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In Search Of A Perfect Fifth

We have now developed a simple method of determining appropriate frequencies for each note on our keyboard.  We are all set to compose an entire repertoire of masterpieces, no??  Well, almost.  The above example takes for granted that we will divide the octave into 12 intervals, which is true of our western system today.  But we were creating our system from scratch, if you recall, and we still haven't learned of any particular reason to choose this number.  So we must still determine how many intervals to place between the octaves of our imaginary keyboard.  To do that, we need to first take a bit of a detour to learn something about which notes will work well together.

For example, if you try playing any two adjacent keys of a piano at the same time, you will find the resulting sound is none too pleasant.  So which notes do sound good when played together??  The ancient Greeks decided it was those that are integer multiples of the lowest tone played.  This includes the octave, of course, but also many other multiples such as those with a 3:1, 5:1, 6:1, etc. frequency ratio.

There is actually good reason for this.  Most sound sources do not produce a pure sine wave of a single frequency, but rather a composite tone made up of various harmonics (or overtones).  Besides having a characteristic envelope (attack, sustain, and decay time), the unique combination of overtones present within each sound (including whether they are predominantly even or odd, for example), along with their associated relative amplitudes, gives each particular sound source its own "fingerprint" – a distinctive tonal quality, or timbre – thus separating a gunshot from a girl's voice or a dog from a bird, for instance.  One can easily see, then, that since this condition is prevalent in nature, why these simple multiples of the root frequency will sound pleasing when they are played together on an instrument.

Forget for a moment our previously established frequency designations, and let us instead build up a keyboard using another approach.  Assume a starting tone of 100 Hz.  If we put together a set of tones that includes all octaves and whole integer overtones of that initial value, we will then have the following notes: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, etc.  The reader may notice that when playing a composition using these notes, one will at some point inevitably include two tones in the sequence which are integer multiples of the first tone (100 Hz), but not of each other.  The most audible of these will be the lowest two, 200 Hz and 300 Hz, which are each whole number multiples of 100 but are related to each other by a 3:2 ratio.  This interval is known as a fifth, for reasons that will become apparent later, and it is quite pleasant sounding.   Indeed, this and other similar simple non-integer intervals (such as 4:3 and 5:4) are also compatible musically, and are typically present as harmonics on, say, a plucked string.

As it happens, not only is 3:2 the simplest non-integer ratio, having the smallest possible values for both numerator and denominator, but it is generally considered to be the best sounding.  This is because, as a rule, the larger the numbers necessary to describe the ratio, the more dissonant the sound.  For instance, we would find that two tones separated by a ratio of 243:179 would be quite harsh.

Therefore, due to the extreme sweetness of this interval, it would seem to be very desirable to add as many fifths to the line-up as possible.  One way to accomplish this is by creating what is known as a "circle of fifths", and as an added bonus we will find that doing so will also help us decide how many tones to include between each octave in our hypothetical keyboard.

In building our circle of fifths, we will restrict ourselves to those tones that occur within a single octave – calculating each fifth by multiplying the previous tone by 3/2, starting with the original.  However, in order to stay within a single octave, we will divide by two whenever we reach a frequency that is outside our desired range.  We can do this because the note we are dividing down would actually be an octave of the calculated lower frequency when we expand our keyboard later to include additional range beyond just one octave.

For example, consider an actual octave found on the piano, say the one between 110 Hz and 220 Hz.  The first fifth occurs at 165 Hz (1.5 x 110, which we keep); but the next one, at 247.5 Hz (1.5 x 165), is too high.  So we divide this by two to obtain 123.75 Hz, and add this to our collection.  One fifth above that is 185.625 Hz, so this becomes one of our tones.  The next fifth falls in at 278.4375 – again out of range.  One-half of that value is 139.21875 (another keeper).

If we continue in this fashion, taking fifths upon fifths, and dividing by two whenever the result falls outside the intended octave range, we will eventually come upon a note that is very near the original tone at some point.   This occurs after dividing by two the frequency that is fairly close to an octave (220 Hz) at the twelfth step of the process.  The circle is thus closed, as we have arrived back at the starting frequency – or at least close to it.  The results of this exercise are provided below:

110 → 165.00 → 123.75 → 185.625 → 139.21875 → 208.828125 → ≈156.621 →

≈111.5 ← ≈148.67 ← ≈198.2236 ← ≈132.15 ← ≈176.2 ← ≈117.4658 ←

This method provides us with an obvious suggestion as to the appropriate number of intervals we should divide our octave into if we want to include as many fifths as possible.  The results may be represented as shown in the figure (we will talk about the 3rds later), where our modern western letter assignments for each tone are shown instead of frequencies.  Each adjacent letter as one travels clockwise around the circle represents a musical fifth.

But wait a minute...we have also learned that there are other ratios (such as 4:3 and 5:4) that have a pleasing sound as well.  What about them??  More importantly, the reader may notice that most of these frequencies are not the same as we had calculated previously.  If this process actually provides intervals that produce superior tonal quality, why do we use the frequencies we do (i.e. equal temperament) instead??

The answer to both questions will be discovered as we proceed to review the evolution of temperament through the years towards our present configuration.

Do You Have A Bad Temperament??

It turns out that the path followed by history in designating the tunings for instruments over the years pointed in the opposite direction from that which we took.  We derived our original values by taking a mathematical approach, basing it on our discovery of how the human system handles intervals such as the octave.  We then began to investigate how we might add pleasant sounding intervals to the mix.  However, the development of our current western arrangement actually began from what our predecessors felt sounded good, and only recently went kicking and screaming into the realm of equal temperament.

The truth is that most early musicians were masters at math, and would have had no problem at all in soon arriving at the correct formula for making our current note assignments once the number of tones per octave was determined.  And the circle of fifths did a decent job of pointing this number out.  Furthermore, as discussed in the next section, there is an even more elaborate mathematical technique that can be used to verify without any doubt what the most appropriate number should be, based on whether we want to build our system on perfect fifths or on some other interval.  But history had another idea in mind.  Mysticism and religion, it as it happens, played an overriding role in establishing the early scales.

The details of these scales will be covered later, but suffice it to say for now that the harmonic series of 2:1, 3:2, 4:3 (the first three harmonics of a plucked string) was considered holy due to the 1–2–3 sequence.  Adding fuel to the fire, it was also noted that the latter two fit perfectly within the octave, as 3/2 X 4/3 = 2.  This reverential attitude to mathematics and harmony (both words being derived from the same Greek root) persisted well into the 18th century – and is still embraced in some circles today.  The French Academie at Notre Dame held as late as the 14th century that only perfect fifths could be used to construct a scale because the three in the 3:2 ratio represented the trinity!!

As a matter of fact, because they were based on natural harmonics, these early systems consisting only of pure intervals or tones (as opposed to those that have been adjusted, or "tempered") actually offered something that our system doesn't.  The sweetness of these original scales simply cannot be approached in equal temperament, where this absolute purity has been compromised.

That being the case, then, why tamper (or perhaps I should say "temper") with perfection at all??

Well, we have one huge problem with our circle of fifths.  Remember that the final tone came out close to, but not exactly equal to the initial one??  The human ear, it seems, is very sensitive to the smallest discrepancies when it comes to octave tones.  If we left the final octave value derived above at ≈223 Hz instead of rounding it down to 220, the resulting combination when both octave tones were played together, would be quite noticeable.  And the error would continue to grow as we expanded or keyboard to include higher octaves until eventually the overtone wouldn't even sound like an octave.  Therefore, the last tone must be held at the correct octave frequency of 220 Hz.

This means that the last fifth (just below the final octave tone in our example above) would actually be tuned shy of a true fifth.  Since they're supposed to be a fifth apart, these frequencies should hold to a ratio of 3:2.  But if tuned this way, the last one wouldn't; the ratio would work out to about 2.96:2 instead.  That is, the final tones would be a little too close, so that the combination would sound flat.  This doesn't appear to be a big deal on the surface, but it actually does present quite a dilemma.

The difference between the final interval obtained using the circle of fifths and a true fifth became known as the Pythagorean Comma, after the philosopher/mathematician who first pointed it out.  And how it might be best dealt with would become the source of much consternation and manipulation throughout several millennia!!  Only when eventually forced into it by the failure of any other means (and believe me, many were tried) to mitigate the enormous conflicts presented by the Pythagorean Comma, what many consider to be a huge compromise – equal temperament – was finally adopted to resolve the problem.

The following discussion attempts to point out not only how this tempered fifth was such a problem to composers, but will also provide further insight as to why the change to equal temperament was resisted so stubbornly.

You see, following along the series of partials present when a string is plucked (i.e. 2:1, 3:2, 4:3, 5:4, etc), one might notice that each successive interval decreases in size.  The progressively decreasing size of the intervals between these partials insures that when two notes are played together, somewhere in their harmonics will be a frequency that is common to both.  This is important because it is by these shared higher overtones that the tonal natures or colors of intervals are formed.  If these intervals are de-tuned in order to incorporate some of the Pythagorean Comma (as they are in equal temperament), the intervals will then actually be defined by more complex rational numbers, and the associated color becomes distorted – or perhaps lost completely.

This happens not only because the common overtones are no longer common, but also in part because another very important acoustical phenomenon occurs as well.  Consider the just (i.e. pure) fifth with fundamental tones at 100 and 150 Hz.  If we decide to lower the fundamental of the upper note to 148 Hz, its fourth partial now becomes 592 Hz instead of 600 Hz.  When combined with the 600 Hz overtone of the 100 Hz note, which used to be a common harmonic, these two partials are now mismatched and produce a wavering vibrato effect (called a beat frequency) at 600 - 592 = 8 cycles per second.  This beating (or the lack thereof), is another defining characteristic of the interval's musical nature.

When the interval is pure or just, there is no beating, and the general feeling is one of harmony.  But as the notes are tuned further apart, the beating speeds up, and the feeling evolves into something more sinuous.  When the beating becomes too fast, the interval is perceived as being totally out of tune...and generally quite displeasing.

The earliest composers, however, preferring not to compromise tonal color and purity, or the sanctity of the harmonics, left the entire comma to fall between one single terribly out-of-tune interval, rather than splitting it up and adding a portion to each fifth.  Lore has it that the term "wolf interval" originated in medieval Europe, and was coined because it was said that the horrible intervals encountered in ancient tuning schemes often evoked the howling of wolf packs.  Their solution, then, was simply to avoid using these wolf intervals.  Because of this, they were typically limited as to which key signatures they could use, and by extension were very much restricted in their capability to transpose music or to modulate (i.e. change keys mid-piece) freely.

The reader may see how this puts us between a rock and a hard place.  On the one hand, there is the encumbrance of having to perpetually avoid the wolf intervals and accept the severe constraints in modulation.  On the other hand, a system full of compromises in both sound quality and spiritual beliefs is offered as the alternative.  And the latter option also meant compromising the ideal of including as many perfect fifths as possible within the octave scale.  Instead each would be tempered so that it was close to – but not exactly equal to – a 3:2 ratio.   In addition, what about the other simple-integer ratios??  It turns out that our equal-tempered thirds, for example, also sound much worse than those in many prior tunings.  To top it all off, along with the purity of fifths and thirds which is lost with equal temperament, what has been described as distinct differences in color between each key signature would have to be given up as well.

It should now be apparent why the promised benefits of equal temperament did not appear for many years to be sufficient reason for our ancestors to give up the purity of the remaining fifths provided by their existing arrangements.  Especially when the limited technology of the time prevented precision tuning, and tended to cover over less obvious cases of dissonance when using certain portions of the wolf interval.  Considering the numerous concessions musicians unwillingly, but inevitably, had to make, it is understandable that mankind spent many years exploring all other available options before eventually being forced into acceptance of equal temperament.

We will follow that trail in an upcoming section.  But for the present, please permit me to mention a couple of definitions pertaining to temperament.  First, it should be clarified that a "tuning" is laid out with nothing but pure intervals, leaving the comma to fall as it must, whereas a "temperament" involves deliberate de-tuning of some intervals to obtain a distribution of the comma.  Temperaments can be grouped into two classes: "regular" and "irregular".  In the former, all fifths but the wolf fifth are tempered the same way, while the quality of the fifths around the circle changes in irregular temperaments – generally so as to make the more commonly-used keys more consonant.

Temperaments may further be classified as circulating (closed) if they allow unlimited modulation (equal temperament and most irregular temperaments), or non-circulating (open) if they do not (tunings and most regular temperaments).

 

This page was last updated on  February 12, 2023

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