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 = [Graphics:Images/index_gr_1.gif]

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 = [Graphics:Images/index_gr_2.gif]instead.  An octave has r = 3:2.

Each well-tempered note is [Graphics:Images/index_gr_3.gif] 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 [Graphics:Images/index_gr_4.gif] and [Graphics:Images/index_gr_5.gif], the interval ratio [Graphics:Images/index_gr_6.gif]is the slope of a line.  If the notes have width, the lines get fat:

[Graphics:Images/index_gr_7.gif]

[Graphics:Images/index_gr_8.gif]

[Graphics:Images/index_gr_9.gif]


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 [Graphics:Images/index_gr_10.gif]of the reserved band-- does just that.

[Graphics:Images/index_gr_11.gif]



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 [Graphics:Images/index_gr_12.gif]; if too narrow, we lose the E flat and the A:

[Graphics:Images/index_gr_13.gif]


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):

[Graphics:Images/index_gr_14.gif]

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:

[Graphics:Images/index_gr_15.gif]

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

[Graphics:Images/index_gr_16.gif]

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.
fig60a

  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. 

[Graphics:Images/index_gr_17.gif]

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.