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From Plomp & Levelt 1965, Tonal Consonance and Critical Bandwidth: "The critical frequency is lower for more consonant intervals. This behavior reflects the musical practice to avoid thirds at low frequencies and to use mostly octaves or wider intervals."

Is it true that musical practice tends to avoid thirds at lower frequencies? Is there an objective measurement of such behavior, or is it just something noticed personally after seeing a lot of music?

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    Just sit at a piano and try it. Play a simple tune with an accompaniment a third away. Above middle C, it will probably sound nice but the lower you go, the less nice it will sound. I wouldn't dare name an exact limit, each to his own. – badjohn May 19 '17 at 23:41
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    Isn't seeing a lot of music and noting an absence of thirds in lower registers an objective measure? – Todd Wilcox May 20 '17 at 5:18
  • Something to do with the harmonics of each note, rather than the notes fundamentals? So, it will vary to an extent between instruments. But still not good on many. – Tim May 20 '17 at 7:22
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    They way to get an "objective measurement" is to look at a well-defined corpus of music and do a statistical analysis. That has become a lot easier now there are big collections of music that are encoded accurately enough to be useful, both in MIDI and audio format. For example, from a MIDI file you could select a note, and then measure the proportion of time when the next nearest note was at different intervals (2nd, 3rd, 4tn....) above it, and compare the results for different pitched notes over a sample of say 10,000 or even 100,000 scores. – user19146 May 20 '17 at 18:52
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    @ToddWilcox It also depends on the timbre of the instrument. For example most of the acoustic energy produced by an oboe is in the 4th and 5th harmonics (two octaves above what you perceive as the "pitch of the note") not in the fundamental. But the 4th and 5th harmonics are a just intonation major third apart, not an equal tempered third - hence the well known orchestration rule-of-thumb NEVER to write passages for two oboes in parallel thirds or sixths, unless you WANT them to sound out of tune! (The "rule" was known long before anybody had measured the oboe's sound spectrum, BTW) – user19146 May 20 '17 at 18:58
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+100

There is a trick that organists do, which is to play parallel fifths on low notes in the pedals. The notes match the 2nd and 3rd harmonic of a lower note. The fundamental of that lower note is only present as "beats" produced by the two notes, but you "hear" it nevertheless. The low note is called a "resultant" and the pitches you hear are an octave lower than any that the organ can actually produce. The effect doesn't work at higher pitches.

This is really the only thing that very low fifths are good for. They don't improve the harmonies in any way; they just allow you to fake a 32-foot rank that's not there.

If you play thirds in the pedals you hear the beats as fast beats rather than as a low tone. Musically that's just about useless.

  • The more I read this answer, the more comprehensive it becomes. The last paragraph about beats is so true, just disarmingly concise. – lauir Jun 5 '17 at 23:04
  • The fundamental of that lower note is only present as "beats" produced by the two notes ...unless they pass through some kind of non-linearity... – endolith Mar 19 '18 at 17:26
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The answer is due to two factors: (a) harmonic series and (b) the formation of beats.

If you pluck a string or vibrate the air in a wind instrument, you predominantly hear one note or frequency. But in reality, there are additional notes/frequencies being produced. The note we predominantly hear is called the "fundamental frequency," and the additional frequencies produced are called "harmonic frequencies." The harmonic frequencies are all integer multiplies of the fundamental frequency. (For example, an A at 440 Hz will have as its harmonic frequencies these pitches: 440 x 2 = 880 Hz, 440 x 3 = 1320 Hz, 440 x 4 = 1760 Hz.) So playing a single note on the piano actually produces a series of frequencies, which is called the "harmonic series" in physics. Harmonic series are pleasing to the ear, and these harmonics are not as loud and higher in frequency than the fundamental frequency.

Now that we understand harmonic series, we can consider the second important phenomenon: beats. If you simultaneously play two different pitches with frequencies fA and fB, then a third frequency fbeat = fBfA can be heard. This "virtual" frequency fbeat indicates how often the "crest" of one sound wave fA overlaps with the "trough" of the other sound wave fB in such a way as to cancel each other out. For example, we could draw the two frequencies fA and fB in blue and red, overlap them, and see that there are times they will cancel. The resultant wave/combination of the two is shown on the bottom:

http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/imgsou/beat4.gif

This phenomenon of the waves periodically destroying each other is referred to as "beats," and it arises from something in physics called the superposition principle. When we say that the beat frequency is 20 Hz, we mean that, 20 times a second, the two waves cancel each other out and we hear nothing. Because the beat frequency fbeat is given by the difference between the two notes fA and fB, the beat frequency fbeat is always lower than the two notes fA and fB.

(In reality, sound waves do not have crests and troughs, but instead have compressions and rarefactions/expansions. Also, beats won't always be audible to humans, but we'll ignore that point and focus only on beats that are above 20 Hz and thus audible to human ears.)

Here's what we have so far:

  1. harmonic frequencies are integer multiples of a fundamental frequency and are pleasing to the ear;
  2. when you play two notes together, a third frequency is produced

Consonant intervals are two notes just far enough apart in frequency that the virtual third tone (the beat frequency) completes a harmonic series with the two real tones. For example, if you play two notes fA = 100 Hz and fB = 150 Hz, then the beat frequency will be fbeat = 150 – 100 = 50 Hz. This beat frequency completes a harmonic series because tone A, 100 Hz, is 2 x 50 Hz and tone B, 150 Hz, is 3 x 50 Hz. Both notes (100 Hz and 150 Hz) are integer multiples of the beat frequency, and so they are consonant intervals. And what exactly are these notes, 100 Hz and 150 Hz? They are roughly Ab and Eb (a perfect fifth apart) in the lower register:

https://mrandmrs55.files.wordpress.com/2012/05/keyboardnotes.jpg

So the trick for producing consonant intervals is to have the beat frequency fbeat produce a virtual fundamental frequency that completes a harmonic series with the two real notes. Stated differently, a consonant interval is produced whenever the two notes being played have a difference in frequency fbeat = fBfA, where fB ÷ fbeat and fA ÷ fbeat are both integers (or close to integers).

As it turns out, if you pick a particular register of the piano, there is a particular interval that tends to satisfy this condition to produce a virtual fundamental frequency. For example, in the lower register, perfect fifths tend to satisfy the condition, exactly as our example above shows. (100 Hz and 150 Hz occur in the lower register and are roughly a perfect fifth apart.) In higher registers, smaller intervals tend to satisfy the condition.

6

The lower you go, the larger the range of intervals to avoid.

Above treble clef, for instance, piccolos need not even fear harmony as narrow as a whole step.

Below bass clef, some “basses,” as in tuba and viol parts, dare octaves but nothing tighter. (I have printed examples but haven’t found online ones.)

This plays on the trade-off between hearing individual notes, at up to 10 or so per second, and tonal/chordal combinations, at 30 Hertz or so and above. Between those rates we hear a disruptive pattern of beating, which can be annoying because our ears try to interpret it as speech, not music.

Down below the bass clef, say C2 at 65 Hz, a third would be E2 at 82 Hz and produce 82 − 65 = 17 beats per second. Ow!

When beats are much slower than that, they can sound warm. When rapid enough, they become chords! But there is a middle ground where their warbling is just plain distractingly destructive.

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The other reason to avoid low thirds: thirds are much more out of tune in equal temperament than fourths, fifths, or octaves, and sound especially grungy down low, where the beats are slow.

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    Very true, but I'm not sure the argument makes sense here. Actually I think the bad quality of 12-edo thirds is more problematic in higher registers, because the beat frequencies become really audible. E.g. C5-E5 has a disturbance of 21 Hz at the first match of overtones, whereas G1-B1 merely has a subtle 4Hz beat. (The lower tones obviously also have faster beats in their higher overtones, but those are obscured by the low components and by other effects like string-inharmonicity.) – leftaroundabout May 20 '17 at 23:56
  • Hmmm.... I'll have to try it out sometime. – Scott Wallace May 22 '17 at 9:35
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As a bass guitar player, I can definitely confirm this. The "octave-fifth-root" pattern is very common, partly because it's something you can play without any risk of causing serious dissonance. Not coincidentally, these notes are also what power chords on the guitar are made up of.

When playing multiple notes simultaneously, many bassists even avoid fifths because they end up sounding muddy. Octaves, on the other hand, are more common.

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I'd rather agree with the approach to "to avoid thirds at low frequencies", but then I'd go with "and to use mostly fifths or wider intervals".

Indeed, @jdjazz physical descriptions of "what happens with the beats" is correct to me, and, as a (rookie) bass and piano player I'd agree with @Lee White (that's why I replaced octaves with fifths).

BTW: That's for chords, but what about solos/melodies?

Physically, given what happens with frequency-vs-time in the Laplace's transform, it would take a "longer time" for your ears to "recognize" a "lower frequency".

Because of this is a simple property of waves, it would be harder to "understand" (i.e. "recognize the single notes played") a solo by Jaco on a bass, than a solo by Paco de Lucia on a guitar.

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