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Frequency ratio: Difference between revisions

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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.
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).  


[[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).
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.
 
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…


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 ==
== Conversion ==
{{See also| Cent #Conversion | Monzo #Examples }}
=== 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
<math>\displaystyle r = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \cdot \ldots</math>
=== Cents to ratio ===
=== Cents to ratio ===
To find the ratio ''c'' for an interval of ''s'' [[cent]]s, apply
To find the ratio ''r'' for an interval of ''s'' cents, apply


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


=== Monzo to ratio ===
== Extended frequency ratio (EFR) ==
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
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 ===
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.


<math>\displaystyle c = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \ldots </math>
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.


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


=== Harmonic (ascending) EFRs ===
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.
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.)


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 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.


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.
=== 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 or 7:6:5:4.  


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.
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 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.
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.


=== Subharmonic (descending) EFRs, aka SEFRs ===
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).
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".


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 between EFR from and to SEFR, do the following:


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.
* [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


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.
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.  


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


=== 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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