Pure tones are an abstraction-- all real-world sounds have *bandwidth.* Bells
make
nearly-pure
narrow
sounds;
drums
make
wide
ones. How
do
we measure interval ratios of notes with width?

We really need to know bandwidth relative to the note. For example, the octave-sized interval {220hz, 440hz} has a width of 220hz, but the octave {440hz, 880hz} takes up 440hz. They're both octaves and should have the same "width", so we use the ratio r = instead. An octave has r = 3:2.

Each well-tempered note is of an octave, so each reserves a band with r = 18:1. Excepting glissando, they don't normally use the whole width. We can estimate how much they actually use:

For frequencies and , the interval ratio is the slope of a
line. If the notes have width, the lines get fat:

The interval ratio could be the slope of any line going through the black box, for example, either of these:

The **lowest denominator rule** says the interval ratio is the
most consonant possible ratio.

Here's a sanity test for the rule. A piano evokes interval
ratios even though the notes are well-tempered. The prediction is
that for some bandwidth, the well-tempered notes take the interval
ratios they're supposed to. In fact,
a frequency-to-bandwidth ratio of 100:1-- about of the reserved
band-- does just that.

The diagram shows all possible integer ratios, with increasing
denominator on the y-axis. The target ratios are marked in
green.

If the bandwidth is too wide, the B flat becomes ; if too narrow, we lose the E flat and the A:

Using the lowest-denomionator rule, wide tones allow fewer distinct
intervals. This is why you can't get harmony
out of drums: wherever the center frequency is, there tends to be a
consonance nearby. The interval ratio for frequencies *near*
a ratio of 3:2 will *sound like* (by the rule) they *are*
3:2. A graph of possible denominators might look like this
(y-axis is denominator, x-axis is frequency ratio):

So denominators even as high as 16 (a semitone) aren't possible at
this bandwidth.

Here is the graph for widths typical of a piano. The y-axis now
goes much higher:

Extremely narrow bands makes many ratios available, but each consonance has a tiny catchment basin:

Historical note: in "On the Sensations of Tone," Helmholtz calculated
figure 60a to show the relative 'roughness' of arbitrary
intervals. 'Roughness' is his measure of the number of beats
produced by simultaneous tones; basically, dissonance. The
lowest-denominator rule seems to provide a closely related measurement.

Helmholtz: On the Sensations of Tone, page 193.

The idea of 'available ratios' is summarized in this
diagram. At the bottom are the wideband ratios; as you move
up the trunks, the bands thin and new harmonies become available as the
tones are purified. Black is consonance, white is
dissonance.

With wide bands, *only* consonance is possible-- like drum
melodies, which roughly spell out a tune but can't create
harmony. Higher on the y-axis, as the bands narrow, more
intervals become possible.