How did tetrachords become whole and half steps?

Question

How did the ancient Greek concept of tetrachords evolve into the whole- and half-step model familiar today?

Answer 1

I'll try to summarize the main bits of this development.

First, let's note that our modern idea there were only three types of tetrachord (diatonic, chromatic, enharmonic) is oversimplified and really more based on the 16th-century revitalization and reinterpretation of ancient Greek practices from a few specific treatises. In actual Greek music theory, there are literally dozens of possible divisions of tetrachords. Yes, some treatises classified various tunings into different categories, but I think it's important to recognize just how flexible tetrachord tunings could be.

How many tunings were actually used in musical practice is unclear, but the essential concept of a tetrachord for the ancient Greeks seemed to be to take a perfect fourth (4:3 ratio), then build down two intervals from the top note, and then generally end up with a "leftover bit" on the bottom of the interval that was typically a dissonance and usually had a much more complicated mathematical ratio. The first "step" down from the top note could be anything from a whole tone (9:8 ratio) to a pure major third (5:4 ratio) or lots of other options.

(Note: these ratios were conceptualized by the Greeks typically as ratios of string lengths. For example, if you have two strengths of the same diameter, under the same tension, and one is 4 units long while the other is 3 units long, they'd sound a 4:3 perfect fourth musically.)

So where did whole tones come from?

The first issue the Greeks had was to divide the octave, the simplest ratio (2:1). The Pythagoreans discovered that 3:2 perfect fifths and 4:3 perfect fourths sounded good as consonances too. So, if we divide an octave, we could put the 5th on top and the 4th on bottom. Or we could put the 4th on top and the fifth on bottom.

Imagine an E to E octave. (I'm just going to assume modern pitch letter names rather than dealing with Greek nomenclature for notes.) In the first division, we'd end up with E-A-E (going down -- the Greeks always liked to think of scalar divisions going down). In the second division, with 4th on top, we'd get E-B-E.

If we combine these two divisions, we get an E-B-A-E division of the octave. And the ratio of the two middle notes (B to A) is 9:8, a whole tone.

This way of breaking up an octave is found in the earliest known Pythagorean sources, like Philolaus. That's essentially the four-note "skeleton" we're building a division of the octave on. And theoretically, you could fill out the two perfect fourths there (E to B and A to E) with tetrachords in any manner as I described above.

However, very early on, the Pythagoreans also decided that it might be good to just utilize that 9:8 whole tone to build other notes of the scale and fill in the tetrachords. If we build down two 9:8 whole tones from each top note (E-D-C and A-G-F), we now get a full octave scale: E-D-C-B-A-G-F-E.

That later became known as a "diatonic" scale, where "dia-tonic" literally means "through [whole] tones." The C-B and F-E intervals were the "leftover bits" which in this case have mathematical ratios 256:243. Again, while the theoretical treatises would note such ratios eventually, the Greeks seemed often to care more about tuning the two middle notes in the tetrachord in relation to the top note or each other, while leaving that bottom "leftover" bit to be some random ratio.

Fairly early on in Greek music theory, they noticed that this 256:243 ratio was rather close to half of the 9:8 whole tone in size. Not exactly, of course. The first is about 90 cents in size, while the 9:8 ratio is about 204 cents. But it's close to half.

Aristoxenus was one early Greek theorist who then essentially wondered if these small tuning differences always mattered in musical practice. Different performers were known to tune the tetrachords slightly differently in practice, and Aristoxenus considered these disagreements to be simply a matter of taste perhaps, rather than an argument that one was definitely right or wrong. And what if the results were so close that one could barely tell the difference by ear?

By extension, what if we just assumed that the "bottom leftover bit" in the diatonic tetrachord was really "about half" of the other whole tones?

Well, in that case, each tetrachord would be roughly 5 "leftover bits." (That is, two whole tones plus that leftover bit = 2+2+1 = 5.) So two tetrachords plus the middle whole tone (B-A in my scale above, which was equal to about 2 "leftover bits") would be 5+2+5 = 12 "leftover bits" in an octave. Which means that there were roughly 6 tones (i.e., whole tones) per octave.

However, the Greeks realized quickly that the numbers didn't quite work out. If you take six 9:8 whole tones and stack them up together, you get (9:8)^6 = 531441:262144, which is slightly different from 2:1 as an octave ratio. Specifically, it's off by about 23.5 cents from an octave, an interval sometimes known as the Pythagorean comma.

To Aristoxenus and those who followed his arguments, this little discrepancy didn't matter that much practically. Essentially if you just shave a tiny bit off of each of those 6 whole tones, an octave is approximately 6 whole tones. And those other "leftover bits" in the scale are approximately half a whole tone in size, so we might think of them as "half of a tone" or a "semitone." (The development and standardization of nomenclature occurred later, but the basic concept of an approximation for half of a tone emerged early with this 12-fold division of the 2:1 octave.)

The Pythagorean faction among Greek theorists found this troubling, as the actual construction of such divisions required things like square roots and "irrational" numbers. ("Irrational" in this sense literally comes from not conforming to a numerical musical ratio like 2:1 or 3:2 or 9:8.)

One thing that should be made clear, however, is that there was no real sense of a 12-note "chromatic scale" at this time. Aristoxenus was really just noting that one could mathematically view the octave as approximately in size equal to "12 parts." And if we combine 2 of those small parts together, we get a very close approximation to the 9:8 whole tone, an interval the Pythagoreans already found important.

With this imprecision/approximation of mathematics came an imprecision in language about musical intervals too. A word like "tonos/tonus/tone" could mean specifically a 9:8 ratio, but in other contexts could perhaps refer to another close interval, like a 10:9 ratio. (That interval could occur, for example, in dividing up a 5:4 major third into two intervals, 9:8 and 10:9. For the Pythagoreans, this division was "rational" even if uneven; whereas an Aristoxenean perspective might just say "make them both about half of the interval" and not worry about the exact ratio for tuning.) Similarly, when the term "semitone" first occurred, it could mean lots of different possible intervals, all approximately half the size of a 9:8 "tone."

This tension between exact tuning of intervals according to ratios vs. approximate division of the octave according to the ear continued for many centuries. But the basic idea of whole tones and semitones (with various names) as approximate intervals dates back to the ancient Greeks. This eventually led to the modern nomenclature of "whole steps" and "half steps." (I assume you're less interested in the various linguistic developments in these terms over the centuries than the musical justification, so I think I'll leave it here.)

Once equal temperament with the 12-fold chromatic division of the octave became gradually accepted in the 16th-19th centuries, all the subtleties of different tunings for different variants of "tones" and "semitones" were gradually forgotten, and it all became just "whole and half steps."