma consistently, so that octave periodicity is respected, and do not obtain another MOS, then we obtain a MODMOS. We can restrict this definition by requiring that no more than a certain percentage of the notes may be adjusted, or requiring that the monotonic ascending ordering of the notes by size be retained, or relaxing the condition that notes are adjusted by a single chroma so that they can be adjusted by any number of them, but we will take this as the basic definition of a MODMOS.&lt;br /&gt;
 &lt;br /&gt;
-The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ∓N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N if we limit ourselves to note adjustments of one chroma.&lt;br /&gt;
+The name comes from the fact that if the period is an octave and the number of notes in the MOS scale is N, then a chroma moves a note along the chain of generators by ±N, so that a MODMOS, reduced modulo N on the generator chain, becomes a MOS. In all cases the Graham complexity of the chroma c is N, since this is the number of generator steps needed to reach c times P, the number of periods to an octave. If the steps of the MODMOS have a complexity no more than N, we may call it a chromatic MODMOS. Otherwise, the steps may be of complexity greater than N, for instance by having steps of size d = |s-c|; these we may call enharmonic MODMOS. In no case can the complexity be more than 2N if we limit ourselves to note adjustments of one chroma.&lt;br /&gt;
 &lt;br /&gt;
 One particular way of generating MODMOS is via the Melisse series. For a MOS of size N with octave period, we may put the 1/1 at the base of the generator chain, so that it can be represented by 0, 1, 2 ... N-1. The Melisse series now consists of the N-1 MODMOS 0, 1, 2 ... N-1-k, 2N-k, 2N-k+1 ... 2N-1 for each k 0 &amp;lt; k &amp;lt; N. Since the 1/1 can be any note, we may then pick the desired note and subtract generators so as to make that note correspond to 0, giving the desired mode of the MODMOS.&lt;br /&gt;