"beat frequency" math question

[earthtonesaudio]

The beat frequency of any two notes is trivial: f_beat = f₁ - f₂. 

But what about 3 or more notes, or all 12?

I want to find the beat frequency for all notes of a chromatic scale combined (where the ratio between two frequencies is f_n+1 = f_n*2^1/12)?

[Jazznoise]

The answer is that you'll have multiple beat frequencies for 12 sine tones in a chromatic scale.

C3 (261.6 HZ)
Db/C#3 (277.2 HZ)
D3 (293.7 HZ)
Eb/D#3 (311.1 HZ)
E3 (329.6 HZ)
F3 (349.2 HZ)
Gb/F#3 (370 HZ)
G3 (392 HZ)
Ab/G#3 (415.3 HZ)
A4 (440 HZ)
Bb/A#4 (466.2 HZ)
B4 (493.8 HZ)
C4 (523.2 HZ)

Get each semitone, then each whole tone, then each minor third etc. You'll have 12 of each. But then there's also the beating between each beating tone since every semitone is not the same size in Hz.

This is assuming no harmonics. Playing with beating in complex signals can get silly complex. Is this for an electroacoustic composition or something like that?

[artifus]

http://www.soundonsound.com/sos/dec10/articles/celemony.htm

[earthtonesaudio]

I'm using Excel to generate a top octave wavetable, with the key feature that looping back from tn to t0 for any frequency does not produce a discontinuity.  So if f1 is a sine wave starting at zero at time t0, then at time tn it should be just below zero and increasing.  Same for f2, f3, etc.

   t0   t1   t2   t3   t4   t...   tn
f1                     
f2                     
f3                     
f4                     
f5                     
f6                     
f7                     
f8                     
f9                     
f10                     
f11                     
f12                     

[earthtonesaudio]

D'oh.  Realized that what I am really after is the period of a complex waveform, whose frequency components are f1, f2, ..., fn.

...And the period of any complex waveform is the least common multiple of the periods of its components, therefore:

T_{complex waveform}= LCM(T_f1, T_f2, ... T_fn).

[R.G.]

Note that the ratio between two adjacent semitones in the even tempered scale is the twelfth root of two.

But the beat between any two notes is the arithmetic difference between the tones. These are quite different, and also different for each pair of semitones compared.

[amptramp]

sin A + sin B = sin A+B + sin A-B

sin A * sin B = 0.5 (cos A-B - cos A+B)

If you are adding or multiplying these frequencies, use trig identities where angle is replaced by 2*pi*f and this is equal to A or B.  A perfect multiplier gives you sum and difference frequencies.  There are other trig identities for increasing the order of a multiplication.  In the second example above, you can see that if A = B, you have a DC component plus a signal at double the frequency of the incoming signal.  You can have fun with the stuff here:

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

The double and triple angle formulae and the power reduction formulae can provide chords.

[earthtonesaudio]

error: overflow. 

[Tony Forestiere]

sin A + sin B = sin A+B + sin A-B
sin A * sin B = 0.5 (cos A-B - cos A+
If you are adding or multiplying these frequencies, use trig identities where angle is replaced by 2*pi*f and this is equal to A or B.  A perfect multiplier gives you sum and difference frequencies.

I am going to step out on a limb here and show my ignorance (or the "aha" moment I foolishly think I just had).
Would a "perfect multiplier" be akin to a "perfect" ring modulator? If the carrier frequencies corresponded to the two fundamental frequencies being compared, the output frequency (sum and difference of/between the fundamentals) should be the beat frequency?

Thinking hurts. I hope I got it. 

[puretube]

sin A + sin B = sin A+B + sin A-B
sin A * sin B = 0.5 (cos A-B - cos A+
If you are adding or multiplying these frequencies, use trig identities where angle is replaced by 2*pi*f and this is equal to A or B.  A perfect multiplier gives you sum and difference frequencies.

I am going to step out on a limb here and show my ignorance (or the "aha" moment I foolishly think I just had).
Would a "perfect multiplier" be akin to a "perfect" ring modulator? If the carrier frequencies corresponded to the two fundamental frequencies being compared, the output frequency (sum and difference of/between the fundamentals) should be the beat frequency?

Thinking hurts. I hope I got it. 

You asked for it...
including this,
this,
this,
and this...

[amptramp]

sin A + sin B = sin A+B + sin A-B
sin A * sin B = 0.5 (cos A-B - cos A+
If you are adding or multiplying these frequencies, use trig identities where angle is replaced by 2*pi*f and this is equal to A or B.  A perfect multiplier gives you sum and difference frequencies.

I am going to step out on a limb here and show my ignorance (or the "aha" moment I foolishly think I just had).
Would a "perfect multiplier" be akin to a "perfect" ring modulator? If the carrier frequencies corresponded to the two fundamental frequencies being compared, the output frequency (sum and difference of/between the fundamentals) should be the beat frequency?

Thinking hurts. I hope I got it.  

Similar, but not quite the same.  The ring modulator takes in one analog channel A but uses the other channel B to switch its polarity at the output.  The analog input A is modulated by the switching which is no longer a single-frequency sine wave but a square wave.  The square wave contains the fundamental, 1/3 of the third harmonic, 1/5 of the fifth harmonic, 1/7 of the seventh harmonic etc.  The result is, you have the trig products of A and B, A and 3B, A and 5B etc. so you have A ± B, A ± 3B, A ± 5B, A ± 7B etc.  Some of these things are interesting musically but others may not be.  You can get some interesting effects with harmonics.  Everyone knows that if you double the frequency, you go up an octave, but not as many know that if you triple the frequency of a note, say A at 110 Hz, you get 330 Hz which is close but not quite equal to E 329.6 Hz on the evenly-tempered scale.  You then get subtle (and sometimes not so subtle) modulation caused by the difference frequency of 330 and 329.6 Hz, giving you a tremolo with a 0.4 Hz or 2.5 second period.  This would be noticeable if signal A was a chord with 110 Hz and some 329.6 Hz.

Take an Advil and keep reading:

If you watch muscal instruments on a scope, you can lock the trigger on one part of the waveform and see other parts running back and forth since most musical instruments do not have a true harmonic relation between each other.  One item of test data I am familiar with is to pluck an acoustic guitar string at 300 Hz.  You will have partials (slightly off-frequency harmonics) at 595.4 Hz, 897 Hz, 1198.1 Hz and 1500 Hz.  The detuning effect of delivering energy to the hole under the strings in an acoustic guitar is greatest for the partials that put maximum deflection over the hole.  The ratios are 1.00, 1.985, 2.99, 3.984 and 5.00 in this case.  If the partials were actual harmonics, the sound would be like a low-grade electronic organ with dividers for each note.  We all know how non-musical that sounds.  It is the slight detuning of the partials from true harmonic multiples that make an instrument come alive with subtle nuanced beat notes that give it character.

[Jazznoise]

AmpTramp +1 for the acoustics theory.

The harmonicity of a string does mess with all of this a little. A guitar is a very complex source with alot of partials, so I tend to find fundamentals and 1st harmonics the easiest to play off of with a Ring Modulator. If you use a scale closely matched to the Harmonci Series like a Just Intonation Lydian Dominant (An ET Lydian Dominant will do fine) you'll find the sidebands are often quite harmonic and generate melodies of their own.

Which makes you wonder how we heard the harmonic series and made scales out of it, and how exactly is our brain relating this information back to itself!