Beat (acoustics)/Amplitude beats





Prerequisite skills for this lesson
This is an advance resource that requires proficiency in a wide variety of subjects. The following terms are fully discussed in these Wikipedia articles:
 * Beat usually refers to an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies. Here we extend this definition to include the case where the difference in pitches is not small, but instead have a ratio of frequencies differs slightly from that of just intonation.
 * An Interval is a difference in pitch between two sounds.
 * Just intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies.
 * A Harmonic is a wave with a frequency that is a positive integer multiple of the fundamental frequency, the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the 1st harmonic, the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a harmonic series.
 * A Fourier series: is a summation of harmonically related sinusoidal functions, also known as components or harmonics.

Lots and Stone suggest that the mechanism by which beats can be heard between two notes of a consonant interval is not fully understood. In fact, beats associated with musical notes actually sung or played on an instrument exhibit complexities that render them difficult to mathematically model. A commonly used simplification called Fourier analysis has permitted some progress to be made. On this page we introduce Fourier analysis and explain how it can explain beats between consonant musical intervals using an argument similar to that proposed by Helmholtz in 1877. The mathematical justification for this Helmholtz model for beats is based on the well known and easily understood amplitude beats that occur between two sinusoidal waves of nearly equal frequency.

Derivation of amplitude beats among harmonics
Let p and q be relatively prime integers, and suppose one of the frequencies is slightly detuned, for example, with $$f_p=pf_0$$ and $$f_q\approx qf_0$$. Let the integers, $$(m, n)$$ a matching pair of harmonics with nearly equal frequency, so that $$mf_p\approx nf_q.$$ Since our goal is to find the set of all such nearly matching pairs $$mf_p\approx nf_q\approx gf_0$$. In the limit of very slow beats, we may drop the inequalities, so that dividing by $$f_0$$ yields: $$mp=nq=g.$$ The most obvious such match occurs when,  $$m=q\quad\&\quad n=p$$. Define $$g_1=pq$$ as the most lowest patching pair (assuming $$p$$ and $$q$$ are relatively prime.) This yields:


 * $$\frac p q = \frac n m\quad\Longrightarrow\quad n=iq\quad\&\quad m=ip\quad $$ (for $$i=1,2,3,\ldots$$)

The beating is between $$ipf_q $$ and $$iqf_p.$$ Now define $$f_p^\prime$$ to be a slightly from $$f_p=iqf_p.$$  Use the fact that $$\Delta (qf_p^\prime )=q\Delta f_p^\prime$$ to obtain $$f_B=$$ To first order, we  can combine both effects if $$f_q$$ is also detuned (note that when both notes in an exactly tuned interval are detuned by the same factor, the interval is still exactly tuned.) This explains the minus sign in:

${}^i\!f_B= i\cdot\left

Why these intervals were selected


Here we examine plots for three just intervals: • perfect fifth ($6/5$ ratio) • minor sixth ($5/4$ ratio) • tritone ($4/3$ ratio)

These intervals were selected because all are within a half-step of the perfect fifth, which might facilitate comparison of graphical representation of intervals with entirely different natures. Figure 4 establishes the fifth (3:2) as the most fundamental consonant interval between the unison and the octave. The figure is taken from a proof of the countability of rational numbers, modified to include only fractions between $3/2$ and $8/5$, with ratios that are not prime written in red. The smaller integers tend to occupy the upper left hand corner, where ratios associated with consonant intervals can be found.

Hearing the beats
All files play the beat at 90 beats per minute and are arranged in 3/4 time:
 * 1) Four bars with no metronome: Listen for the beats
 * 2) Four bars with metronome. Think: click-2-3 | 2-2-3 | 3-2-3 | 4-2-3
 * 3) Four bars with no metronome. Count: rest-2-3 | 2-2-3 | 3-2-3 | 4-2-3
 * 4) One bar with metronome. Listen: click-2-3

The challenge occurs at step 3: Try to count the beats as if they were a four measure rest in a waltz. If you are the clicker, you would come on the first beat of the fifth bar after that rest. After a while you might be able to start counting at the beginning. Think of yourself as a clicker, who has four bars of rest, four bars of clicks, four bars of rest, and one bar of clicks.