Lower interval limits

I've heard about this concept of lower interval limits: An arbitrary limit to how low any given pattern of harmonic intervals can be played before it starts to sound muddy. It generally predicts that more consonant intervals can be played lower (but with exceptions), so for example octaves can be played as low as one wants, and fifths are exceptionally low, but major sevenths and minor seconds cannot be played very low. I've got a few questions:

1) Why does the muddiness depend on absolute frequencies for any given ratio (not just ratio)?

2) Does timbre affect the lower interval limits? I'd presume intervals of sine waves would be able to be played lower than more complex waveforms, but I'm not sure.

3) Also if anyone has like a cool graphic of how low specific intervals can go that would be nice :)

• Your statement that "the muddiness depend[s] on absolute frequencies and not the ratio between them" contradicts with your statement that "octaves can be played as low as one wants, and fifths are exceptionally low, but major sevenths and minor seconds cannot be played very low". Note that octaves, fifths, sevenths, and seconds all denote ratios between frequencies: the notes in an octave are in a 2:1 ratio, for example. Dec 5, 2018 at 7:15
• Why do you say that the limit is "arbitrary"? I don't have a graphic but I recommend a text book, Physics and the Sound of Music by Rigden. This has a whole chapter on these bands and predicting pitch discrimination.
– user50691
Dec 5, 2018 at 12:23
• @Dekkadeci I meant that for any given interval, the actual frequencies will be the factor in muddiness. Dec 5, 2018 at 15:55

Why does the muddiness depend on absolute frequencies and not the ratio between them?

It does depend on the ratio between them, as Dekkadeci commented. That's the whole reason why different intervals have different lower limits!

But it shouldn't really be surprising that absolute frequency also plays a role. That's not only for lower interval limits, even a single note can't be played arbitrarily low without getting muddy. As cool as 64' infrasound organ stops are, they can't be musically used in all but some eerie sound-FX settings (but the same could also be said about a dissonant interval on the low strings of a double bass).

Does timbre affect the lower interval limits?

Yes, but just as much on musical context. You're right that sine waves can sometimes be used at extremely low frequency, even when they're dissonant with other bass instruments. But I wouldn't agree that two dissonant low sine waves are less likely to be muddy than two dissonant tones with richer spectrum each – for the sine wave, the interval just won't be distinguishable at all anymore, you only get an uncomfortable beat. Whereas for two, say, double-bassoons playing a dissonance, you'll get a very rough, yet clear texture. Both can have their musical use, but in either case you need to be very careful. The problems are, respectively:

• If there's too little overtone information (sine waves, tubas, organ pedals) then the ear has trouble latching to anything in the sound. You'll just hear there's something going on in the bass register, but for sufficiently dissonant intervals it'll be hard to tell whether it's really tones at all or just hard-filtered brown noise. Which, again, can be used, but you should be sure that's what you want. I would mostly use it as a “subconscious uncomfort” effect below higher-pitched material that should also work on its own.
• If there's too much overtone information, you basically pollute the entire spectrum with incoherent rasp. Here I'd give the opposite advice: don't try fighting the rasp with higher instruments, rather emphasize on the low ones and allow the listener to fully focus on them.

There's one thing I would mention that's perhaps too little known and/or used, and quite relevant to the question – there are two fundamentally different families of overtone spectrums:

• Most instruments feature all overtones of the fundamental, i.e. when the fundamental frequency is ν they have also 2×ν, 3×ν, 4×ν, 5×ν etc.. Some bass examples: bassoon, tuba, double bass, piano, sawtooth synth, diapason organ stops.
• Some instruments have only (or at least predominantly) the odd overtones, i.e. 3×ν, 5×ν, 7×ν etc.. Examples: bass clarinet, bass flute, clean Rhodes e-piano, square or triangle synth, gedackt organ stops. This can also be achieved on harp and electric bass by plugging the strings exactly in the middle of the sounding length.

The latter family is in principle able to be used lower, due to the sparser spectrum (so the intervals can be further down distinguishable without polluting the midrange that much). But here it's crucial that you do not use octaves (which would effectively add back the even overtones). Rather you should “octavate” at the duodecime. The Bohlen-pierce tuning/scale is built entirely around this idea.

if anyone has like a cool graphic of how low specific intervals can go

Well, I don't think much of that precisely because it depends so much on both timbre and context. What I could give are lower limits that I would consider uncontroversial in most settings and with most timbres. I.e. these are conservative estimates, you may often find you can go considerably deeper but you shouldn't take it for granted.

• Fifth: C2-G2 65.4 Hz - 98.0 Hz, resultant: C1=32.7 Hz (even 3-limit)
• Major sixth: D2-B2 73.4 Hz - 123.4 Hz, resultant: G0=24.5 Hz (odd 5-limit)
• Major third: F2-A2 87.3 Hz - 110.0 Hz, resultant: F0=21.8 Hz (even 5-limit)
• Minor third: G2-B♭2 98.0 Hz - 116.5 Hz, resultant: E♭0=19.4 Hz (even 5-limit)
• Tritone: G2-C♯3 98.0 Hz - 138.6 Hz, resultant: ⁷D♯0=19.6 Hz (odd 7-limit)
• Minor sixth: G2-E♭3 98.0 Hz - 116.5 Hz, resultant: E♭0=19.4 Hz (even 5-limit)
• Minor seventh: G2-F3 98.0 Hz - 174.6 Hz, resultant: F0=21.8 Hz (even 3-limit)
• Major seventh: A2-G♯3 110.0 Hz - 207.6 Hz, resultant: A-1=13.9 Hz (even 5-limit)
• Major second: C3-D3 130.8 Hz - 146.8 Hz, resultant: C0=16.4 Hz (even 3-limit)
• Minor second: E3-F3 164.8 Hz - 174.6 Hz, resultant: F-1=10.9 Hz (even 5-limit)

Notice how these intervals have a resultant – i.e. frequency at which the combined signal repeats – that's still in the audible range, or almost so. That's perhaps not necessary, but it certainly helps the ear to make sense of the sound (interpreting both tones as effective overtones to a single, even deeper bass note).