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Added: Mode matrix, interval matrix, and... degree matrix?
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== Producing an interval matrix from a mode matrix ==
The notion of an [[interval matrix]] is already well-described, but not so much the idea of a mode matrix nor producing an interval matrix from a mode matrix. This is based on the idea of sorting the strings for a mos's modes in lexicographic order to equivalently sort its modes by modal brightness, so pulling from that section, we start with the modes of 5L 2s sorted by modal brightness as an example:
{| class="wikitable"
!'''Binary'''
!'''UDP'''
!'''Mode name'''
!'''Scale string'''
|-
|1110110
|<nowiki>6|0</nowiki>
|Lydian
|LLLsLLs
|-
|1101110
|<nowiki>5|1</nowiki>
|Ionian
|LLsLLLs
|-
|1101101
|<nowiki>4|2</nowiki>
|Mixolydian
|LLsLLsL
|-
|1011101
|<nowiki>3|3</nowiki>
|Dorian
|LsLLLsL
|-
|1011011
|<nowiki>2|4</nowiki>
|Aeolian
|LsLLsLL
|-
|0111011
|<nowiki>1|5</nowiki>
|Phrygian
|sLLLsLL
|-
|0110111
|<nowiki>0|6</nowiki>
|Locrian
|sLLsLLL
|}
A mode matrix for this is is a 7x7 matrix, consisting of only a single L or a single s in each entry, where each row vector corresponds to one of the mos's modes.
{| class="wikitable"
|+
!Scale string
!'''Mode name'''
!Step 1
!Step 2
!Step 3
!Step 4
!Step 5
!Step 6
!Step 7
|-
!LLLsLLs
!Lydian
|L
|L
|L
|s
|L
|L
|s
|-
!LLsLLLs
!Ionian
|L
|L
|s
|L
|L
|L
|s
|-
!LLsLLsL
!Mixolydian
|L
|L
|s
|L
|L
|s
|L
|-
!LsLLLsL
!Dorian
|L
|s
|L
|L
|L
|s
|L
|-
!LsLLsLL
!Aeolian
|L
|s
|L
|L
|s
|L
|L
|-
!sLLLsLL
!Phrygian
|s
|L
|L
|L
|s
|L
|L
|-
!sLLsLLL
!Locrian
|s
|L
|L
|s
|L
|L
|L
|}
An interval matrix can be defined as the following: for an nxn mode matrix, its column matrix consists of n+1 columns and n rows. For our example, our interval matrix contains 8 columns and 7 rows. Recall that L and s not only stand for characters in a string, but are also in place for actual numbers. Each column vector in the interval matrix represents the sum of several column vectors from the mode matrix; specifically, if the mode matrix's column vectors are enumerated as c1, c2, to cn, then the column vectors of the interval matrix are c1, c1+c2, c1+c2+c3, and so on to c1+c2+c3+...+cn. There is an additional column vector c0 for the interval matrix whose values are all zero; this represents the roots of the scales and are unisons; likewise, the last column vector represents the octaves (or equivalence intervals).
For the mode matrix above, the interval matrix can then be calculated as this:
{| class="wikitable"
!String
!Mode
!c0
!c1
!c2
!c3
!c4
!c5
!c6
!c7
|-
!LLLsLLs
!Lydian
|0
|L
|2L
|3L
|3L + s
|4L + s
|5L + s
|5L + 2s
|-
!LLsLLLs
!Ionian
|0
|L
|2L
|2L + s
|3L + s
|4L + s
|5L + s
|5L + 2s
|-
!LLsLLsL
!Mixolydian
|0
|L
|2L
|2L + s
|3L + s
|4L + s
|4L + 2s
|5L + 2s
|-
!LsLLLsL
!Dorian
|0
|L
|L + s
|2L + s
|3L + s
|4L + s
|4L + 2s
|5L + 2s
|-
!LsLLsLL
!Aeolian
|0
|L
|L + s
|2L + s
|3L + s
|3L + 2s
|4L + 2s
|5L + 2s
|-
!sLLLsLL
!Phrygian
|0
|s
|L + s
|2L + s
|3L + s
|3L + 2s
|4L + 2s
|5L + 2s
|-
!sLLsLLL
!Locrian
|0
|s
|L + s
|2L + s
|2L + 2s
|3L + 2s
|3L + 4s
|5L + 2s
|}
Note that the enumeration of column vectors lines up with the TAMNAMS convention of referring to intervals as some quantity of mossteps (seconds are 1-steps, thirds are 2-steps, and so on); in fact, the column vectors describe mos intervals.
Curiously, since the mode matrix consists of only two values, this makes it a logical (or binary) matrix. Likewise, the interval matrix can be converted into a logical interval matrix as such: for each column vector (except for the first and last), the larger of the two values is replaced with 1 and the smaller with 0. The first column vector is all zeros, and the last all ones (though this convention is arbitrary as these two columns are technically not needed). This in turn describes scale degrees as being major or minor, or in the case of the generating intervals, augmented, perfect, or diminished. (The unison and equivalence interval are both perfect.)
{| class="wikitable"
!String
!Mode
!d0
!d1
!d2
!d3
!c4
!c5
!c6
!c7
|-
!LLLsLLs
!Lydian
|''0''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|-
!LLsLLLs
!Ionian
|''0''
|'''1'''
|'''1'''
|''0''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|-
!LLsLLsL
!Mixolydian
|''0''
|'''1'''
|'''1'''
|''0''
|'''1'''
|'''1'''
|''0''
|'''1'''
|-
!LsLLLsL
!Dorian
|''0''
|'''1'''
|''0''
|''0''
|'''1'''
|'''1'''
|''0''
|'''1'''
|-
!LsLLsLL
!Aeolian
|''0''
|'''1'''
|''0''
|''0''
|'''1'''
|''0''
|''0''
|'''1'''
|-
!sLLLsLL
!Phrygian
|''0''
|''0''
|''0''
|''0''
|'''1'''
|''0''
|''0''
|'''1'''
|-
!sLLsLLL
!Locrian
|''0''
|''0''
|''0''
|''0''
|''0''
|''0''
|''0''
|'''1'''
|}
This also describes a property of the generating intervals: they appear in one specific size in all but one mode; for 5L 2s's perfect 4th, it appears as the smaller size except in the lydian mode where it appears as an augmented 4th, and for the perfect 5th, it appears as the larger size except in the locrian mode where it's a diminished 5th.
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