Partial Math

by Robert Morss, RPT

A convenient method for understanding why a certain test works for a particular interval.

Let’s start with the simple 2:1 octave, where the goal is to make the 2nd partial of the lower note beatless with the fundamental of the upper note. We all learned that the test for a 2:1 octave is the 10th-17th — i.e. the major-10th formed beneath the lower note of the octave should have the same beat rate as the major-17th formed below the upper note of the octave. This is shown in the figure below:

2:1 Octave ~ 10th-17th Test

Red dots are the octave, the green dot is the test note.

If the 10th and the 17th have the same beat rate, the octave is 2:1.

Why does this test prove a 2:1 octave? Partial math to the rescue!

• Major 10th = 5:2
• Major 17th = 5:1
• We “cancel out” the 5’s and form a ratio of the remaining numbers, which gives us 2:1 — the interval we are testing for.
• Q.E.D. ~ if the 10th and the 17th have the same beat rate, you have a 2:1 octave.

The expression “Major 10th = 5:2” means that the ratio of the fundamental frequencies between the notes of a pure major-10th is five-to-two. Thus, if the lower note of a major-10th has a fundamental of 200 cycles per second, the upper note fundamental is 500 cps. If you need help determining interval ratios, try using the Partial Slide-Rule. Note that the first set of coincident partials between two notes also tells us the ratio of the fundamental frequencies between the notes.

The “canceling out” step is where a little hocus-pocus occurs. What’s happening is that the test note has one partial that overlaps the coincident partials being tested for. For example, in the 10th-17th test (above), partial math cancels out the 5’s because the fifth partial of the test note matches both the second partial of the lower note and the fundamental of the upper note. Thus we see that test notes allow us to isolate an exact set of coincident partials (which is, in fact, the whole point of test notes in the first place).

To read the math behind all this, see The Equal-Beating Theorem by Michael Wathen, RPT.

Here are some more for you to work out on your own.

4:2 Octave ~ 4th-5th Test

Red dots are the octave, the green dot is the test note.

Perfect 4th = 4:3
Perfect 5th = 3:2

4:2 Octave ~ 3rd-10th Test

Red dots are the octave, the green dot is the test note.

Major 3rd = 5:4
Major 10th = 5:2

6:3 Octave ~ 3rd-6th Test

Red dots are the octave, the green dot is the test note.

Minor 3rd = 6:5
Major 6th = 5:3

Perfect Fifth ~ 6th-10th Test

Red dots are the fifth, the green dot is the test note.

Major 6th = 5:3
Major 10th = 5:2

The ratio of a perfect fifth is 3:2.

Because fifths are narrowed when tuning equal temperament, the Major-6th should beat slightly faster than the Major-10th. If they have the same beat rate the fifth is pure.

Perfect Fourth ~ 3rd-6th Test

Red dots are the fourth, the green dot is the test note.

Major 3rd = 5:4
Major 6th = 5:3

The ratio of a perfect fourth is 4:3.

Because fourths are widened when tuning equal temperament, the Major-3rd should beat slightly slower than the Major-6th. If they have the same beat rate the fourth is pure.

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Updated: 16 November 2002