Ratio math - Revision history

Sintel: Remove personal note. (This page needs some love!)

2026-01-28T15:52:13Z

Remove personal note. (This page needs some love!)

← Older revision		Revision as of 15:52, 28 January 2026
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	{{Beginner}}		{{Beginner}}
	This page describes the way we manipulate [[ratios]] arithmetically. Some of it is just math, and some of it is specific to music.		This page describes the way we manipulate [[ratios]] arithmetically. Some of it is just math, and some of it is specific to music.

	[[Jake Freivald\|I]] am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there.

	== About ratios ==		== About ratios ==

Sintel: beginner template

2025-03-06T17:49:46Z

beginner template

← Older revision		Revision as of 17:49, 6 March 2025
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			{{Beginner}}
	This page describes the way we manipulate [[ratios]] arithmetically. Some of it is just math, and some of it is specific to music.		This page describes the way we manipulate [[ratios]] arithmetically. Some of it is just math, and some of it is specific to music.

FloraC: Style

2024-01-09T10:03:23Z

Style

← Older revision		Revision as of 10:03, 9 January 2024
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	[[Jake Freivald\|I]] am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there.		[[Jake Freivald\|I]] am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there.

	== About ~~Ratios~~ ==		== About ratios ==

	Ratios are just one way of writing various kinds of numbers.		Ratios are just one way of writing various kinds of numbers.

	1/1, 2/1, 3/1, 4/1, 5/1, and so on are the same as 1, 2, 3, 4, 5, and so on. Put differently, when there is no bottom number (denominator), it's as if there was a denominator equal to 1.		1/1, 2/1, 3/1, 4/1, 5/1, and so on are the same as 1, 2, 3, 4, 5, and so on. Put differently, when there is no bottom number (denominator), it is as if there was a denominator equal to 1.

	Ratios can be reduced by dividing both top and bottom by the same numbers until you ~~can't~~ reduce them anymore. So:		Ratios can be reduced by dividing both top and bottom by the same numbers until you cannot reduce them anymore. So:

	54/24		54/24
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	= 9 / 4 [dividing top and bottom by 3]		= 9 / 4 [dividing top and bottom by 3]

	~~...~~so 54/24 can be reduced to 9/4.		… so 54/24 can be reduced to 9/4.

	(You can do this faster if you get used to prime factorization:		(You can do this faster if you get used to prime factorization:
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	All of these numbers mean the same thing: 2, 2/1, 4/2, 8/4, 16/8. The top (numerator) is double the bottom (denominator).		All of these numbers mean the same thing: 2, 2/1, 4/2, 8/4, 16/8. The top (numerator) is double the bottom (denominator).

	It's legitimate to write things like		It is legitimate to write things like
	2/1 = 2		2/1 = 2
	or		or
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	----		----

	When we~~'re~~ finding the difference between intervals, we're actually dividing their ratios.		When we are finding the difference between intervals, we're actually dividing their ratios.

	Remember that dividing by a ratio is the same as multiplying by its reciprocal (flip, then multiply).		Remember that dividing by a ratio is the same as multiplying by its reciprocal (flip, then multiply).
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	= 6/5.		= 6/5.

	== Octave ~~Reduction~~ ==		== Octave reduction ==
			{{Main\| Octave reduction }}

	When a number is larger than the octave, which is 2/1 (or equivalently just "2"), we octave-reduce it by dividing by 2/1.		When a number is larger than the octave, which is 2/1 (or equivalently just "2"), we octave-reduce it by dividing by 2/1.

	9/4 is greater than 8/4, so it's greater than 2.		9/4 is greater than 8/4, so it is greater than 2.

	To octave-reduce, divide by 2.		To octave-reduce, divide by 2.
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	----		----

	Let's do it again, using 8/3.		Let us do it again, using 8/3.

	(8/3) / (2/1)		(8/3) / (2/1)
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	----		----

	Let's do it again, using 5/12.		Let us do it again, using 5/12.

	5/12 * 2 = 10/12, which is still less than 1.		5/12 * 2 = 10/12, which is still less than 1.
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	----		----

	Whichever way you~~'re~~ going, keep going until the ratio is between 1 and 2.		Whichever way you are going, keep going until the ratio is between 1 and 2.


	[[Category:~~Math~~]]		[[Category:Ratio]]
	[[Category:Elementary math]]		[[Category:Elementary math]]

Sintel: categories

2022-02-03T20:14:50Z

Jdfreivald: Added links

2019-08-26T18:35:18Z

Added links

← Older revision		Revision as of 18:35, 26 August 2019
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	This page describes the way we manipulate ratios arithmetically. Some of it is just math, and some of it is specific to music.		This page describes the way we manipulate [[ratios]] arithmetically. Some of it is just math, and some of it is specific to music.

	I ~~(Jake)~~ am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there.		[[Jake Freivald\|I]] am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there.

	== About Ratios ==		== About Ratios ==

Jdfreivald: Created page

2019-08-26T18:32:13Z

Created page

New page

This page describes the way we manipulate ratios arithmetically. Some of it is just math, and some of it is specific to music.

I (Jake) am not trying to format this particularly nicely, since a lot of these kinds of discussions come up on Facebook and similar places, and you may as well see it ugly here so you understand it when it's ugly there.

== About Ratios ==

Ratios are just one way of writing various kinds of numbers.

1/1, 2/1, 3/1, 4/1, 5/1, and so on are the same as 1, 2, 3, 4, 5, and so on. Put differently, when there is no bottom number (denominator), it's as if there was a denominator equal to 1.

Ratios can be reduced by dividing both top and bottom by the same numbers until you can't reduce them anymore. So:

54/24
= 27 / 12 [dividing top and bottom by 2]
= 9 / 4 [dividing top and bottom by 3]

...so 54/24 can be reduced to 9/4.

(You can do this faster if you get used to prime factorization:

54/24
= 3*3*3*2 / 3*2*2*2
= 3*3 / 2*2 [by dividing top and bottom by 3*2]
= 9/4.)

All of these numbers mean the same thing: 2, 2/1, 4/2, 8/4, 16/8. The top (numerator) is double the bottom (denominator).

It's legitimate to write things like
2/1 = 2
or
54/24 = 9/4

== Stacking ratios and finding their differences ==
When we "add intervals", or stack them, we're actually multiplying their ratios.

Multiplying ratios means multiplying the tops (numerators) and the bottoms (denominators).

(x/y) * (a/b)
= xa / yb

So if we took a 5/4 major third and stacked a 6/5 minor third on top of it, the outer interval would be

(5/4) * (6/5)
= 5*6 / 4*5
= 30 / 20
= 3/2.

3/2 is a perfect fifth, so that should make sense: The major third C-E with a minor third E-G stacked on top of it encompasses the perfect fifth C-G.

----

When we're finding the difference between intervals, we're actually dividing their ratios.

Remember that dividing by a ratio is the same as multiplying by its reciprocal (flip, then multiply).

(x/y) / (a/b)
= (x/y) * (b/a)
= xb / ya

When we take the difference between the perfect fifth, 3/2, and the major third, 5/4, we should get a minor third.

(3/2) / (5/4)
= (3/2) * (4/5)
= 12/10
= 6/5.

== Octave Reduction ==
When a number is larger than the octave, which is 2/1 (or equivalently just "2"), we octave-reduce it by dividing by 2/1.

9/4 is greater than 8/4, so it's greater than 2.

To octave-reduce, divide by 2.

(9/4) / (2/1)
= (9/4) * (1/2)
= 9*1 / 4*2
= 9/8.

----

Let's do it again, using 8/3.

(8/3) / (2/1)
= (8/3) * (1/2)
= (8*1) / (2*3)
= 8/6
= 4/3.

As you can see, you can divide the top number in half (8/3 goes to 4/3) or double the bottom number (9/4 goes to 9/8) to have the same effect and avoid the annoying arithmetic.

----

Same kind of thing when a number is smaller than 1: Multiply by 2 until you get somewhere between 1 (the root) and 2 (the octave).

2/3 is less than 1.

2/3 * 2/1
= 2*2 / 3*1
= 4/3.

As you can see, you could also just double the numerator (top).

----

Let's do it again, using 5/12.

5/12 * 2 = 10/12, which is still less than 1.

10/12 * 2 = 20/12, which is between 1 (12/12) and 2 (24/12).

Reduce 20/12 to get 5/3.

As you can see, you could just cut the denominator in half.

----

Whichever way you're going, keep going until the ratio is between 1 and 2.

← Older revision		Revision as of 20:14, 3 February 2022
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	Whichever way you're going, keep going until the ratio is between 1 and 2.		Whichever way you're going, keep going until the ratio is between 1 and 2.


			[[Category:Math]]
			[[Category:Elementary math]]