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Subject: Re: Musical instrument tuners (sound frequency comparison)

Date: 09/05/00 at 8:02 PM
Posted by: John Dunn - Consultant
E-mail: ambertec@ieee.org
Message Posted:

In Reply to: Re: Musical instrument tuners (sound frequency comparison) posted by Andrew Freer on 09/01/00 at 10:57 AM:

Hi, Andrew.

I'm not a professional musician, but as I understand the topic of musical tuning, things kind of go like this:

Western music is based on a twelve tone scale. In any scale, the interval between two adjacent notes is called a half tone while the interval between any two notes separated from each other by a third note in the middle is called a whole tone. Therefore, imagining a piano keyboard in your mind, the interval from C to C-sharp is a half tone while the interval from C to D is a whole tone.

In every scale, there is a note called the "root note" which we refer to as "Do" as in Do-Re-Mi-Fa-Sol-La-Ti-Do.

For a major scale, the interval from Do to Re is a whole tone, from Re to Mi is a whole tone, From Mi to Fa is a half tone, from Fa to Sol is a whole tone, from Sol to La is a whole tone, from La to Ti is a whole tone and, finally, from Ti to Do is a half tone.

The final "Do" in this sequence is the root tone again, but it is said to be one octave above the first "Do". Now, hold that thought!

The Greek mathematician Pythagoras, being one very astute fellow indeed, figured out that two tones sounding at the same time are especially pleasant to listen to when the frequencies of the tones are in particular ratios to each other. He didn't have access to a frequency counter, but he inferred the ratios from relative lengths of vibrating strings.

The two most pleasing ratios are when the two frequencies are in the ratio of 1.25:1 and 1.5:1. By today's notation, the nominal ratio of the frequency of "Mi" to "Do" is 1.25:1 and the nominal ratio of the frequency of "Sol" to "Do" is 1.5:1. Note that I said the "nominal ratio" and not simply the ratio. This is very, very important, so hold this thought too!

When a scale's "Do" happens to be the note C, the scale for that "Do" is in the key of C. Likewise, when a scale's "Do" happens to be the note G, the scale for that "Do" is in the key of G. The name of the note corresponding to "Do" is the name of the key.

The interval from "Do" to "Sol" in any key is called a "fifth". The interval from C to G in the key of C is a fifth. The "Sol" of the key of C then becomes the "Do" of the key of G.

In the exact same fashion, in the key of G, the interval from G to D is a fifth which means that D is the Sol of the key of G and that D becomes the "Do" of the key of D.

This process goes on and on in a relationship called the "circle of fifths". Ideally, after you do this process twelve times, the frequency of your last derived "Do" should be 128 times the frequency of the "Do" you began with, but there is a problem!

If you keep multiplying by 1.5 to get the "Sol" for each "Do", you wind up, after twelve multiplications, at the number 129.746 instead of 128. The error is actually cumulative as you work your way up through the octaves. The progression of values you get is, starting from a normalized frequency value of unity: 1.000, 1.500, 2.250, 3.375, 5.063, 7.594, 11.391, 17.086, 25.629, 38.443, 57.665, 86.498 and finally 129.746.

Violinists like to tune their instruments to perfect pitch, i.e., using the 1.5 factor. However, a violin doesn't cover many octaves in its frequency range, so Pythagoras' comma doesn't cause a problem. A piano on the other hand, has an enormous frequency range and cannot be tuned with the factor of 1.5 because it will sound terrible!

For pianos, a tempered scale must be used. A tempered scale is a scale in which the frequency of any note is the frequency of the note a just below, multiplied by the twelfth root of two, or 1.059463094....

Kind of nasty looking, isn't it!

The perfect fifth numbers which were:

1.000, 1.500, 2.250, 3.375, 5.063, 7.594, 11.391, 17.086, 25.629, 38.443, 57.665, 86.498, 129.746.

1.000, 1.498, 2.245, 3.364, 5.048, 7.551, 11.314, 16.951, 25.398, 38.055, 57.017, 85.430, 128.000.

Tuned this way, a piano sounds good and a violin sounds not quite as good as it could sound alone. However, when a pianist and a violinist play together, that is the compromise!

When you do your tuner project, I suggest that you address yourself to the tempered scale. The ratio between notes is a constant and that will make things easier.

Just be aware that the tempered scale isn't the be-all end-all of musical tuning.

Good luck.

John Dunn - President
Ambertec, Inc.
ambertec@ieee.org

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