Modal UDP notation: Difference between revisions

Given a monotone periodic scale ''S'', suppose it is also a [[MOS]] or DE scale. Let the generator {{nowrap|''S''(''m'') {{=}} ''g''}} be such that {{nowrap|''g'' &ge; ''S''(''i'' + ''m'') − ''S''(''i'')}} for all ''i''. If ''Q'' is the period of ''S'', let ''u'' be the largest integer such that {{nowrap|0 &le; ''u'' &lt; ''Q''}} and {{nowrap|''S''(''m''{{dot}}''u'') {{=}} ''g''{{dot}}''u''}}, and ''d'' the largest integer such that {{nowrap|0 &le; ''d'' &lt; ''Q''}} and {{nowrap|''S''(−''m''{{dot}}''d'') {{=}} −''g''{{dot}}''d''}}. If {{nowrap|''S''(''P''{{dot}}''Q'') {{=}} octave}}, so that ''P'' is the number of periods to an octave, let {{nowrap|''U'' {{=}} ''P''{{dot}}''u''}} and {{nowrap|''D'' {{=}} ''P''{{dot}}''d''}}. Then the UDP notation for the given mode is is <math>U|D(P)</math>. If <math>P = 1</math> we may omit it and just write <math>U|D</math>.

For example, consider the quasiperiodic function <math>\text{Ionian}(i) = V[(i+3 \bmod 7)+1] + 31 \left \lceil{\frac{n+4}{7}-49} \right \rceil</math>, where <math>V = [5, 10, 15, 18, 23, 28, 31]</math>. This has period 7, and <math>\text{Ionian}(7) = 31</math>, where the tuning is [[31edo]] so that 31 represents an octave. Going up from 0, it has values <math>0, 5, 10, 13, 18, 23, 28, 31, 36, 41...</math> corresponding to <math>0, 1, 2, 3, 4, 5, 6, 7, 8, 9...</math>, and going down from 0, it gives <math>0, -3, -8, -13...</math> corresponding to <math>0, -1, -2, -3...</math>. This gives the Ionian, or major, mode of the diatonic scale. Then <math>\text{Ionian}(4) = 18</math>, the fifth, and <math>18 ≥ \text{Ionian}(i+4)-\text{Ionian}(i)</math> for all ''i''. We have <math>\text{Ionian}(4) = 18</math>, <math>\text{Ionian}(8) = 36</math>, <math>\text{Ionian}(12) = 54</math>, <math>\text{Ionian}(16) = 72</math> and <math>\text{Ionian}(20) = 90</math>. However, <math>\text{Ionian}(4·6) = \text{Ionian}(24) = 106</math>, which is less than <math>6·18 = 108</math>. Hence the largest value for which <math>\text{Ionian}(4·u)</math> and <math>18·u</math> are equal is <math>u = 5</math>. Similarly, <math>\text{Ionian}(-4) = -18</math>, but <math>\text{Ionian}(-8) = -34</math>, not -36, and so <math>d = 1</math>. Since <math>­\text{Ionian}(7) = 31</math>, which is the octave, <math>P = 1</math>, so <math>U = u = 5</math>, <math>D = d = 1</math>, and the UDP notation for Ionian is 5|1(1), or simply 5|1.

For example, the proper generator for meantone[7] is the perfect fifth, because it's larger than the other specific interval it shares a class with, the diminished fifth. Consequentially, meantone[7]'s Ionian mode is 5|1(1), which is 5|1 for short, because it contains five chroma-positive generators up from the root and one down, as in the diagram F-[C]-G-D-A-E-B for C Ionian. This also means it has five "sharper" scale degrees - the second, third, fifth, sixth, and seventh - and one "flatter" scale degree - the fourth. If we want to sharpen the fourth to turn it into an augmented fourth, we arrive at 6|0 or [C]-G-D-A-E-B-F#. Conversely, Aeolian mode, with only two sharp scale degrees - the second and fifth - is 2|4. We can add accidentals as well, so that meantone's harmonic minor is 2|4 #7.

 

* The ascending melodic minor scale is 5|1(1) b3, abbreviated 5|1 b3 for short, but could also be 3|3(1) #7, abbreviated 3|3 #7 for short.

* Paul Erlich's standard pentachordal major for Pajara[10] is 4|4(2) #8, or alternatively 6|2(2) b3.

* Porcupine[7] Lssssss mode, but altered with 7/4 instead of 11/6, is 6|0 b7.

** The major pentatonic scale is a subset of the major scale. The genchain is simply shortened on either end. Thus one would expect the major pentatonic scale contained within Meantone[7] 5|1 to be Meantone[5] 4|0. But instead it's Meantone[5] 0|4. This is because changing the size of the MOS scale often changes the generator to its octave inverse. Meantone[5]'s bright generator is the 4th not the 5th.

** Because the MOS type that a temperament's MOS represents is tuning-dependent, the choice of bright generator can be tuning-dependent. For example, assume you are using approximately but not exactly 700{{c}} for dominant meantone. Then Dominant[12] is either [[7L 5s]] (if generator < 700{{c}}), in which case the bright generator is the fourth, or [[5L 7s]] (if generator > 700{{c}}), meaning that the bright generator is the fifth. As a result, Dominant[12] 7|4 is ambiguous.

* It's impossible to determine the bright generator in a non-MOS scale like Meantone[8], thus Meantone[8] 5|2 is ambiguous.