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 |
Using step sizes
Working with a scale described as a sequence of steps, such as a mos, means that the cent values or JI ratios may not be known beforehand. However, it's still possible to generate an interval matrix as follows.
Consider the familiar diatonic scale (or 5L 2s), represented as the string LLsLLLs, for example. (WWHWWWH also works, but to be general, L and s are used instead.) The intervals between the scale's root and any other scale degree can be considered as being the number of L's and s's from a substring of "LLsLLLs" that starts at the first character (or step) and ends at any other character, including itself; for example, a 2nd is "L", a 3rd is "LL" (or 2L), a 4th is "LLs" (or 2L + s), and s on. The order of L's and s's is not important, rather the number of L's and s's. In other words, what makes a perfect 5th a perfect 5th is that it's reached by going up from the root by 3 large steps and 1 small step (or 3L + 1s), no matter what the order of steps are. Note that a unison, or 1st, corresponds to a substring consisting of zero characters, or an empty string, and thus its sum of L's and s's is zero.
The first row of the matrix can then be populated as such:
| 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Substr. | Sum | Substr. | Sum | Substr. | Sum | Substr. | Sum | Substr. | Sum | Substr. | Sum | Substr. | Sum | |
| LLsLLLs | empty-string | 0 | L | L | LL | 2L | LLs | 2L + s | LLsL | 3L + s | LLsL | 4L + s | LLsLLL | 5L + s |
To find the next row, we need to rotate the the scale (by moving the first step to the end of the sequence) and find the substrings that start at the first step of that rotated scale. Repeating this process finds the intervals sizes for each of the scale's modes. Since LLsLLLs represents the ionian mode, shifting this way produces the dorian mode (LsLLLsL), then the phrygian mode (sLLLsLL), and so on. The completed matrix is shown below:
| String | Mode | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th |
|---|---|---|---|---|---|---|---|---|---|
| LLsLLLs | Ionian | 0 | L | 2L | 2L + s | 3L + s | 4L + s | 5L + s | 5L + 2s |
| LsLLLsL | Dorian | 0 | L | L + s | 2L + s | 3L + s | 4L + s | 4L + 2s | 5L + 2s |
| sLLLsLL | Phrygian | 0 | s | L + s | 2L + s | 3L + s | 3L + 2s | 4L + 2s | 5L + 2s |
| LLLsLLs | Lydian | 0 | L | 2L | 3L | 3L + s | 4L + s | 5L + s | 5L + 2s |
| LLsLLsL | Mixolydian | 0 | L | 2L | 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 |
| sLLsLLL | Locrian | 0 | s | L + s | 2L + s | 2L + 2s | 3L + 2s | 3L + 4s | 5L + 2s |
The column of firsts consists of only the unison, which is zero units above the root, hence the entire column is zero. The column of eights consists of the entire string, hence the entire column is 5L + 2s. Note that this matrix is for any arbitrary L and s; any other properties, such as the size of each interval in cents or whether the scale forms a constant structure, cannot be known unless L and s are known.
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.)