MODMOS scale: Difference between revisions

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__FORCETOC__

=Introduction=

== Introduction ==

A scale is considered to be ~~an '~~''[[MOS scale~~|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.

A scale is considered to be a ''[[MOS scale]]'' 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.

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.

This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished~~~~ by the sense that they are ~~~~[[~~Periodic_scale~~|epimorphic]]~~~~, and hence of special musical interest. ~~~~However, alterations by other intervals may also be useful.

This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished by the sense that they are [[Periodic scale|epimorphic]], and hence of special musical interest. However, alterations by other intervals may also be useful.

=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|periodic scale]] by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.

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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.

=Examples=

== Examples ==

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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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#.)

Of course, MODMOS are not confined to scales of meantone. If we take the [[~~Hobbits|~~hobbit scale]] [[~~prodigy11|~~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 [[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].

[[Category:modmos]]

[[Category:mos]]

[[Category:scales]]

@@ Line 1: / Line 1: @@
 __FORCETOC__
-=Introduction=
+== Introduction ==
-A scale is considered to be an '''[[MOS scale|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.
+A scale is considered to be a ''[[MOS scale]]'' 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.
-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.
-This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished<span style="line-height: 1.5;"> by the sense that they are </span><span style="line-height: 1.5;">[[Periodic_scale|epimorphic]]</span><span style="line-height: 1.5;">, and hence of special musical interest. </span>However, alterations by other intervals may also be useful.
+This choice of chromatic alteration interval is so common that the term MODMOS has generally come to be associated only with those scales being altered by chroma. In the exposition below, we give a formal treatment of MODMOS's that looks only at chroma-altered scales. These scales are distinguished by the sense that they are [[Periodic scale|epimorphic]], and hence of special musical interest. However, alterations by other intervals may also be useful.
-=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 &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 [[Periodic_scale|periodic scale]] by conjoining P copies of the MOS scale inside R so as to produce a MOS scale for the whole octave.
@@ Line 16: / Line 16: @@
 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.
-=Examples=
+== Examples ==
 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.
@@ Line 23: / Line 23: @@
 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#.)
-Of course, MODMOS are not confined to scales of meantone. If we take the [[Hobbits|hobbit scale]] [[prodigy11|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 [[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].
 [[Category:modmos]]
 [[Category:mos]]
 [[Category:scales]]