Explain why chords sound pleasant mathematically?

[brandy]

Main Question or Discussion Point

can anyone explain with mathematics/physics why chords or notes thirds (c & e, e&g, d&f, etc) or octaves sound better than say two consecutive tones???

 

[James3849]

mathematically and physically it is quite simple. consider the simple wave form of the sound of a note, say middle c. now consider its frequency and wavelength (440Hz for middle c). now if you add another note, be it a semitone or a tone (i.e c-sharp (466Hz) or d) and add these two waveforms together you will see an odd looking curve. you will still have a peaks and troughs like a normal sinusoid, however there will be a slight spike near these peaks.

because sound is travels through air by minute pockets of higher and lower pressure air, the two notes are assualting you air at 440 beats per second for middle c and 466 beats per second for c-sharp. like with all conflicting chords, the gap between the peak and the spike creates another sort of internal frequency that is unpleasant to the ear. it is sort of like a beating sound withing a chord. it is hard to hear with a piano due to the diminishing volume of a chord strike however if you hear two brass or woodwing players deliberatly playing off notes, the 'beating sound' is quite pronounced.

with chords major 3rds, 5th's etc. the addition of the two waveforms to note do not create peaks that are close together and hence create pleasent sounds where the 'beating' is eliminated

 

[brandy]

so what would be the conditions in order for something to be "pleasant"

 

[BryanP]

so what would be the conditions in order for something to be "pleasant"

based off what he's saying, you'd want to have a combination of sound waves that do not have peaks that create internal frequencies (the spikes he is referring to) that causes odd freq sounds to occur

 

[Phrak]

But you asked 'why', not what is pleasant. I think you need to ask some rocket surgons, or brain chemists, or something.

What sounds pleasant are simple ratios between notes like 1 to2 (an octave), 2 to 3, and 3 to 5. Things like 6 to 7 begin to sound .... unpleasant. Two notes side by side on a piano are in the ratio of about 11 to 12.

 

[Redbelly98]

The actual ratio between adjacent piano keys is

$$2^{1/12} = 1.059...$$

or roughly 89/84. :yuck:

12/11 is 1.09..., about 1 and one-half piano keys apart.

Contrast that to a major C chord:

C & E are a factor of 2^(4/12) = 1.260, pretty close to 5/4
C & G are a factor of 2^(7/12) = 1.498, pretty close to 3/2

 

[rcgldr]

Contrast that to a major C chord:
C & E are a factor of 2^(4/12) = 1.260, pretty close to 5/4
C & G are a factor of 2^(7/12) = 1.498, pretty close to 3/2

This is true on a piano, but an ochestra using multiple instruments can play a true C chord where E 5/4 above C and G is 3/2 above C.

A is defined to be 440hz for most forms of music.

Link to page showing integer multiples as opposes to the equal tuning of a piano.

http://www.music.sc.edu/fs/bain/atmi02/hs/index-noaudio.html

 

[Phrak]

The actual ratio between adjacent piano keys is

$$2^{1/12} = 1.059...$$

or roughly 89/84. :yuck:

Yuck indeed. A little calculator work tells me 18:17 is close. 1.0588., with .06% error. Still :yuck:

12/11 is 1.09..., about 1 and one-half piano keys apart.

Contrast that to a major C chord:

C & E are a factor of 2^(4/12) = 1.260, pretty close to 5/4
C & G are a factor of 2^(7/12) = 1.498, pretty close to 3/2

Doh! you got me.

Suprize!, it isn't 2^{0/12}:2^{1/12} either.

Pianos are retuned from the mathematical 1:2^{1/12} in a compromise, so that cords played in the most common keys fall closer to the simple ratios as you just noted. The interval between each pair of keys is slightly different than the next.

 

[turbo]

Tough one, because some musical traditions are more accepting of dissonance than others, and to some listeners, some tonal intervals are quite well accepted that in other traditions might be rejected.

 

[Phrak]

Tough one, because some musical traditions are more accepting of dissonance than others, and to some listeners, some tonal intervals are quite well accepted that in other traditions might be rejected.

In some traditions (ages zero through 5 or 8 or 10 and on up) dissonance is quite pleasant and intently practiced. It gets you a bottle, your diapers changed, etc.

Did I fail to say that genre informs dissonance, or that pleasantness speaks to out ethnic harmonic being?

So what gives Turbo? Have you suddenly developed a taste in postmodernism?

 

[rbj]

can anyone explain with mathematics/physics why chords or notes thirds (c & e, e&g, d&f, etc) or octaves sound better than say two consecutive tones???

there's an old USENET post of mine that speaks to this issue. i would repost it, but i was actually depending on the use of "ASCII art" to illustrate it and PF won't let me precede a line of text with spaces. be sure to read it with a mono-spaced font, otherwise intervals in the illustrations don't line up right.

 

[Redbelly98]

This is true on a piano, but an ochestra using multiple instruments can play a true C chord where E 5/4 above C and G is 3/2 above C.

Thanks, I have sometimes wondered if musicians ever do that.

 

[HallsofIvy]

Any musician playing an instrument that is not "tempered", a violin say, does that. Since a violin does not have frets, there is, for example, a difference between "A sharp" and "B flat".

 

[rcgldr]

Just intonation is one where all the notes differ by ratios of whole numbers (integers)

http://en.wikipedia.org/wiki/Just_intonation

Equal temperament is a comprimise where every pair of adjacent notes has an identical frequency ratio, typically 2^1/12

http://en.wikipedia.org/wiki/Equal_temperament

Using the standard of A4 = 440hz, then C4 is 264hz with just intonation, and 261.626hz with equal temperament. Some composers dislike equal temperament.

A concert or pedal harp is an example of a just intonation instrument. There are 7 pedals, each of which controls all the octaves of a certain note, for example the "C" pedal shifts all the "C" strings to C flat, C normal, or C sharp. With no pedals down, the harp is tuned to C flat (or A flat minor, same notes). With all pedals in the middle position, the harp is tuned to C major (or A minor, same notes), and there are no strings that are sharp or flat.

http://en.wikipedia.org/wiki/Pedal_harp

High end synthesizers can be programmed to use just innotation, by including key specific (A major, C major) programming in addition to playing notes. Organs, even old style pipe organs, also have "stops" (hand or foot operated buttons) to change the key and allow them to be just intonation intstuments. I'm not aware of any pianos that have stops, but a piano could be tuned to a specific key for a concert.

 

[pkleinod]

But you asked 'why', not what is pleasant. I think you need to ask some rocket surgons, or brain chemists, or something.

What sounds pleasant are simple ratios between notes like 1 to2 (an octave), 2 to 3, and 3 to 5. Things like 6 to 7 begin to sound .... unpleasant. Two notes side by side on a piano are in the ratio of about 11 to 12.

Yes, and there are some intervals that can sound both pleasant or
unpleasant depending on the context. The interval C to G#, an
augmented fifth, is dissonant, whereas C to A-flat, a minor sixth, is
consonant. This is so even on a piano, where these intervals are exactly
the same. To hear this, prepare your ear in the key of A-flat major by
playing an E-flat seventh chord followed by the tonic
chord, A-flat. Now play the interval C to A-flat (holding C and A-flat
simultaneously); it should sound consonant. Next, prepare your ear in the
key of C major by playing the G-seventh chord followed by the tonic chord
C major. Now play the interval C to G#, which, this time, should grate on
the ear, although it is physically exactly the same interval as before.

 

[jim mcnamara]

A is defined to be 440hz for most forms of music.

Since about 1900. Try playing js bach as he would have heard it with a=~410.
Most recorders (the musical instrument) allw you to tune to different A's, for example.
One of the authorities on tuning and history of tunings is Margo Shulter. Hopefuly she is still with us. There have been a lot of tunings.

Instead of me blathering read someone who knows, Margo:
http://www.medieval.org/emfaq/harmony/pyth.html

 

[rcgldr]

Try playing js bach as he would have heard it with a=~410.

On another historical note (bad pun), how long ago did organs have presets to set them for "just intonation"? How long ago did organs have stops (affects multple presets with a single switch, commonly a foot operated button)?

 

[Andy Resnick]

can anyone explain with mathematics/physics why chords or notes thirds (c & e, e&g, d&f, etc) or octaves sound better than say two consecutive tones???

"American Scientist" just had an article about this exact question:

http://www.americanscientist.org/issues/feature/2008/4/the-psychoacoustics-of-harmony-perception

It was rather interesting: there is a quantitative metric regarding the percieved dissonance of a triad.