Frequency ratio - Revision history

FloraC: + redirect notice

2025-12-07T09:43:02Z

+ redirect notice

← Older revision		Revision as of 09:43, 7 December 2025
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	\| ja = 周波数比率		\| ja = 周波数比率
	}}		}}
			{{Redirect\|Ratio\|step ratios in scales\|Step ratio}}

	A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).		A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).

Nick Vuci: Added details about conventions for writing ratios with more than two terms in main body.

2025-11-25T16:28:46Z

Added details about conventions for writing ratios with more than two terms in main body.

← Older revision		Revision as of 16:28, 25 November 2025
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	When the larger number is written first (''note''/''base''), this usually signifies a note being played ''above'' some base tone (perhaps the starting note of a scale). When the smaller number is written first (''base''/''note''), this usually signifies the note being played ''below'' that base tone.		When the larger number is written first (''note''/''base''), this usually signifies a note being played ''above'' some base tone (perhaps the starting note of a scale). When the smaller number is written first (''base''/''note''), this usually signifies the note being played ''below'' that base tone.

	[[~~Chord~~]]s ~~with three or more notes can also be expressed as ratios~~. For example, the just intoned major triad in root position is 4:5:6. Chords can also be written as a string of intervals, such as 1/1–5/4–3/2. (4:5:6 can be viewed as a shorthand for 4/1:5/1:6/1 or 4/4:5/4:6/4).		Ratios with more than two terms (sometimes called '''extended ratios''') can be used to express [[chord]]s and [[Scale\|scales]]. For example, the just intoned major triad in root position is 4:5:6. Chords can also be written as a string of intervals, such as 1/1–5/4–3/2. (4:5:6 can be viewed as a shorthand for 4/1:5/1:6/1 or 4/4:5/4:6/4).

	The [[harmonic series]] can be represented as the infinite ratio 1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:16:17…		The [[harmonic series]] can be represented as the infinite ratio 1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:16:17… Segments of the harmonic series are commonly written in abbreviated form with a double colon. For example, 8:9:10:11:12:13:14:15:16 is commonly written as 8::16.

	In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".		In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".

Eufalesio: Added information about self-inverses and improved EFR-SEFR conversion description

2025-11-25T13:16:41Z

Added information about self-inverses and improved EFR-SEFR conversion description

← Older revision		Revision as of 13:16, 25 November 2025
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	== Extended frequency ratio (EFR) ==		== Extended frequency ratio (EFR) ==
	An extended ratio is a ratio of more than 2 numbers. For example, a recipe might combine cups of flour, milk and sugar in a 5:3:4 ratio. Xenharmonic music uses an extended ratio to indicate the various frequency ratios between three or more notes. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous unordered example, EFRs are ordered, with harmonic EFR typically written in an ascending order and subharmonic EFR typically written in a descending order.		An extended ratio is a ratio of more than 2 numbers. For example, a recipe might combine cups of flour, milk and sugar in a 5:3:4 ratio. Xenharmonic music theory uses an extended ratio to indicate the various frequency ratios between three or more notes. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous unordered example, EFRs are ordered, with harmonic EFR typically written in an ascending order and subharmonic EFR typically written in a descending order.

	=== Harmonic EFR ===		=== Harmonic EFR ===
	For example, consider a [[just intonation]] major triad on {{w\|A440}} with a ratio list of 1/1–5/4–3/2. The three frequencies are 440, 550 and 660. The EFR is 440:550:660, which simplifies to 4:5:6, spoken as "four five six". Had the root been other than A440, the EFR would be the same.		For example, consider a [[just intonation]] major triad on {{w\|A440}} with a ratio list of 1/1–5/4–3/2. The three frequencies are 440, 550 and 660. The EFR is 440:550:660, which simplifies to [[4:5:6]], spoken as "four five six". Had the root been other than A440, the EFR would be the same.

	To convert an EFR to a ratio list, simply divide every harmonic by the first harmonic. For example, (4:5:6)/4 is 4/4–5/4–6/4, which simplifies to 1/1–5/4–3/2.		To convert an EFR to a ratio list, simply divide every harmonic by the first harmonic. For example, (4:5:6)/4 is 4/4–5/4–6/4, which simplifies to 1/1–5/4–3/2.

	The EFR directly indicates the interval between any pair of notes in the chord. In the harmonic seventh chord 4:5:6:7, the interval between the third and the fifth is 6/5, that between the third and seventh is 7/5, etc.		The EFR directly indicates the interval between any pair of notes in the chord. In the harmonic seventh chord [[4:5:6:7]], the interval between the third and the fifth is 6/5, that between the third and seventh is 7/5, etc.

	The EFR also indicates where in the harmonic series the chord occurs. 4:5:6:7 occurs as [[harmonic]]s 4, 5, 6 and 7. Thus an EFR is also a list of harmonics.		The EFR also indicates where in the harmonic series the chord occurs. 4:5:6:7 occurs as [[harmonic]]s 4, 5, 6 and 7. Thus an EFR is also a list of harmonics.
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	=== Subharmonic EFR, a.k.a. SEFR ===		=== Subharmonic EFR, a.k.a. SEFR ===
	Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Some writers omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.		Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Some writers omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as /7:6:5:4 or 7:6:5:4.

	To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.		To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.
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	To convert a list of ratios to a list of subharmonics, divide each ratio by the LCM of the numerators. For example, 1/1–6/5–3/2 has numerators 1, 6 and 3, with an LCM of 6. Dividing each ratio by 6 makes 1/6–1/5–1/4. Extract the ones to get 1/(6:5:4).		To convert a list of ratios to a list of subharmonics, divide each ratio by the LCM of the numerators. For example, 1/1–6/5–3/2 has numerators 1, 6 and 3, with an LCM of 6. Dividing each ratio by 6 makes 1/6–1/5–1/4. Extract the ones to get 1/(6:5:4).

	To convert an EFR to a ~~SEFR or vice versa~~, first ~~convert it~~ to ~~a ratio list~~.		To convert between EFR from and to SEFR, do the following:

			* [to SEFR] 10:12:15 (write as reciprocals) → 1/(1/10:1/12:1/15) → (multiply by LCM and reduce) → 60/(1/10:1/12:1/15) = 1/(6:5:4)
			* [to EFR] 1/(6:5:4) (write as reciprocals) → 1/6:1/5:1/4 → (multiply by LCM and reduce) → 60*(1/6:1/5:1/4) = 10:12:15

			Any SEFR can be written as a EFR and viceversa. In theory, 1/(7:6:5:4) could be written as [[60:70:84:105]], and so could 4:5:6:7 be written as 1/(105:84:70:60). The first option is more common, as seen in the available link. Ultimately, the choice depends on clarity.

			Some chords are [[Self-invertible chord\|self-inverses]], making SEFRs unnecessary. Examples include [[8:10:12:15]] (pental major seventh), [[10:12:15:18]] (pental minor seventh), [[12:14:18:21]] (septimal minor seventh), and [[14:18:21:27]] (septimal major seventh). These chords often have [[Homonym\|homonyms]], and their inversions may simply be other rotations of the same set. For example, 9:10:12:15 (pental major sixth, 2nd inversion) inverts to 12:15:18:20 (idem, root position), both rotations of 10:12:15:18.

	=== Alternative forms ===		=== Alternative forms ===

FloraC: + link to where you might be interested

2025-11-25T11:54:13Z

+ link to where you might be interested

← Older revision		Revision as of 11:54, 25 November 2025
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	== Conversion ==		== Conversion ==
			{{See also\| Cent #Conversion \| Monzo #Examples }}

	=== Monzo to ratio ===		=== Monzo to ratio ===
	To find the ratio ''r'' for an interval of monzo '''m''' = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> ''m''<sub>3</sub> … }}, apply		To find the ratio ''r'' for an interval of monzo '''m''' = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> ''m''<sub>3</sub> … }}, apply

FloraC: /* Conversions */ the other conversion are in Cent and Monzo pages

2025-11-25T11:46:45Z

Conversions: the other conversion are in Cent and Monzo pages

← Older revision		Revision as of 11:46, 25 November 2025
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	In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".		In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".

	== ~~Conversions~~ ==		== Conversion ==

	~~=== Ratio to monzo ===~~
	Factor both the numerator and the denominator into primes. Express the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. Enter the exponents into the monzo.

	~~<math>\displaystyle 15/8 = (3 \cdot 5) / (2 \cdot 2 \cdot 2) = 2^{-3} \cdot 3^{1} \cdot 5^{1} = </math> {{monzo\|-3 1 1}}~~

	~~If any primes smaller than the largest prime don't appear, include them using a zero exponent.~~

	~~<math>\displaystyle 28/27 = (2 \cdot 2 \cdot 7) / (3 \cdot 3 \cdot 3) = 2^{2} \cdot 3^{-3} \cdot 7^{1} = 2^{2} \cdot 3^{-3} \cdot 5^{0} \cdot 7^{1} = </math> {{monzo\|2 -3 0 1}}~~

	=== Monzo to ratio ===		=== Monzo to ratio ===
	To find the ratio '''r''' for an interval of [[monzo]] '''m''' = {{monzo\| ~~a b c … }}, apply~~		To find the ratio ''r'' for an interval of monzo '''m''' = {{monzo\| ''m''<sub>1</sub> ''m''<sub>2</sub> ''m''<sub>3</sub> … }}, apply

	<~~math~~>~~\displaystyle r =~~ 2~~^{a} \cdot~~ 3~~^{b} \cdot 5^{c} \ldots~~ </~~math~~>

	~~=== Ratio to cents ===~~		<math>\displaystyle r = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \cdot \ldots</math>
	~~To find the cents '''c''' for a ratio '''r''', apply~~

	<math>\displaystyle c = ~~1200~~ \cdot ~~log(r) / log(2)</math>~~

	~~Logarithms of any base (base 10, base 2, base ''e'', etc.) can be used.~~

	~~<math>~~\~~displaystyle c = 1200~~ \cdot ~~ln(r) / ln(2)~~</math>

	=== Cents to ratio ===		=== Cents to ratio ===
	To find the ratio '''r''' for an interval of '''c'~~'' [[cent]]s~~, apply		To find the ratio ''r'' for an interval of ''s'' cents, apply

	~~<math>\displaystyle r = 2^{c/1200}</math>~~

	~~The result will be in decimal form, and will only be as exact as '''c''' is. For example, 702 cents yields 1.500038989.., which is approximately 1.5, which is 3~~/2.		<math>\displaystyle r = 2^{s/1200}</math>

	== Extended frequency ratio (EFR) ==		== Extended frequency ratio (EFR) ==

TallKite: /* Conversion */ added sections for ratio to cents and ratio to monzo, rewrote the other sections

2025-11-24T22:23:49Z

Conversion: added sections for ratio to cents and ratio to monzo, rewrote the other sections

← Older revision		Revision as of 22:23, 24 November 2025
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	In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".		In the context of just intonation, ratios are almost always used to label and identify intervals and chords. However, the use of ratios to identify intervals and chords in tempered scales is also common - in these cases, it is implied that the notes are in the ''approximate'' ratio indicated. For example, a common shorthand expression might be "4:6:7:9:11 chords in [[17edo]]", which really means "the chords in which the notes are in the approximate ratio of 4:6:7:9:11 in 17edo".

	== ~~Conversion~~ ==		== Conversions ==
	=== ~~Cents~~ to ~~ratio~~ ===
	~~To find~~ the ratio ~~''c'' for~~ an ~~interval of ''s'~~' ~~[[cent]]s~~, ~~apply~~		=== Ratio to monzo ===
			Factor both the numerator and the denominator into primes. Express the entire ratio as a product of primes, each raised to an exponent. For primes appearing in the denominator, these exponents will be negative. (A prime never appears in both the numerator and the denominator.) Arrange the primes in ascending order. Enter the exponents into the monzo.

			<math>\displaystyle 15/8 = (3 \cdot 5) / (2 \cdot 2 \cdot 2) = 2^{-3} \cdot 3^{1} \cdot 5^{1} = </math> {{monzo\|-3 1 1}}

			If any primes smaller than the largest prime don't appear, include them using a zero exponent.

	<math>\displaystyle c = 2^{~~s/1200~~}</math>		<math>\displaystyle 28/27 = (2 \cdot 2 \cdot 7) / (3 \cdot 3 \cdot 3) = 2^{2} \cdot 3^{-3} \cdot 7^{1} = 2^{2} \cdot 3^{-3} \cdot 5^{0} \cdot 7^{1} = </math> {{monzo\|2 -3 0 1}}

	=== Monzo to ratio ===		=== Monzo to ratio ===
	To find the ratio ''c'' for an interval of [[monzo]] '''m''' = {{monzo\| m<~~sub~~>1</~~sub~~> m<~~sub~~>2</~~sub~~> m<~~sub~~>3</~~sub~~> ~~… }}~~, apply		To find the ratio '''r''' for an interval of [[monzo]] '''m''' = {{monzo\| a b c … }}, apply

			<math>\displaystyle r = 2^{a} \cdot 3^{b} \cdot 5^{c} \ldots </math>

			=== Ratio to cents ===
			To find the cents '''c''' for a ratio '''r''', apply

			<math>\displaystyle c = 1200 \cdot log(r) / log(2)</math>

			Logarithms of any base (base 10, base 2, base ''e'', etc.) can be used.

			<math>\displaystyle c = 1200 \cdot ln(r) / ln(2)</math>

			=== Cents to ratio ===
			To find the ratio '''r''' for an interval of '''c''' [[cent]]s, apply

			<math>\displaystyle r = 2^{c/1200}</math>

	~~<math>\displaystyle~~ c ~~= 2^{m_1} \cdot~~ 3~~^{m_2} \cdot 5^{m_3} \ldots <~~/~~math>~~		The result will be in decimal form, and will only be as exact as '''c''' is. For example, 702 cents yields 1.500038989.., which is approximately 1.5, which is 3/2.

	== Extended frequency ratio (EFR) ==		== Extended frequency ratio (EFR) ==
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	=== Subharmonic EFR, a.k.a. SEFR ===		=== Subharmonic EFR, a.k.a. SEFR ===
	Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". ~~Confusingly, some~~ writers ~~tend to~~ omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.		Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Some writers omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.

	To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.		To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.

FloraC: Rework cuz "7:6:5:4" is unacceptably confusing.

2025-11-24T10:11:12Z

Rework cuz "7:6:5:4" is unacceptably confusing.

FloraC: These forms aren't codes

2025-11-24T09:31:59Z

These forms aren't codes

← Older revision		Revision as of 09:31, 24 November 2025
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	A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).		A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).

	All [[interval]]s can be expressed as ratios, and they can be rational or irrational. Although mostly written in the form ~~<code>~~larger/smaller~~</code>~~ throughout this wiki, they may be written in several ways:		All [[interval]]s can be expressed as ratios, and they can be rational or irrational. Although mostly written in the form ''larger''/''smaller'' throughout this wiki, they may be written in several ways:
			* 2/1, 2:1, 1/2, 1:2 for the [[octave]];
			* 3/2, 3:2, 2/3, 2:3 for the [[3/2\|just perfect fifth]].

	* 2/1, 2:1, 1/2, 1:2 ([[octave]])		When the larger number is written first (''note''/''base''), this usually signifies a note being played ''above'' some base tone (perhaps the starting note of a scale). When the smaller number is written first (''base''/''note''), this usually signifies the note being played ''below'' that base tone.
	* 3/2, 3:2, 2/3, 2:3 ([[3/2\|just fifth]])

	When the larger number is written first (~~<code>~~note/base~~</code>~~), this usually signifies a note being played ''above'' some base tone (perhaps the starting note of a scale). When the smaller number is written first (~~<code>~~base/note~~</code>~~), this usually signifies the note being played ''below'' that base tone.

	[[Chord]]s with three or more notes can also be expressed as ratios. For example, the just intoned major triad in root position is 4:5:6. Chords can also be written as a string of intervals, such as 1/1–5/4–3/2. (4:5:6 can be viewed as a shorthand for 4/1:5/1:6/1 or 4/4:5/4:6/4).		[[Chord]]s with three or more notes can also be expressed as ratios. For example, the just intoned major triad in root position is 4:5:6. Chords can also be written as a string of intervals, such as 1/1–5/4–3/2. (4:5:6 can be viewed as a shorthand for 4/1:5/1:6/1 or 4/4:5/4:6/4).

FloraC: There's no expert page and the title is clear enough that this isn't about step ratio

2025-11-24T09:29:58Z

There's no expert page and the title is clear enough that this isn't about step ratio

← Older revision		Revision as of 09:29, 24 November 2025
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	{{interwiki		{{interwiki
			\| en = Frequency ratio
	\| de =		\| de =
	~~\| en = Ratio~~
	\| es = Radio		\| es = Radio
	\| ja = 周波数比率		\| ja = 周波数比率
	}}		}}
	~~{{beginner}}~~
	~~''For the L/s ratio, see [[Step ratio]].''{{Wikipedia\|Interval ratio}}~~
	A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).		A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).

TallKite: added a section on extended frequency ratios

2025-11-24T09:09:02Z

added a section on extended frequency ratios

@@ Line 6: / Line 6: @@
 }}
 {{beginner}}
-{{Wikipedia|Interval ratio}}
+''For the L/s ratio, see [[Step ratio]].''{{Wikipedia|Interval ratio}}
 A '''frequency ratio''' (often shortened to '''ratio''') is the relationship between the frequencies of the [[pitch]]es of two or more notes. For example, a piano string vibrating at 110 Hz (110 times per second) and a piano string vibrating at 220 Hz are in a 2:1 ratio (since 220/110 reduces to 2/1).
@@ Line 33: / Line 33: @@
 <math>\displaystyle c = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \ldots </math>
 [[Category:Ratio| ]] <!-- main article -->
 [[Category:Notation]]
 [[Category:Terms]]

← Older revision		Revision as of 10:11, 24 November 2025
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	== Extended frequency ratio (EFR) ==		== Extended frequency ratio (EFR) ==
	An extended ratio is a ratio of more than 2 numbers. For example, a recipe might combine cups of flour, milk and sugar in a 5:3:4 ratio. Xenharmonic music uses an extended ratio to indicate the various frequency ratios between 3 or more notes. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous ''unordered'' example, EFRs are ''ordered~~'' either~~ ascending or descending~~. The ascending form is much more common~~.		An extended ratio is a ratio of more than 2 numbers. For example, a recipe might combine cups of flour, milk and sugar in a 5:3:4 ratio. Xenharmonic music uses an extended ratio to indicate the various frequency ratios between three or more notes. This is called an '''extended frequency ratio''' or '''EFR'''. Unlike the previous unordered example, EFRs are ordered, with harmonic EFR typically written in an ascending order and subharmonic EFR typically written in a descending order.

	=== Harmonic ~~(ascending) EFRs~~ ===		=== Harmonic EFR ===
	For example, consider a [[just intonation]] major triad on ~~A-440~~ with a ~~'''~~ratio list~~'''~~ of 1/~~1 - 5~~/~~4 - 3~~/2. The three frequencies are 440, 550 and 660. The EFR is 440:550:660, which simplifies to 4:5:6, spoken as "four five six". (Had the root been other than ~~A-440~~, the EFR would be the same.)		For example, consider a [[just intonation]] major triad on {{w\|A440}} with a ratio list of 1/1–5/4–3/2. The three frequencies are 440, 550 and 660. The EFR is 440:550:660, which simplifies to 4:5:6, spoken as "four five six". Had the root been other than A440, the EFR would be the same.

	To convert an EFR to a ratio list, simply divide every ~~number~~ by the first ~~number~~. For example, (4:5:6)/4 is 4/~~4 - 5~~/~~4 - 6~~/4, which simplifies to 1/~~1 - 5~~/~~4 - 3~~/2.		To convert an EFR to a ratio list, simply divide every harmonic by the first harmonic. For example, (4:5:6)/4 is 4/4–5/4–6/4, which simplifies to 1/1–5/4–3/2.

	The EFR directly indicates the interval between any pair of notes in the chord. In the ~~JI dom7~~ chord 4:5:6:7, the interval between the ~~3rd~~ and the ~~5th~~ is 6/5, that between the ~~3rd~~ and ~~7th~~ is 7/5, etc.		The EFR directly indicates the interval between any pair of notes in the chord. In the harmonic seventh chord 4:5:6:7, the interval between the third and the fifth is 6/5, that between the third and seventh is 7/5, etc.

	The EFR also indicates where in the [[harmonic series]] the chord occurs. 4:5:6:7 occurs as [[~~Harmonic\|harmonics~~]] 4, 5, 6 and 7. Thus an EFR is also a list of harmonics.		The EFR also indicates where in the harmonic series the chord occurs. 4:5:6:7 occurs as [[harmonic]]s 4, 5, 6 and 7. Thus an EFR is also a list of harmonics.

	To convert a list of ratios to a list of harmonics, multiply each ratio by the LCM of the denominators. For example, 1/~~1 - 6~~/~~5 - 3~~/2 has denominators 1, 5 and 2, with an LCM of 10. Multiplying each ratio by 10 makes 10/~~1 - 12~~/~~1 - 15~~/1. Remove the ones to get 10:12:15.		To convert a list of ratios to a list of harmonics, multiply each ratio by the LCM of the denominators. For example, 1/1–6/5–3/2 has denominators 1, 5 and 2, with an LCM of 10. Multiplying each ratio by 10 makes 10/1–12/1–15/1. Remove the ones to get 10:12:15.

	=== Subharmonic ~~(descending) EFRs~~, ~~aka SEFRs~~ ===		=== Subharmonic EFR, a.k.a. SEFR ===
	Consider the melodic inversion of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is~~. While~~ it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR ~~is a ''descending'' EFR~~, in this case 7:6:5:4. This list of subharmonics is spoken as "seven six five four".		Consider the [[melodic inversion]] of 4:5:6:7. The ratio list is 1/1, 7/6, 7/5 and 7/4, a min7flat5 chord. The EFR is 60:70:84:105. These large numbers make the chord seem more complex than it actually is because while it occurs quite high in the harmonic series, it occurs quite low in the [[subharmonic series]] as subharmonics 7, 6, 5 and 4. The subharmonic EFR or SEFR represents it as reciprocals of harmonics, in this case 1/(7:6:5:4). This list of subharmonics is spoken as "one over seven six five four". Confusingly, some writers tend to omit the numerator in SEFRs. For example, 1/(7:6:5:4) is written as 7:6:5:4.

	To convert an SEFR to a ratio list, simply ~~divide~~ the first number ~~by every number~~. For example, 7/(7:6:5:4) is 7/~~7 - 7~~/~~6 - 7~~/~~5 - 7~~/4, with 7/7 simplifying to 1/1.		To convert an SEFR to a ratio list, simply replace the numerator with the first subharmonic number, and put it inside each subharmonic. For example, 1/(7:6:5:4) is 7/7–7/6–7/5–7/4, with 7/7 simplifying to 1/1.

	The SEFR directly indicates the interval between any pair of notes in the chord. In 7:6:5:4, the interval between the ~~3rd~~ and the ~~5th~~ is 6/5, that between the ~~3rd~~ and ~~7th~~ is 6/4 which is 3/2, etc.		The SEFR directly indicates the interval between any pair of notes in the chord. In 1/(7:6:5:4), the interval between the third and the fifth is 6/5, that between the third and seventh is 6/4 which is 3/2, etc.

	To convert a list of ratios to a list of subharmonics, divide each ratio by the LCM of the numerators. For example, 1/~~1 - 6~~/~~5 - 3~~/2 has numerators 1, 6 and 3, with an LCM of 6. Dividing each ratio by 6 makes 1/~~6 - 1~~/~~5 - 1~~/4. ~~Remove~~ the ones to get 6:5:4.		To convert a list of ratios to a list of subharmonics, divide each ratio by the LCM of the numerators. For example, 1/1–6/5–3/2 has numerators 1, 6 and 3, with an LCM of 6. Dividing each ratio by 6 makes 1/6–1/5–1/4. Extract the ones to get 1/(6:5:4).

	To convert an EFR to a SEFR or vice versa, first convert it to a ratio list.		To convert an EFR to a SEFR or vice versa, first convert it to a ratio list.

	=== ~~Alternate~~ forms ===		=== Alternative forms ===
	Both ratio lists and EFRs can indicate the voicing of a chord. For example, a 4:5:6 major triad in [[Kite's thoughts on hi-lo notation\|hi3add8 voicing]] is 1/~~1 - 3~~/~~2 - 2~~/~~1 - 5~~/2 or 2:3:4:5.		Both ratio lists and EFRs can indicate the voicing of a chord. For example, a 4:5:6 major triad in [[Kite's thoughts on hi-lo notation\|hi3add8 voicing]] is 1/1–3/2–2/1–5/2 or 2:3:4:5.

	Contiguous harmonics such as n:n+1:n+2:n+3 can be written with a double colon as n::n+3. Likewise for contiguous subharmonics. This is especially common for scales like 8::16.		Contiguous harmonics such as ''n'':(''n'' + 1):(''n'' + 2):(''n'' + 3) can be written with a double colon as ''n''::(''n'' + 3). Likewise for contiguous subharmonics. This is especially common for scales like 8::16.

	~~SEFRs are sometimes written not as a:b:c:d but as 1/(a:b:c:d).~~
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