Frequency 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 for the octave;
• 3/2, 3:2, 2/3, 2:3 for the just perfect 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 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 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.

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 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 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.

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 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. 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.

Alternative 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.