5 ms·

> Most Western instruments produce “harmonic” sounds that, when analyzed as Fourier described, have relatively strong lower overtones f, 2f, 3f, 4f. The overton

> Most Western instruments produce “harmonic” sounds that, when analyzed as Fourier described, have relatively strong lower overtones f, 2f, 3f, 4f. The overtones of several of these sounds will match when their fundamental frequencies are related by simple whole-number ratios. I've always had problems with existing consonance theories along these lines, probably out of ignorance. I believe that consonance probably has something to do with partial relationships. But it always has seemed the existing theory regarding consonance and partials is half baked at best. Harmonic sounds have partials at f, 2f, 3f, 4f, and so on. So let's take C and G, a perfect fifth. Will say f = C. So C's sawtooth has partials of f 2f 3f 4f 5f 6f ... G, at a perfect fifth, is 3/2 the frequency of C. So it has partials at 3/2f 3f 9/2f 6f 15/2f 9f 21/2f 12f ... The only overlaps are 3f, 6f, 9f, 12f. That's pretty thin gruel given how fast these partials drop of in amplitude (for a sawtooth, a partial at xf drops off as 1/x) Now consider E, a major 3 and also considered highly consonant. This is 5/4. So we have 5/4 5/2 15/4 5 25/4 15/2 35/4 10 ... That's an overlap of just 5, 10, ... with a super fast dropoff. Now consider Eb, a minor 3 and also consonant. This is, wait, Eb can't be approximated in rational values at all, um.... And if you just have sine waves, rather than sawtooth or whatnot, they consist of a single partial so there's never any overlap -- you're back to the pythagorean square 1.

deleted 3y ago
[deleted]
pcc 3y ago
With acoustic instruments, sympathetic resonance can play a big role in the overall colour, because eg unplayed undamped strings also respond according to their harmonic likeness to the presence of other excited frequencies. For example, on a piano if you undamp but don't sound one C, by setting the key down gently and holding it, then strike the C an octave below, the higher C will start ringing quite audibly and add its own partials, the same happens also of course to a lesser degree if you say undamp the G an octave + 5th above the C you're striking. This happens all the time during normal piano playing, because pianists purposefully hold long-duration keys down, and the sustain pedal keeps undamped the notes already played. It can also work in the "downward" direction for example on the violin where the open strings are E A D G with E the highest, the well known Vivaldi A minor concerto (eg Suzuki book 4) opens with shifting on that E string to eg 3rd position to play the A which is an octave higher than the open A string. To the extent you get this perfectly in tune, presuming a properly tuned violin of sufficient quality, the open A string ie an octave lower will start "ringing" as will the D below that to a lesser degree; and if you then subsequently damp those open strings, it changes the sound significantly. This "ringing" effect is quite important to violinists in developing their sense of tuning and calibrating exactly where to set their fingers.
whiddershins 3y ago
It doesn’t stop with are these partials the same as other partials on a different note. The ratio of the partials creates waves of differing complexity and periodicity. This makes the wave easier or harder to comprehend by the auditory system.
codeulike 3y ago
As well as overtones from an instrument, there are also combination tones that manifest in the cilia (tiny vibrating hairs) in our ears. https://en.wikipedia.org/wiki/Combination_tone https://en.wikipedia.org/wiki/Combination_tone ... So even if you play two pure sine waves with no overtones, say C and G, the cilia in your ears will still generate a bunch of other tones based on the relationships between the two notes, and I think a lot of the 'pleasing' nature of 'pairs of notes with simple frequency rations' (Octaves and Fifths etc) comes from that.
- tzs 3y ago
  There is also interaction of different notes somewhere in the processing in the brain. On speakers play a sine wave in the left channel and a sine wave of a slightly different frequency in the right channel. You'll hear a beat frequency due to the actual physical interaction of the two waves in the air. Then do the same thing except using headphones with good isolation so each ear only hears one of the sine waves. You still hear a beat frequency even though the two waves are no longer interacting in their air.
dspig 3y ago
I think what's going on with consonance is that there is an implied lower pitch that all the partials are harmonics of - that also works when there are just sine waves.
PaulDavisThe1st 3y ago
Here's a little offering for you that can take you on a deeper dive: https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html
mrob 3y ago
William Sethares published a model of dissonance, where dissonance of a sound is an amplitude-weighted sum of the dissonance of all pairs of partials, with dissonance of individual pairs a function of their frequencies according to Plomp and Levelt's dissonance curve. This model is a good approximation of how humans perceive dissonance, and it generalizes to inharmonic sounds: Local Consonance and the Relationship Between Timbre and Scale https://sethares.engr.wisc.edu/paperspdf/consonance.pdf https://sethares.engr.wisc.edu/paperspdf/consonance.pdf Informal explanation of the paper: https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html

deleted 3y ago

[deleted]

pcc 3y ago

With acoustic instruments, sympathetic resonance can play a big role in the overall colour, because eg unplayed undamped strings also respond according to their harmonic likeness to the presence of other excited frequencies. For example, on a piano if you undamp but don't sound one C, by setting the key down gently and holding it, then strike the C an octave below, the higher C will start ringing quite audibly and add its own partials, the same happens also of course to a lesser degree if you say undamp the G an octave + 5th above the C you're striking. This happens all the time during normal piano playing, because pianists purposefully hold long-duration keys down, and the sustain pedal keeps undamped the notes already played. It can also work in the "downward" direction for example on the violin where the open strings are E A D G with E the highest, the well known Vivaldi A minor concerto (eg Suzuki book 4) opens with shifting on that E string to eg 3rd position to play the A which is an octave higher than the open A string. To the extent you get this perfectly in tune, presuming a properly tuned violin of sufficient quality, the open A string ie an octave lower will start "ringing" as will the D below that to a lesser degree; and if you then subsequently damp those open strings, it changes the sound significantly. This "ringing" effect is quite important to violinists in developing their sense of tuning and calibrating exactly where to set their fingers.

whiddershins 3y ago

It doesn’t stop with are these partials the same as other partials on a different note. The ratio of the partials creates waves of differing complexity and periodicity. This makes the wave easier or harder to comprehend by the auditory system.

codeulike 3y ago

As well as overtones from an instrument, there are also combination tones that manifest in the cilia (tiny vibrating hairs) in our ears. https://en.wikipedia.org/wiki/Combination_tone https://en.wikipedia.org/wiki/Combination_tone ... So even if you play two pure sine waves with no overtones, say C and G, the cilia in your ears will still generate a bunch of other tones based on the relationships between the two notes, and I think a lot of the 'pleasing' nature of 'pairs of notes with simple frequency rations' (Octaves and Fifths etc) comes from that.

tzs 3y ago
There is also interaction of different notes somewhere in the processing in the brain. On speakers play a sine wave in the left channel and a sine wave of a slightly different frequency in the right channel. You'll hear a beat frequency due to the actual physical interaction of the two waves in the air. Then do the same thing except using headphones with good isolation so each ear only hears one of the sine waves. You still hear a beat frequency even though the two waves are no longer interacting in their air.

dspig 3y ago

I think what's going on with consonance is that there is an implied lower pitch that all the partials are harmonics of - that also works when there are just sine waves.

PaulDavisThe1st 3y ago

Here's a little offering for you that can take you on a deeper dive: https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html

mrob 3y ago

William Sethares published a model of dissonance, where dissonance of a sound is an amplitude-weighted sum of the dissonance of all pairs of partials, with dissonance of individual pairs a function of their frequencies according to Plomp and Levelt's dissonance curve. This model is a good approximation of how humans perceive dissonance, and it generalizes to inharmonic sounds: Local Consonance and the Relationship Between Timbre and Scale https://sethares.engr.wisc.edu/paperspdf/consonance.pdf https://sethares.engr.wisc.edu/paperspdf/consonance.pdf Informal explanation of the paper: https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html