Interval matrix: Difference between revisions
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An ''interval matrix'' is a tabular representation of all possible [[interval]]s in a [[ | An '''interval matrix''' is a tabular representation of all possible [[interval]]s 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. | To create an interval matrix, start with a table with (at least) as many rows and columns are there are pitches in your scale. | ||
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Let's call the pitches of the scale "a" (1/1), "b" (second pitch), "c" (third), etc. | 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. | 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= | == Examples == | ||
=== JI === | |||
==JI== | |||
Take this common JI pentatonic scale: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1 | Take this common JI pentatonic scale: 1/1 - 9/8 - 5/4 - 3/2 - 5/3 - 2/1 | ||
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''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 .'' | ''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== | === Cents === | ||
Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents | Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents | ||
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== Using step sizes == | === Using step sizes === | ||
Working with a [[mos]] or any scale in a [[TAMNAMS|temperament-agnostic]] sense 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. | Working with a [[mos]] or any scale in a [[TAMNAMS|temperament-agnostic]] sense 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. | ||
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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. | 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== | === Scala === | ||
To show the interval matrix of the current scale using [[Scala|Scala]]: | To show the interval matrix of the current scale using [[Scala|Scala]]: | ||
* command line: "show/line intervals" | |||
* graphical interface: View & Interval matrix | |||
* keyboard shortcut: Shift+Alt+I | |||
For example, Scala will display for this scale: | For example, Scala will display for this scale: | ||
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(Note that Scala omits the "1/1" column, and the column numbers are offset by 1 relative to the other examples above.) | (Note that Scala omits the "1/1" column, and the column numbers are offset by 1 relative to the other examples above.) | ||
[[Category:Scale]] | |||