Interval matrix: Difference between revisions

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== Using step sizes ==

Working with a [[mos]] 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.

Consider the diatonic ~~mos~~ [[5L 2s]], represented as the string LLsLLLs, for example. The intervals between the scale's root and any other scale degree can be considered as being 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", a 4th is "LLs", and s on. The order of L's and s's is not important, rather the number of L's and s's.

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 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", a 4th is "LLs", 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, 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:

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|+

! rowspan="2" |

! colspan="2" |1st

! colspan="2" |2nd

! colspan="2" |3rd

Line 144:

Line 145:

! colspan="2" |7th

|-

!Substr.

!Sum

!Substr.

!Sum

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|-

!LLsLLLs

|empty-string

|0

|L

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|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 ~~and does not reveal~~ any ~~information with regards to tuning or~~ other properties.

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==

@@ Line 129: / Line 129: @@
 == Using step sizes ==
-Working with a [[mos]] 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.
-Consider the diatonic mos [[5L 2s]], represented as the string LLsLLLs, for example. The intervals between the scale's root and any other scale degree can be considered as being 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", a 4th is "LLs", and s on. The order of L's and s's is not important, rather the number of L's and s's.
+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 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", a 4th is "LLs", 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, 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:
@@ Line 137: / Line 137: @@
 |+
 ! rowspan="2" |
+! colspan="2" |1st
 ! colspan="2" |2nd
 ! colspan="2" |3rd
@@ Line 144: / Line 145: @@
 ! colspan="2" |7th
 |-
+!Substr.
+!Sum
 !Substr.
 !Sum
@@ Line 158: / Line 161: @@
 |-
 !LLsLLLs
+|empty-string
+|0
 |L
 |L
@@ Line 262: / Line 267: @@
 |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 and does not reveal any information with regards to tuning or other properties.
+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==