Interval matrix - Revision history

Ganaram inukshuk: /* Using step sizes */ More succinct description on finding a step sequence's next mode

2022-12-03T00:26:51Z

Using step sizes: More succinct description on finding a step sequence's next mode

← Older revision		Revision as of 00:26, 3 December 2022
Line 131:		Line 131:

	=== Using step sizes ===		=== Using step sizes ===
	Working with a ~~[[mos]] or any~~ scale described as a sequence of steps 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 scale described as a sequence of steps, such as a [[MOS scale\|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.		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.
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	\|LLsLLL		\|LLsLLL
	\|5L + s		\|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:
	To find the next row ~~means finding the substrings starting at the second L and ending at any other step after it. However~~, ~~this is equivalent~~ to ~~rotating~~ the the scale (~~specifically~~ moving the first L to the end) and ~~finding~~ the substrings that start at the first step of that rotated scale. Since LLsLLLs represents the ionian mode, shifting this way produces the dorian mode (LsLLLsL), ~~and the mode after that is~~ the phrygian mode (sLLLsLL), so ~~populating the rest of the interval matrix means finding the quantities of L's and s's for every substring for every mode~~. The completed matrix is shown below:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

Ganaram inukshuk: /* Using step sizes */ Swapped out "looking at scales in a temperament agnostic sense" with "looking at scales described as sequences of steps"

2022-07-28T07:58:09Z

Using step sizes: Swapped out "looking at scales in a temperament agnostic sense" with "looking at scales described as sequences of steps"

← Older revision		Revision as of 07:58, 28 July 2022
Line 131:		Line 131:

	=== 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 described as a sequence of steps 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.		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.

Fredg999: Markup, categories

2022-07-20T21:05:52Z

Markup, categories

← Older revision		Revision as of 21:05, 20 July 2022
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	An ''interval matrix'' is a tabular representation of all possible [[interval]]s in a [[~~scale\|~~scale]].		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.
Line 5:		Line 5:
	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.

	~~<ul><li>~~In the first row, list the pitches of the scale as-is: a, b, c, ...~~</li><li>~~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.)~~</li><li>~~In the third row, list the intervals: (c-c), (d-c), ..., (a-c), (b-c), (c-c). (Reduce (a-c) and (b-c).)~~</li><li>~~etc.~~</li></ul>~~		* 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]]:

	~~<ul><li>~~command line: "show/line intervals"~~</li><li>~~graphical interface: View &~~gt;~~ Interval matrix~~</li><li>~~keyboard shortcut: Shift+Alt+I~~</li></ul>~~		* 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:
Line 294:		Line 298:

	(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]]

Ganaram inukshuk: /* Using step sizes */ even more clarification

2022-07-20T02:06:08Z

Using step sizes: even more clarification

← Older revision		Revision as of 02:06, 20 July 2022
Line 131:		Line 131:
	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.

	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.		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:		The first row of the matrix can then be populated as such:

Ganaram inukshuk: /* Using step sizes */ Some clarification

2022-07-20T02:03:12Z

Using step sizes: Some clarification

← Older revision		Revision as of 02:03, 20 July 2022
Line 129:		Line 129:

	== Using step sizes ==		== 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:		The first row of the matrix can then be populated as such:
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	\|+		\|+
	! rowspan="2" \|		! rowspan="2" \|
			! colspan="2" \|1st
	! colspan="2" \|2nd		! colspan="2" \|2nd
	! colspan="2" \|3rd		! colspan="2" \|3rd
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	! colspan="2" \|7th		! colspan="2" \|7th
	\|-		\|-
			!Substr.
			!Sum
	!Substr.		!Substr.
	!Sum		!Sum
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	\|-		\|-
	!LLsLLLs		!LLsLLLs
			\|empty-string
			\|0
	\|L		\|L
	\|L		\|L
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	\|5L + 2s		\|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==		==Scala==

Ganaram inukshuk: /* Using step sizes */ Wasn't sure if "rotate to the right" was the same as "shift right" (I think it's supposed to be shift left); edited just in case.

2022-07-19T04:34:29Z

Using step sizes: Wasn't sure if "rotate to the right" was the same as "shift right" (I think it's supposed to be shift left); edited just in case.

← Older revision		Revision as of 04:34, 19 July 2022
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	\|5L + s		\|5L + s
	\|}		\|}
	To find the next row means finding the substrings starting at the second L and ending at any other step after it. However, this is equivalent to rotating the the scale ~~to the right~~ (moving the first L to the end) and finding the substrings that start at the first step of that rotated scale. Since LLsLLLs represents the ionian mode, shifting ~~to the right~~ produces the dorian mode (LsLLLsL), and the mode after that is the phrygian mode (sLLLsLL), so populating the rest of the interval matrix means finding the quantities of L's and s's for every substring for every mode. The completed matrix is shown below:		To find the next row means finding the substrings starting at the second L and ending at any other step after it. However, this is equivalent to rotating the the scale (specifically moving the first L to the end) and finding the substrings that start at the first step of that rotated scale. Since LLsLLLs represents the ionian mode, shifting this way produces the dorian mode (LsLLLsL), and the mode after that is the phrygian mode (sLLLsLL), so populating the rest of the interval matrix means finding the quantities of L's and s's for every substring for every mode. The completed matrix is shown below:
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

Ganaram inukshuk: /* Examples */ Added an interval matrix example for a temperament-agnostic situation, where no cent values or JI ratios are known but instead the number of L's and s's that make up each interval

2022-07-19T04:30:36Z

Examples: Added an interval matrix example for a temperament-agnostic situation, where no cent values or JI ratios are known but instead the number of L's and s's that make up each interval

@@ Line 1: / Line 1: @@
-An ''interval matrix'' is a tabular representation of all possible [[interval|interval]]s in a [[scale|scale]].
+An ''interval matrix'' is a tabular representation of all possible [[interval]]s in a [[scale|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.
@@ Line 127: / Line 127: @@
 | | 1200.0
 |}
 ==Scala==

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>spt3125: Imported revision 511025642 - Original comment:

2014-05-24T19:27:25Z

**Imported revision 511025642 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference: 
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-05-24 19:27:25 UTC</tt>. 
: The original revision id was <tt>511025642</tt>. 
: The revision comment was: <tt></tt> 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it. 
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">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.
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:
[[code]]
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
[[code]]

this interval matrix:
[[code]]
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
[[code]]

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

[[media type="custom" key="26024358"]]</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>interval matrix</title></head><body>An interval matrix is a tabular representation of all possible <a class="wiki_link" href="/interval">interval</a>s in a <a class="wiki_link" href="/scale">scale</a>. 
 
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 &quot;a&quot; (1/1), &quot;b&quot; (second pitch), &quot;c&quot; (third), etc. 
 
<ul><li>In the first row, list the pitches of the scale as-is: a, b, c, ...</li><li>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.)</li><li>In the third row, list the intervals: (c-c), (d-c), ..., (a-c), (b-c), (c-c). (Reduce (a-c) and (b-c).)</li><li>etc.</li></ul> 
Optionally (as in the examples below), you may have a &quot;header row&quot; of degrees (1, 2, ...) and a &quot;header column&quot; of the original scale pitches. 
 
 
<h1 id="toc0"><a name="Examples"></a>Examples</h1>
 
<h2 id="toc1"><a name="Examples-JI"></a>JI</h2>
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: 

<table class="wiki_table">
<tr>
<td> 
</td>
<td>1 
</td>
<td>2 
</td>
<td>3 
</td>
<td>4 
</td>
<td>5 
</td>
<td>(6) 
</td>
<td>comments 
</td>
</tr>
<tr>
<td>1/1 
</td>
<td>1/1 
</td>
<td>9/8 
</td>
<td>5/4 
</td>
<td>3/2 
</td>
<td>5/3 
</td>
<td>2/1 
</td>
<td>original scale 
</td>
</tr>
<tr>
<td>9/8 
</td>
<td>1/1 
</td>
<td>10/9 
</td>
<td>4/3 
</td>
<td>40/27 
</td>
<td>16/9 
</td>
<td>2/1 
</td>
<td>10/9 is the interval between 5/4 and 9/8; 4/3 = 3/2 - 9/8; etc. 
</td>
</tr>
<tr>
<td>5/4 
</td>
<td>1/1 
</td>
<td>6/5 
</td>
<td>4/3 
</td>
<td>8/5 
</td>
<td>9/5 
</td>
<td>2/1 
</td>
<td>6/5 = 3/2 - 5/4; etc.; 9/5 = (2/1 + 9/8) - 5/4 
</td>
</tr>
<tr>
<td>3/2 
</td>
<td>1/1 
</td>
<td>10/9 
</td>
<td>4/3 
</td>
<td>3/2 
</td>
<td>5/3 
</td>
<td>2/1 
</td>
<td> 
</td>
</tr>
<tr>
<td>5/3 
</td>
<td>1/1 
</td>
<td>6/5 
</td>
<td>27/20 
</td>
<td>3/2 
</td>
<td>9/5 
</td>
<td>2/1 
</td>
<td> 
</td>
</tr>
</table>

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 . 
 
<h2 id="toc2"><a name="Examples-Cents"></a>Cents</h2>
Here is an example with a tempered scale: 0.0 - 226.3 - 486.8 - 713.2 - 939.5 - 1200.0 cents 

<table class="wiki_table">
<tr>
<td> 
</td>
<td>1 
</td>
<td>2 
</td>
<td>3 
</td>
<td>4 
</td>
<td>5 
</td>
<td>(6) 
</td>
</tr>
<tr>
<td>0.0 
</td>
<td>0.0 
</td>
<td>226.4 
</td>
<td>486.8 
</td>
<td>713.2 
</td>
<td>939.5 
</td>
<td>1200.0 
</td>
</tr>
<tr>
<td>226.4 
</td>
<td>0.0 
</td>
<td>260.5 
</td>
<td>486.8 
</td>
<td>713.2 
</td>
<td>973.6 
</td>
<td>1200.0 
</td>
</tr>
<tr>
<td>486.8 
</td>
<td>0.0 
</td>
<td>226.4 
</td>
<td>452.7 
</td>
<td>713.2 
</td>
<td>939.5 
</td>
<td>1200.0 
</td>
</tr>
<tr>
<td>713.2 
</td>
<td>0.0 
</td>
<td>226.4 
</td>
<td>486.8 
</td>
<td>713.2 
</td>
<td>973.6 
</td>
<td>1200.0 
</td>
</tr>
<tr>
<td>939.5 
</td>
<td>0.0 
</td>
<td>260.5 
</td>
<td>486.8 
</td>
<td>747.3 
</td>
<td>973.6 
</td>
<td>1200.0 
</td>
</tr>
</table>

 
<h2 id="toc3"><a name="Examples-Scala"></a>Scala</h2>
To show the interval matrix of the current scale using <a class="wiki_link" href="/Scala">Scala</a>: 
<ul><li>command line: &quot;show/line intervals&quot;</li><li>graphical interface: View &gt; Interval matrix</li><li>keyboard shortcut: Shift+Alt+I</li></ul> 
For example, Scala will display for this scale: 

<style type="text/css">
</style><pre class="text"> 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</pre>

 
this interval matrix: 

<style type="text/css">
</style><pre class="text"> 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</pre>

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