Ratio

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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:

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

$\displaystyle c = 2^{s/1200}$

Monzo to ratio

To find the ratio c for an interval of monzo m = [m1 m2 m3, apply

$\displaystyle c = 2^{m_1} \cdot 3^{m_2} \cdot 5^{m_3} \ldots$