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At this point we have several options.  We could just be satisfied, as was Pythagoras, with this condition whereby the final fifth is tuned extremely flat while all others are perfect fifths.  Or we could fudge some or all of the intervals to some degree.  Since we have already considered these courses of action without finding any ideal way out of the mess, let us instead consider a third possibility – going around the circle once again (or perhaps even several times) – just to see if we might eventually end up exactly on the octave.

Unfortunately, this plan will never work.  As long as we insist the frequencies be in a strict 3:2 ratio, we'll never get the circle to close exactly.  In fact, as we introduce more and more tones describing increasingly complex ratios, they will begin to sound worse and worse.  The simpler the ratio, the better the sound, right??

The problem is simply that no power of 3/2 is ever a whole number of octaves (or indeed any integer multiple of the first frequency).  What we are looking for here can be expressed as 3/2 = 2p/q.  This should look familiar, as we have simply altered the coefficients from our previous equation because we are now attempting to obtain fifths instead of octaves.  Those who are not so handy at manipulating exponents using logarithms will have to take my word for it that the following demonstrates that 3/2 isn't equal to 2p/q for any whole numbers "p" and "q":

3/2 = 2p/q

log 3/2 = (p/q) log 2

log 3 - log 2 = (p/q) log 2

(log 3 - log 2)/log 2 = p/q

log 3/log 2 = (p/q) + 1

but log 3/log 2 = 1.584962501... (irrational)

The bad news is that the above proof indicates the ratio we desire is irrational, and therefore cannot be described by use of any rational number, no matter how complex.  Thus we can never precisely fit a circle of fifths into an even number of octaves.  The good news is that we are now in a position to demonstrate how it may be determined that using twelve intervals per octave is indeed the ideal approach for our situation.

The task at hand requires us to find a rational number that is roughly equal to 3/2 (since it can't be exact) so that some small power of this ratio is also a power of 2.  In other words, we must find a good fractional approximation for the irrational number log 3/log 2 derived above.  Mathematicians use a process called "continued fractions" for this.

The technique of "continued fractions" promises to produce a series of all fractions approximating our real number, with the property that each successive fraction come closer to approximating that real number, and is a better approximation than any other fraction you might try with a smaller denominator.  The theory of continued fractions requires us to form the continued fraction expansion of this real number, stopping at certain points to re-evaluate the fraction which will approximate said real number (log3/log2 in this case).  There is a lot known about this process (just not by me!!), but only one other fact is useful here: the approximations are unreasonably good if and only if we stop just before a big term shows up in the continuous fraction expansion series.

So here is the series expansion: log3/log2 => [1,1,1,2,2,3,1,5,2,23...].  I would have no clue how to use this, but apparently it gives the following optimal approximations: 1/1, 2/1, 3/2, 8/5, 19/12, 65/41, 84/53, 485/306, 1054/667.  We stop here because the next term is relatively large.  The next best approximation would have an extremely huge denominator (as if 667 isn't large enough!!).  And check it out for yourself...each of these fractions does come closer to approximating 3/2 than the one before it does!!

Musically, this tells us that when building pure fifths we get closer and closer to the exact octave by increasing the number of tones per octave used, basing our choice on the quantities suggested by moving further up the denominator values.  This can be represented as the series: 1, 2, 5, 12, 41, 53, etc.  And we are assured that other numbers of tones-per-octave offer no particular advantage.

Notice that the 5-tone pentatonic scale is represented here, accounting for its widespread use in the east.  In breaking the octave up into 5 tones, a span of three of these intervals yields a pretty good fifth (that is, 23/5 = 1.516 is fairly close to the ideal 1.5).

More importantly, notice also that it would require either 41 or 53 tones per octave to improve on our present system of 12!!  In the first case, the 24th tone would have a relative frequency of 1.50041943 -- which is really quite close to a true fifth.  Relative to an A440, the true and false fifths would produce a beat frequency of something like 12 seconds.

But that's a lot of keys!!  Worse yet, can you imagine having 53 x 7 = 371 keys on a piano??  Interestingly enough, it has been reported that "two harmoniums with 53 notes to the octave were built" in the mid-19th century, prior to when Bach made well-tempered tuning famous.  They must have been a keyboard tuner's nightmare!!  Apparently, the mechanisms were such that playing certain keys (the upper ones) forced various side keys to come down as well.

One can imagine that this might work very well for harmony, but the 21/53 intervals, which would mean tones that were ≈1.013164 Hz apart, would be far too small to be distinguishable when used for playing a melody.

Moving on though, let's look at how close the fifth is to ideal in our system.  The seventh tone of our scale will actually make a pretty good fifth.  So we can verify how close the result 7/12 is to the preferred irrational number log 3/log 2 as follows:

log 3/log 2 = 1.584962501... = (p/q) + 1

p/q = (log 3/log 2) - 1 = 0.584962501...

but also p/q = 7/12 = 0.5833333...

Now that we know fifth consists of seven steps on what we have decided will be a 12-interval-per-octave keyboard, we can calculate the interval of our fifth as: 27/12 ≈ 1.4983

This result is pretty close to the ideal 1.5, and indeed the difference will not be noticed by any but the most astute of musicians.  In fact, we will soon discover that our list of equally tempered values includes fairly close approximations to quite a few simple fractional ratios.  By extension, this means that the traditional western arrangement actually contains a reasonably good number of the more pleasant sounding note combinations.

The development of musical keys and chords will be discussed in the final section of this paper, but first it might be interesting to consider for a moment how other musical systems not based on the fifth would look.

Exploring Our Options
 

If you wanted to include a different number of tones per octave than those mentioned above, you would then either need to use a multiple of these numbers or sacrifice the quality of your fifth (perhaps with the intent of improving another interval, such as a fourth).  The former possibility is really rather tame – it would simply mean interleaving more frequencies into a pentatonic or 12-interval scale.  You could keep existing repertoire and mix in some additional "in-between" tones.  The latter possibility, however, would be an interesting departure from our standard western practice.  It is even possible to attempt a uniform optimization of several intervals, although I won't discuss this here as it involves many additional subjective choices.

The next higher non-integer ratio above 3/2 is 4/3.  But since 4/3 x 3/2 = 2/1, an octave, the scales which provide good fourths are the same as those providing good fifths.  To get a new variant on western music we would have to seek scales allowing some other nice ratio of frequencies.

The next simplest ratios are 5:3 and 5:4.  Emphasizing the 5:3 ratio are 3-tone, 4-tone, and 19-tone scales (when speaking of tones instead of intervals, the final repeated octave note is not counted).  With 19 notes, a wonderful approximation to 5:3 occurs at the 14th tone.  As a bonus, 11 tones up from the root on this scale would make a pretty good fifth (211/19 = 1.494).  To get the next great approximation would require 418 or more tones per octave!!  As for the 5:4 ratio, it would require 3-tone, 28-tone, or 59-tone octaves.  Not much to choose from here.

Subsequent ratios (7:4, 7:5, 8:5, 9:5, 7:6, 11:6, 8:7, 9:7, etc.) seem less likely to be useful as fundamental ratios in a musical composition, as they tend to be harder to hear and more likely to be dissonant.

Much of this theory is based on the assumption that among the ratios of frequencies available will be the 2:1 octave.  If we relax this assumption, we can get sets of tones from which other (obviously more unusual) music can be constructed.  The mathematical theory works just as well with any pair of ratios one wants to approximate.

For example, we could ask that the set of frequencies repeat with factors of three.  In this case, if A440 is included, we replace the octave with a 3:1 interval of 440 to 1320.  The continued fractions method then dictates that we try dividing the scale into 3, 7, 17, or 58 tones.  If we were curious as to how close we might come to also achieving a 2:1 interval within this set, we would find that if the 3:1 "octave" were divided into 17 steps, step number 12 is a good approximation of an A880.

Replacing the 2n octave will cause difficulties when it comes to transposing music, however.   This process requires that if an interval of frequency ratio "r" turns up, then an interval of ratio "r times r" must occur also.  This wouldn't always be the case using a 3:1 or other arrangement.

At the outset of this paper we arrived at our desired keyboard layout rather quickly.  This was due to having assumed a limited set of rather basic criteria.  Among these was our observation regarding how we perceive octave tones, and hear exponential pitch changes linearly, adding in our desire to limit the range of keys to a reasonable number.  However, as this section illustrates, the many restrictions set down in arriving at our arrangement also means a decrease of flexibility that might otherwise have afforded some very interesting and unique compositions!!

This page was last updated on February 12, 2023

Always remember to "Think Green" because good planets are hard to find!!  




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