Interval matrix

An interval matrix is a tabular representation of all possible intervals in a scale.

To create an interval matrix, start with a table with (at least) as many rows and columns are there are pitches in your scale.

Let's call the pitches of the scale "a" (1/1), "b" (second pitch), "c" (third), etc.

• In the first row, list the pitches of the scale as-is: a, b, c, ...
• In the second row, list the intervals: (b-b), (c-b), (d-b), ..., (a-b), (b-b). (Reduce (a-b) by an octave, or whatever the period of the scale is.)
• In the third row, list the intervals: (c-c), (d-c), ..., (a-c), (b-c), (c-c). (Reduce (a-c) and (b-c).)
• etc.

Optionally (as in the examples below), you may have a "header row" of degrees (1, 2, ...) and a "header column" of the original scale pitches.

Examples

JI

Take this common JI pentatonic scale: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1

Here is the interval matrix of this scale:

	1	2	3	4	5	(6)	comments
1/1	1/1	9/8	5/4	3/2	5/3	2/1	original scale
9/8	1/1	10/9	4/3	40/27	16/9	2/1	10/9 is the interval between 5/4 and 9/8; 4/3 = 3/2 - 9/8; etc.
5/4	1/1	6/5	4/3	8/5	9/5	2/1	6/5 = 3/2 - 5/4; etc.; 9/5 = (2/1 + 9/8) - 5/4
3/2	1/1	10/9	4/3	3/2	5/3	2/1
5/3	1/1	6/5	27/20	3/2	9/5	2/1

Note that the distance between (for example) 3/2 and 5/4 is written above as 3/2 - 5/4, as is common for JI intervals, but actually calculated as 3/2 ÷ 5/4 .

Cents

Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents

	1	2	3	4	5	(6)
0.0	0.0	226.4	486.8	713.2	939.5	1200.0
226.4	0.0	260.5	486.8	713.2	973.6	1200.0
486.8	0.0	226.4	452.7	713.2	939.5	1200.0
713.2	0.0	226.4	486.8	713.2	973.6	1200.0
939.5	0.0	260.5	486.8	747.3	973.6	1200.0

Scala

To show the interval matrix of the current scale using Scala:

• command line: "show/line intervals"
• graphical interface: View > Interval matrix
• keyboard shortcut: Shift+Alt+I

For example, Scala will display for this scale:

  0:          1/1               0.000000 unison, perfect prime
  1:          9/8             203.910002 major whole tone
  2:          5/4             386.313714 major third
  3:          3/2             701.955001 perfect fifth
  4:          5/3             884.358713 major sixth, BP sixth
  5:          2/1            1200.000000 octave

this interval matrix:

      1    2     3     4    5  
 1/1: 9/8  5/4   3/2   5/3  2/1
 9/8: 10/9 4/3   40/27 16/9 2/1
 5/4: 6/5  4/3   8/5   9/5  2/1
 3/2: 10/9 4/3   3/2   5/3  2/1
 5/3: 6/5  27/20 3/2   9/5  2/1
 2/1

(Note that Scala omits the "1/1" column, and the column numbers are offset by 1 relative to the other examples above.)