Pure Intonation

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An interval is defined as the ratio between the frequencies of two notes (pitches or tones).  Intervals are found and represented in the following ways.

or 

Frequency (higher) : Frequency (lower)

The most fundamental interval is the octave.  The octave has a ratio of 2:1.  In other words, if one note has a frequency of 100 Hertz (Hz), then the note played an octave higher has a frequency of 200 Hz.  The octave is the most natural sounding interval other than two notes of the exact same frequency (also known as unison).

To find the pure tones that fall between a tone and its octave, one needs to find ratios that fall between 1:1 and 2:1 (between 1 and 2 when the ratios are written as fractions).

Tones with smaller integer ratios sound the most pure, and the most consonant.  This assures us that the unison is the most natural interval, and that the octave is second.  (Note: the ratio 1:2 identifies the octave below any given pitch because the frequency of note in question is half of its starting pitch.)  The next purest intervals between 1:1 and 2:1 are 3:2, 4:3, 5:3, 5:4, 6:5, and so on.  Remember that ratios can be written as fractions, so we can sort these ratios in ascending order.  For example, if the seven purest ratios considered thus far can be placed in the following order:

  1:1   6:5  5:4  4:3  3:2  5:3  2:1

or

1  1.2  1.25  1.33…  1.5  1.66…  2

A list of the most natural tones (and their interval ratios) relating closely to our 12-tone scale are listed here:

Tone

Ratio

Root (starting tone) Unison

1:1

minor 2

16:15

Major 2

9:8

minor 3

6:5

Major 3

5:4

Perfect 4

4:3

Diminished 5

17:12

Perfect 5

3:2

minor 6

8:5

Major 6

5:3

minor 7

9:5

Major 7

15:8

Octave

2:1

We can use these ratios in determining the pure tones on the desired interval between the root.  For example, if we want to find the frequency of the note that is a major third above Concert A (Concert A is the A above middle C on a keyboard and has a frequency of 440 Hz), we would solve the following for x.

So the frequency of the major third above Concert A (C#) has a frequency of 550 Hz when tuned purely.  Found in a similar way, the perfect fifth above Concert A (which happens to be the note E) is

The frequency of any note can be determined as long as a ratio (whether pure or tempered) and one frequency is known.