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.
Contents
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.)