Bandwidth and Interval Ratios

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?

Measuring bandwidth: the ratio r =

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:

The lowest-denominator rule

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.

Checking the rule

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:

Available Ratios for Given Bandwidths

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.

Available Ratios: The dissonance forest

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.