From: Joel Ellis Rea
Date: 2001-12-21, 02:20
Subject: Re: Circle of Fifths
There have been some discussion lately on the Circle of Fifths. One thing about it that applies to Barbershop, though, is the fact that the Circle of Fifths doesn't actually WORK! It's not a circle! Here's why, and what the ramifications are:
The Circle of Fifths was discovered millennia ago by a really smart Greek guy named Pythagoreas. He's the one who figured out that if you held a string really taut and plucked it, you would get a musical sound (an actual pitch), and that the tighter the string, the thinner the string, OR the shorter the string (all else being equal), the higher-pitched the note would be, and vice-versa: looser, thicker, and/or longer strings produced lower-pitched notes.
Well, actually, much of that was already known, but here's the part HE figured out: if you put something kinda sharp (but not sharp enough to cut the string -- think of something sort of like the bridge of a violin) in the exact middle of the string, so that it touched the string, and plucked the string on either side, you would get a different note. That note would be EXACTLY ONE OCTAVE HIGHER. Why did this happen, he wondered? Turns out it was because the bridge-thingie (which we will call a "fulcrum" for the purposes of this discussion -- think of the middle part of a child's see-saw or teeter-totter, the part that holds the board) effectively split the string into TWO strings, each vibrating separately -- as one half went down, the other would go up, and vice-versa (sort of like the afore-mentioend see-saw/teeter-totter [thus the use of the term "fulcrum"], except that the string doesn't move "straight" but rather it curves as it vibrates -- not that that's really important to this). These two strings were each HALF the length of the original string, and so vibrated TWICE as fast, and produced a pitch whose frequency (in sound waves per second) was TWICE as high. This twice-as-high thing is what an Octave IS! It's WHY an octave sounds the way it does, and why even a small child can tell the difference between an octave-type interval and a non-octave-type interval, even though they may have no idea of what those are. They know it when they hear it.
Well, ol' Pythagoreas didn't stop there! What, he wondered, would happen if you split the string into THREE parts? Or four? He soon figured out something really cool: if you placed only ONE fulcrum at a point where the first of a series would have to be if you were to split it into more than two parts, it was the same as if you had all of the fulcrums there: the string would still vibrate in that many parts. So, if you figured out where the exact half-way point was between either end of the string and the exact middle of the string (where the fulcrum would be if you were only splitting the string into two halves) was, and moved the fulcrom to that half-way of a half-way point, the string would vibrate in FOUR parts, each 1/4th of the length of the original string! They would vibrate four times as fast, or twice as fast as that first octave. What was the result? Well, since it was TWICE as fast as the FIRST octave, it was an octave up OF an octave up, thus TWO octaves up, from the whole string's note! Likewise, spliitting the string into eighths would produce a note that was THREE octaves higher, and so on.
But what would happen if you split it into THREE parts? That, he figured, should produce a note that was halfway between the first octave (one octave up) and the second octave (two octaves up from the original), and he was right. It took him a bit of fiddling to get the fulcrum placed in the right spot, since he couldn't just go halfway from the first halfway point, but actually had to figure out what was one-third of the original length (and he didn't even have a calculator -- not even one of those Tibet Instruments-brand abacus thingies! Like I said, he was a REALLY smart Greek guy! Dividing by THREE when you don't even have a good numbering system is NOT EASY!) Anyway, he did it, and he got a new tone that was THREE TIMES the pitch of the original, since the string was now vibrating in three parts, each only one-third the length of the original!
But if TWICE as high is an octave up, and FOUR TIMES as high is TWO octaves up, what is THREE times as high? Well, in our parlance, it would be an octave PLUS a Perfect Fifth! He was the person who discovered the Perfect Fifth interval, and how it worked.
Now, this is where he got REALLY smart: He wondered what would happen if you carried the sequence of Octaves up (splitting the string length in half each time, doubling its pitch) vs. the sequence of Fifths (Octave+5ths actually -- splitting the string length in a third each time, tripling its pitch). Would they ever meet and produce the same pitch? He figured out that if you split a string into thirds twelve times, you would get ALMOST EXACTLY the same pitch as if you split it into halves seven times -- the twelve Fifths came around a sort of "circle" and produced ALMOST EXACTLY the same note as a note a series of seven Octaves! At first he was thrilled -- this seemed to prove to him that this was how the gods made the Universe, and that the whole Universe was based on these "Harmonics" as he and his followers termed them (named after Harmonia, the "good" daughter of Ares, God of War, who was herself the Goddess of Peace and Getting Along with One Another, and opposite of her sister Eris, Goddess of Discord).
But there was one niggling fly in the ointment: that "ALMOST exactly equal" part. Because, well, it just didn't work out that way: the result of the sequence of twelve Perfect Fifths was just a WEE bit sharper than the result of the sequence of seven Octaves. It was SO CLOSE!! But it just wouldn't work out exactly, no matter WHAT he did!
Well, if he'd had a calculator, or at least a decent numbering system (like the ones those Arabs worked out millennia later but still about a millennium ago for us, way back in the Middle Ages, which we still use today, and which makes all of our modern technology, including these new-fangled computer thingies, possible: the place value digital numeration system, with the concept of the Zero as a place-holder -- sure beat those Roman Numeral things of trying to use letters as numbers that Western Civilization was using at the time and had been for centuries -- took Europe awhile to catch on to the value of this, what with bias against the Muslim religion, and all those Crusades and stuff -- but yes, even given what happened on September 11, remember that the Arabs DID accomplish some truly great things), he would've figured out why this was: to get Octaves, he was raising the original frequency by powers of TWO, which (assuming you start with a whole number) ALWAYS gives you an EVEN number: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131052, 262104, 524208, etc. times the original pitch. But when he was going up by Perfect Fifths, he was raising the frequency by powers of THREE, which would always multiply the original frequency by an ODD number: 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, etc. No matter HOW far you carry out the sequence, you will NEVER get them to match up exactly, since an EVEN number can NEVER equal an ODD number!
Tripling the frequency of a note gives you the Perfect Fifth in the next higher octave. Well, to move down an octave, you simply cut the frequency in half (since doubling it moves UP an octave). Half of 3 is 1.5, so to go up to the Perfect Fifth in the SAME octave as the starting note, you multiply by 1.5 instead of by 3. 1.5 can also be written as 1+1/2, or 3/2. The latter is the preferred notation, since it shows the RATIO that the ending note is to the starting note of the interval. Likewise, going up an octave means multiplying by 2, so the RATIO is 2/1.
Anyway, look back at the above sequence. Since we went by 3x for the Perfect Fifths (the true Third Harmonic), we went over an octave each time, and thus went twice as far as we would have by using 1.5x. I used 3x to keep the math simpler, and also because that's what Pythagoreas actually did. Anyway, starting with one note and then raising it to Perfect Fifths twelve times multiplied the original frequency by 531,144. But, starting with the same note and then raising it an Octave eighteen times multiplied the original frequency by 524,208. (Of course, since the human hearing range is only 10 octaves or so, going up 18 octaves, even from the lowest humanly-perceivable note, would still go WAY past our range of hearing. But this is for purposes of illustration.) Going around the Circle of Fifths from C, to G, to D, to A, to E, to B, to F#, to C#, to G#, to D#, to A#, to E#, and then one more produces a B#. On a piano, B# = C, but in reality they're two different notes. The B# is barely noticeably sharper than the C (by a smidgen over 23-and-a-half "cents," where one "cent" is one one-hundreth of the pitch distance between any two adjacent chromatic notes on the piano -- thus, since there are 12 chromatic semitones in one octave on the piano, there are 12 * 100, or 1200, "cents" in an octave). This difference between the result of the Circle of Fifths, and the pure octaves, is called the "Diatonic Comma" or "Pythagorean Comma."
Obviously, its not possible to tune a piano so that the Fifths are all pure Perfect Fifths, and yet all of the Octaves are also pure. And we haven't even gotten into the Thirds, Sevenths, and other intervals yet! Many schemes have been tried over the centuries to attempt to resolve this, but it's mathematically impossible to do with an instrument that cannot be re-tuned instantly on the fly, as chords change. Human voices don't have this limitation, which is why we Barbershoppers can do this easily, and why we don't sing to piano or other fixed-tuning accompaniment (at least not and call it "Barbershop").
Pythagoreas never calculated beyond the Third Harmonic, since it was hard enough for him to figure out how to divide by three with the primitive numeration system of his day. To do a pure Major Third, he would have had to divide the string into fifths (the Fifth Harmonic -- ironic that the Perfect Fifth is the Third Harmonic, but the Major Third is the Fifth Harmonic, and yet pure Dominant Sevenths, Ninths, Elevenths, Thirteenths, etc. are all their own Harmonics -- the Dominant Seventh is the Seventh Harmonic, and so on -- but that's just the way it works out). He DID discover a Major Third, though, but it wasn't the PURE HARMONIC Major Third. What he did was go up four of his beloved Perfect Fifths, and wound up with a note that was close to, but slightly sharper than, the true Major Third. In today's Circle of Fifths, if you went up four Perfect Fifths from C, to G (that's one), to D (that's two), to A (that's three), and then one more, you would get E. That would be a Major Third, but this is called the Pythagorean Major Third, and is NOT the pure perfect Harmonic Major Third we Barbershoppers strive for. The Harmonic Major Third has a ratio of 5/4 (1.25), while the Pythagorean Major Third has a ratio of 81/64 (5/4 = 80/64, so the difference is 81/80). This difference between the Pythagorean Major Third and the Harmonic Major Third is called the "Syntonic Comma."
Now, all of this may sound like really hard math, but your ear, brain, and vocal tract are all hard-wired to do it, since it's all natural harmonics. Everything that vibrates to produce sound also produces harmonics, and these harmonics blend with the fundamental tone to produce the sound we recognize as the nature of the tone. The more and louder these harmonics are relative to the fundamental, the brighter the tone sounds. A violin string, for instance, has lots of harmonics when bowed. It vibrates as a whole, but also in halves (forming a softer tone an octave higher than the fundamental), in thirds (forming a still softer true Perfect Fifth higher than the fundamental, or an Octave plus a Fifth in standard terminology), in fourths (two octaves up), in fifths (a true Harmonic Major Third, or in standard terminology, two Octaves plus a Harmonic Major Third up), in sixths (two octaves and a Perfect Fifth up), in sevenths (two octaves and a Harmonic Dominant [Barbershop] Seventh [ratio of 7/4] up), in eighths (three octaves up), and so on, and so on, theoretically to infinity, all at the same time! A plucked nylon guitar string would also vibrate in harmonics, but they would be softer than the ones for the violin string, thus forming a mellower tone. A steel-stringed guitar would produce more and/or louder harmonics, for a brighter tone. But regardless, the harmonics all form around actual whole multiples of the original fundamental frequency. The string would never vibrate in fractional parts, for instance, or something like "pi" parts. Your vocal folds, too, do this, producing harmonics by vibrating in whole and in multiple parts at the same time.
When multiple people sing, each singing a different note but with precise harmonic relationships to each other, the higher harmonics of each re-inforce each other. For instance, if Bass sings a C, and Bari a G, and Lead another C, and Tenor an E, then the G of the Baritone re-inforces with the Third Harmonic (Perfect Fifth) of the Bass (and the Lead, for that matter). Likewise, the E of the Tenor re-inforces with the Fifth Harmonic (Major Third) of the Bass and Lead. This is what produces the "expanded sound" or "overtones" we strive for, and it CANNOT HAPPEN with the compromised tuning system that pianos, organs, etc. use to try to squoosh the Circle of Fifths to exactly match the Octaves.
Even within a single key or scale, no fixed-tuning system can accommodate the chords that are needed while keeping the harmonic relationships pure. Watch what happens with just the three basic chords of a given scale: I (Tonic Major), IV (Sub-Dominant Major), and V7 (Dominant Seventh). In the Key of C Major, the I would equal C, the IV would be F, and the V7 would be G. The C Major Triad is C, E, and G, forming the Tonic, Major Third, and Perfect Fifth, respectively. Relative to the C, their ratios are 1/1, 5/4, and 3/2, respectively. Adjusting the fractions to have the Least Common Denominator, we get 4/4, 5/4, and 6/4, thus the ratio of the Major Triad Chord is 4:5:6. When sung in this way, it rings brilliantly! Now, let's look at the IV chord, in this case, F. The F Major Triad is F, A, and C. Now, we want the C to stay the same in our scale, so we adjust the tuning so that the F starts almost two cents flatter than its pitch on the piano. Starting with that note, we wind up with the same ratios: 1/1, 5/4, and 3/2, or 4/4, 5/4, and 6/4, again forming a nice ringing 4:5:6 ratio.
Things get hairy with the V7, in this case, G7. We want the G of that chord to match the Perfect Fifth of the C chord, so the G starts up almost two cents sharper than its pitch on the piano. Since we add in the Dominant Seventh, which has a ratio of 7/4 to the Chord Tonic, our ratio is now 4/4, 5/4, 6/4, and 7/4, or 4:5:6:7. This is the classic Barbershop Seventh chord. Problem is, that 7/4 is SUBSTANTIALLY flatter, by over 29 cents, than the equivalent piano note. Yet, it's an F! G, B, D, F! And to get the F to work, we only flattened it by slightly LESS than only TWO cents! That's a difference even an untrained ear can here: it's over an eighth of a tone (a fourth of a semitone)! Obviously, it would not be possible to tune a piano so that the F is correct for both the F Major AND the G7 chord! But singers can handle this situation with ease!
Things get even tricker when you add other common chords into the mix. Popular music often uses the II, II7, IIm, and IIm7 chords. In the Key of C Major, those translate to D, D7, Dm, and Dm7 chords. Now, a D Major chord is D, F#, A. But if the D is the same as the D of a G or G7 chord (the Perfect Fifth of those chords), then the Fifth of the D chord, the A, would NOT be the same as pitch as the A that is the Major Third of the F Major chord! Either the D cannot match the G Perfect Fifth, the A cannot match the F Major Third, or the D chord itself must become dissonant by having the A NOT be a true Perfect Fifth relationship. I can tell you from experience that it sounds HORRIBLE if you try THAT solution! Thus, there are two possible pitches EACH for the notes we call D and A. And we haven't even gotten into Minor 7ths, Augmenteds, Diminished 5ths and 7ths, etc.! That's a whole 'nother can of worms.
I'm working behind the scenes with some very talented music software programmers to help develop some very affordable music composition and notation software for a capella (including especially Barbershop) music that takes all of this into account, automatically. I'm also developing some notation (which the software will be able to print) to help learn a new song easier, helping experienced Barbershoppers see right away what their harmonic responsibility is within each chord, and helping show how to adjust both pitch and loudness to form the perfect ensemble blend. Watch this space for more details as I can reveal more. The notation I could describe now, if anyone's interested (and has slogged this far through my massive essay). :-)
Joel Ellis Rea, Tenor
River Cities Jubilee Chorus