Frequency ratio
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This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. |
For the L/s ratio, see Step ratio.
A frequency ratio (often shortened to ratio) is the relationship between the frequencies of the pitches 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 intervals 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 (octave)
- 3/2, 3:2, 2/3, 2:3 (just fifth)
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.
Chords 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…
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
Cents to ratio
To find the ratio c for an interval of s cents, apply
[math]\displaystyle{ \displaystyle c = 2^{s/1200} }[/math]
Monzo to ratio
To find the ratio c for an interval of monzo m = [m1 m2 m3 …⟩, apply
[math]\displaystyle{ \displaystyle c = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \ldots }[/math]
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.
Harmonic (ascending) EFRs
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.
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 also indicates where in the harmonic series the chord occurs. 4:5:6:7 occurs as harmonics 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.
Subharmonic (descending) EFRs, aka SEFRs
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.
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 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 an EFR to a SEFR or vice versa, first convert it to a ratio list.
Alternate forms
Both ratio lists and EFRs can indicate the voicing of a chord. For example, a 4:5:6 major triad in 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.
SEFRs are sometimes written not as a:b:c:d but as 1/(a:b:c:d).