Music = Math

[Viviuos]

What is the fundamental relationship between music and math? I read on a website, something about this guy wanting to make wind chimes. But then he went into detail about how the original scales equaled 255 but the western scales were 264 or something like that. I’m a computer person and I know 255 (or 256 depending on where you start) is a fundamental number. So assuming my post made at least a little sense to anybody, what is the connection?

[Against Will]

Frequency of notes, division of times, tempo, the list goes on. Some modern composers began using mathematical formulas to write their pieces with.

[peteroberts]

Pat Martino is an example.

[jadesmar]

Well if the discussion was about building wind-chimes. You are probably referring to producing tones using different lengths of windchimes. This relates directly to the overtone series, which I assume is the math that you want to know. So, here's some math stuff (some of it might even be right or interesting)

Temperment
------------
Axioms:
1. An octave is produced by doubling the frequency of a given pitch.
2. "A" is tuned to 25.5, 55, 110, 220, 440 ... Hz
3. The limits of human hearing are from 20-20,000 Hz.
4. Any repeating waveform can be produced by adding a fundimental sinusoidal waveform to a series of waveforms at integer multiples of the frequency of the fundimental. (Fourier Series)

We know that striking a note on a musical instrument does not (for the most part) produce a sign wave, so the waveform that is produced can be heard as a Fourier Series of overtones. Notes in this Fouries Series will be harmonic to the fundimental frequency.

Eg. If you strike a note a 55Hz, it will produce overtones at integer multiples of 55Hz. (i.e. 55Hz (fundimental), 110Hz (first overtone), 165Hz (second overtone). So, we have 2 A's and something like an E. Ok, so, we have a new note. Let's divide by 2 and do that again.

This produces this series (if we keep dividing by 2 until the notes are between 55 and 110)
A - 55
E - 82.5
B - 61.875
F#- 92.8125
C#- 69.609375
G#- 104.4140625
Eb- 78.31054688
Bb- 58.73291016
F - 88.09936523
C - 66.07452393
G - 99.11178589
D - 74.33383942
A- 55.75037956

So, using this method, by the time we get back to A, we are sharp. We need a method to fix this. One method is to keep all of these 5ths perfect except for one of them, which we would call a wolf note. Depending on the key we are playing in, we could change this wolf note so that it would not show up as frequently (eg. put it between G# and Eb). This is known as well temperment.

The method we currently use, however, is to flatten all of these intervals by a constant fixed amount (using the 12th root of two so that 12 notes have the same distance between them on a logarithmic scale). This produces the following. This is even temperment.

A - 58.27047019
Bb- 61.73541266
C - 65.40639133
C#- 69.29565774
D - 73.41619198
Eb- 77.78174593
E - 82.40688923
F - 87.30705786
F#- 92.49860568
G - 97.998859
G#- 103.8261744
A - 110

So, as you can see there is only a slight difference in each 5th but the octave each octave is precise and there are no wolf notes, a fairly decent compromise.

Anyhow, there are probably some errors in there (for example 4ths in a scale are likely not built by going 11 steps into the circle of fifths, but possibly going 1 step backward (i.e. multiplying by 4/3 instead of 3/2 11 times) D would then be 73.3333 Hz, instead of 74.33383942. I'm a little sketch on where to draw the dividing line for going forward and backward in the first tuning I outlined)

[Bob Rogers]

Well since music is sound that follows patterns and since mathematics is the science of patterns there is a pretty basic relationship. My guess is that the numbers you are refering to have to do with the definition of different types of scales: even-tempered, just-tempered, etc. Google mathematics, music, tuning and you will find a huge collection of web sites on this. The basic mathematics is pretty simple, but it also gets into psychoacoustics and that's very interesting.

[Matt Till]

I believe math = metal.

[Brad Barker]

i've read that the music staff is a logarithmic scale.

probably has to do with weber's law, which sets up a differential equation that has a logarithmic solution.

right?

[scorpionldr]

all music is mathamatical, from the frets on our board (and this even applies to those not playing frets) to scales, to frequencies, to meter and time signature and key signature, it's ALL math

[Brendan]

Kind of a side note, there's generally considered three types of prodigies: Mathmatic, Artistic, Mathmatic. Many people argue that math and music are fuctions of the same understanding.

If a complex equation depends on the relationship of variables, and the intereconnecteness of the numbers, how is it that different than a chord progression?

[jadesmar]

Originally Posted by Brad Barker

i've read that the music staff is a logarithmic scale.

probably has to do with weber's law, which sets up a differential equation that has a logarithmic solution.

right?

Not quite.

The music staff can be used to represent the notes in the well-tempered scale as well in which case one of the semi-tones in the scale will have a different frequency ratio. The notes in the even-tempered scale are based on a logarithmic scale (as I explained).

This is due to the fact that ascending one semi-tone is always, in even-temperment equal to multiplying the frequency of the fundimental of the lowest note by the 12th root of two (tone may vary but this is due to the amplitude of the overtones, not thier frequency). This multiplication to get to successive steps is what causes the notes to ascend logarithmically.

[l0calh05t]

Originally Posted by Brendan

[...]
If a complex equation depends on the relationship of variables, and the intereconnecteness of the numbers, how is it that different than a chord progression?

I'd say that the difference is that music usually expresses feelings, but other than that math and music are the same thing. So I'd say music is an artistic form of math (just as so called "fractal art")

[l0calh05t]

Originally Posted by jadesmar

[...]
This multiplication to get to successive steps is what causes the notes to ascend logarithmically.

thus a music staff IS a logarithmic scale

[MJ5150]

John Turner is really smart about this kind of stuff. He helped me understand alot about music when I was at his house a few weeks ago.

-Mike

[jadesmar]

Originally Posted by l0calh05t

thus a music staff IS a logarithmic scale

Only when you are using it to notate the even-tempered scale (which all fretted and most keyboard instruments use).

Fretless instruments can play in well-tempered and just intonation temperments. The same staff is used for that, but the scale is based on ratios of whole numbers and not logarithms.

Don't make me come over there.

[Brad Barker]

Originally Posted by jadesmar

Only when you are using it to notate the even-tempered scale (which all fretted and most keyboard instruments use).

Fretless instruments can play in well-tempered and just intonation temperments. The same staff is used for that, but the scale is based on ratios of whole numbers and not logarithms.

Don't make me come over there.

oh.

[Brad Barker]

Originally Posted by Brendan

Kind of a side note, there's generally considered three types of prodigies: Mathmatic, Artistic, Mathmatic...

you realize that you just repeated yourself repeated?

[Viviuos]

How are the far eastern (original scales) and European scales different, and why is there a difference?

[Brendan]

Originally Posted by Brad Barker

you realize that you just repeated yourself repeated?

D'oh. Proving I'm not one of them. Musical, Artistic, Mathmatic.

[jadesmar]

Originally Posted by Viviuos

How are the far eastern (original scales) and European scales different, and why is there a difference?

Here is why there is a difference. It may even be correct/interesting.

Here is an overtone series for A-55Hz.

This is based on the fact that the overtones of a musical note (those which produce the tone of the note) are integer multiples of the fundimental frequency. Which I took as an axiom earlier.

Since a doubling of frequencies represents an octave of a note, we need to consider only the odd interger multiples.

Overtone 1 - 55Hz
Overtone 3 - 165Hz
Overtone 5 - 275Hz
Overtone 7 - 385Hz
Overtone 9 - 495Hz
Overtone 11 - 605Hz
Overtone 13 - 715Hz
Overtone 15 - 825Hz
Overtone 17 - 935Hz

Now, to give you some idea of the relative volume levels of these overtones, you can play them at the following neck positions:

Overtone 3 - harmonic over the 7th fret of the A string (divides the string in 1/3 - 2/3).
Overtone 5 - harmonic over the 4th fret of the A string
(dividing the string in 1/5 - 4/5)
Overtone 7 - harmonic over the 10th fret of the A string?
(dividing the string in 3/7 - 4/7)

The other harmonics are available at the same integer ratios as their overtone numbers. That is overtone-9 will be heard when you divide a string in n/9 and strike the harmonic there.
I don't have my tape measure with me so, that's basically how to find them.

Using this method you can really see that the power of these harmonics drops off really quickly as you get higher and you need to strike harder to produce an audible tone.

Now, dividing these harmonics so that we are playing in the same range as our even-tempered scale we see the following:

Overtone 1 - 55Hz
Overtone 3 - 82.5
Overtone 5 - 68.75
Overtone 7 - 96.25
Overtone 9 - 61.875
Overtone 11 - 75.625
Overtone 13 - 89.375
Overtone 15 - 103.125

So striking a single note produces the following overtones (and higher).
A - 55Hz
E - 82.5 (a little sharp compared to even-temperment)
C# - 68.75 (a little sharp)
G - 96.25 (a little flat)
Bb - 61.875 (a little flat)
Something about 1/2 way between D and Eb - 75.625
Something about 1/2 way between F and F# - 89.375
G#- 103.125 (a little sharp)

Tuning system basically try and reconcile the three tendancies.
1. 5th are the second most audible note in the overtone series.
2. There should be an integer number of notes in an octave.
3. When striking a note, an overtone series is produced, chords sound pleasant when they do not deviate far from this overtone series.

These are somehow related to "the key" of a piece.

That's the why is there a difference. I'll let someone who has more information on the currently existant scales answer how they differ.

[Tash]

Originally Posted by Viviuos

How are the far eastern (original scales) and European scales different, and why is there a difference?

The simple answer is that they use different tonal and intonation systems in different cultures around the world. There are 11 unique tones in a "scale" in western music (11 half steps). In other tonal systems they divide the same "distance" into 9, or 18, or 21 distinct intervals.

This is why much eastern music sounds "out of tune" to western ears.

Obviously you can make this answer much more complicated if you want to.