Alternative Tuning and How We Hear
Alternative Tuning and How We Hear
There is a trade-off in this scheme. None of the intervals is precisely correct (except octaves). Fifths are slightly off-tune. Thirds are actually quite a bit off-tune.
The rules we want to follow are:
Just to check that it works, let's try comparing two different major third intervals.
The ratio of the frequencies of C and E should be the same as the ratio of the frequencies of E and G#.
A notation system based on this idea is described by David Canright.
If you are planning to experiment with alternate tunings using a synthesizer, you are likely to find that the synthesizer pitch adjustment is expressed in cents.
Another possibility is that you will need to control pitch bending to retune notes.
MIDI pitch bend messages have 14 bits (16384 unique values) that control bending of +/- 2 chromatic steps (400 cents).
This means you have control down to less than 1/40 of a cent - more than enough!
Another way to view the equal tempered scale is to think of the Pythagorean comma being distributed evenly across all the pitches; each pitch is slightly wrong, but the intervals add up consistently.
Thirds are even worse.
Three major thirds should equal an octave.
The out-of-tune major third is probably the worst feature of the 12-tone scale.
But we are immersed in music tuned this way and our ears have become trained to accept it.
A just tuned major third sounds slightly wrong at first, but after just a little while, it sounds great.
Returning to the equal tempered major third then sounds rather jarring.
This document is ©2003 by Bill Grundmann.
note frequency (Hz) pitch (cents) B 493.883 1100 A# Bb 466.164 1000 A 440.000 0900 G# Ab 415.305 0800 G 391.995 0700 F# Gb 369.994 0600 F 349.228 0500 E 329.628 0400 D# Eb 311.127 0300 D 293.665 0200 C# Db 277.183 0100 C 261.626 0000
freq(A#) = freq(A) x R
freq(B) = freq(A#) x R
freq(C) = freq(B) x R
freq(C#) = freq(C) x R
freq(D) = freq(C#) x R
freq(D#) = freq(D) x R
freq(E) = freq(D#) x R
freq(F) = freq(E) x R
freq(F#) = freq(F) x R
freq(G) = freq(F#) x R
freq(G#) = freq(G) x R
freq(A') = freq(G#) x R
or by combining all of them:
2 x freq(A) = freq(A) x R x R x R x R x R x R x R x R x R x R x R x R
this can be rewritten in exponential form:
2 x freq(A) = freq(A) x R^12
and we can solve for R:
R^12 = 2 x freq(A) / freq(A)
R^12 = 2
R = 2^(1/12)
In other words, the ratio is the twelfth-root of two.
The note A above middle-C is defined to be 440 Hz.
From that, all the other pitches can be computed.
329.628 / 261.626 = 1.260
415.305 / 329.628 = 1.260
It works!
MAPPING THE PIANO KEYBOARD TO FREQUENCY
We can define the value n as the number of chromatic steps above A.
Negative numbers would mean the number of steps below A.
This equation lets you compute the frequency of any note:
freq(n) = 440 x 2^(n/12)
MAPPING FREQUENCY TO THE PIANO KEYBOARD
By solving the previous equation we obtain the equation that lets you compute the note from a frequency:
n = 12 * log(freq(n) / 440) / log(2)
LOGARITHMIC VIEW
It might be argued that human hearing perceives pitch on a logarithmic scale.
The same interval in any octave is in some sense the same size.
If we plot the frequencies of a scale on a log graph, we do indeed find that equal intervals appear the same size on the graph.
This is because division is like subtraction in the logarithmic mapping.
log(A/B) = log(A) - log(B)
The log of the frequency is plotted on the vertical axis in the following diagram.
The equal tempered notes are shown and you can see that the vertical distance for any interval will always be the same.
The just tuned notes are also shown.
CENTS
A single chromatic step can be divided into 100 smaller steps called cents.
This is a convenient unit to describe alternate tunings in reference to equal tempered tuning.
One cent is just barely beyond the perception of the best trained musical ear.
COMMA
The comma is the error, or discrepancy, between two different ways of creating the same interval.
The Pythagorean comma points out the problem that perfect fifths don't fit exactly into 7/12 of an octave.
Imagine you step through 12 fifths, each with exactly a 3/2 ratio:
1x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)
= 531441/4096
This ratio should be the same as 7 octaves.
So we compute stepping through 7 octaves, each with exactly a 2/1 ratio:
1x(2/1)x(2/1)x(2/1)x(2/1)x(2/1)x(2/1)x(2/1)
= 128/1
We can rewrite the ratio with the same denominator for easy comparison.
128/1 = 524288/4096 (not the same as 531441/4096)
The discrepancy of about 1.36% is the Pythagorean comma.
This points out that just tuning has a problem with its intervals adding up correctly.
1x(5/4)x(5/4)x(5/4) = 125/64
One octave written with same denominator would be
1x(2/1) = 128/64
The major third comma is about 2.4% - very noticeable even to the untrained ear.
Again, we can view the equal tempered scale as having the major third comma being distributed evenly across all the pitches; each pitch is a bit off, but the intervals add up consistently.
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