Basic Acoustical Concepts

Historically from the very beginning of any understanding of “Acoustics(the Science of Sound) they discovered a tuning problem.  Thru the ages we have adapted to it in various way, yet the problem cannot be solved.  The math does not allow it.  To me, this is the most interesting aspect of Acoustics,  but there are other aspects related to “Tone Quality” that also have interest to singers.

PITCH MEASUREMENTS

Let’s start with how we measure pitch acoustically.  Pitch is measured in two different ways - Frequencies  and in Cents..  (We’ll explain “Cents” later.)  Frequencies are measured in cycles per second (Hertz) and we usually tune to A=440 cycles per second.  What that means is, that if ANYTHING vibrates at a speed of 440 cycles per second, it will produce the pitch of A4 (meaning the 4th octave that we can hear). Since octaves double the frequencies, at A4=440 Hz, then A5=880 Hz,  and A3=220 Hz, etc.  Also A0=27.5 Hz which is about as low as we can distinguish a pitch rather than just hear a flapping sound.  Human hearing range is generally considered to be 20Hz to 20,000Hz - though it varies a great deal with age and other conditions.

THE HARMONIC SERIES

The Harmonic Series is a fundamental acoustical phenomena and is the foundation of two significant aspects of sound:  1) Harmony and 2) Tone Quality.  The Harmonic Series is based on the mathematical ratios of resonance.

We arbitrarily illustrate this below with the fundamental on G and only show it to the 8th partial though the harmonic series can theoretically be extended infinitely.  To simplify the understanding of the “frequencies”, we have chosen round numbers for ease of understanding but they are not far off from the actual frequencies.

The Harmonic Series can be built on any note (or frequency).  For instance, an harmonic series based on C would be. 1=C, 2=C, 3=G, 4=C. 5=E, 6=G, 7=Bb, and 8=C etc.

CHORDS

The distances (intervals) between partials are the basis of chords.  Chords can ring as the different partials reinforce each other.  The

partial numbers form mathematical ratios that can be used in calculating frequencies etc. This gives us Just Intonation(We’ll explain Just Intonation on the next page.)

1 to 2 = The Octave

2 to 3 = the Perfect 5th

3 to 4 = the Perfect 4th

4 to 5 = the Major 3rd

5 to 6 = the Minor 3rd

The most basic interval (aside from the Unison) is the Octave which is double the original note or the ratio of 1 to 2.  Next is the Perfect 5 with the ratio of 2 to 3.  The perfect 4th (P4th) is the inverse of the P5th so we consider them as equivalent. (To illustrate inversion: If we go up a P5th from C we get G but also if we go down a  P4th from C we also get a G. So a P4th is an inversion of a P5th.)   So for tuning, the P5th and the P4th are essentially the same.  Thus, the next basic interval is the Maj.3rd (ratio: 4 to 5)  The competition between these three intervals (Oct., P5th and Maj.3rd)  is where the tuning problem comes (as we will explain on the next page).  Right now we’re talking about the Harmonic Series.

Study the graphic and the ratio listings above until they makes sense.

Note that if you double the partial number or the frequency they are the same syllables (1,2,4,8 are all “Do”) or letter name (3 & 6 are both So’s).  Notice also that partials 4,5,6  make a major chord,  With the addition of the 7th partial (4,5,6,7) it make the “Dominant 7th” chord which in barbershop we call the “Barbershop 7th”.

We can use these ratios to calculate various frequencies in the Harmonic Series.  For instance Middle C4=261.626 Hz., so G4 would be 392.439 Hz - a little bit under A4=440 (261.626 X 3=784.878 / 2=392.439)

BEAT-LESS-NESS

Another very important concept is that of beats between frequencies and beatlessness. Beats are the pulsing of two things that are out of tune so they give sort of an “ah oo ah oo ah oo” sound at a speed dependent on their discrepancies.  When two things (such as strings or voices) produce the same frequency, we say they are beatless (totally in tune).  However if for instance, one string produces an A=440 and a second string produces an A=439 it will give us a beat of one cycle per second “ah oo ah oo ah oo” with the “ah’s” one second apart.  Notice that this use of the term “beat” is NOT what we mean when we talk about a quarter note getting one “beat”.  They’re two different concepts so don’t get them confused.  These frequency beats are what piano tuners use to tune a piano in Equal Temperament by making the P5ths beat at less than a half beat per second (aprox.) narrower than beatlessness (in the mid piano range).  Check it out with a very well tuned piano (or better yet - a synthesizer keyboard).  Count an “ah oo” for each 2 seconds on P5ths.

CENTS

An Equal Temperament half step is 1/12th  of an octave since there are 12 different notes between the Octave (Do, Di, Re, Ri, Mi, Fa, Fi, So, Si, La, Li, and Ti).  To calculate the half step you would take the 12th root of 2 which equals 1.05946309 aprox.  (Recall that the Octave is the ratio of 1 to 2 and we need to divide that into 12 equal parts).

To measure small frequency differences we divide the half step into 100 part called a cent.  So a cent is 1/100th of a half step.  That’s small!  The best ears under ideal conditions may be able to hear only about a 6 cents difference.

If you have any confusions or suggestions,

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