Circle of Fifths
- I have had the circle of fifths explaned to me till I'm blue in the
face. Can someone tell me a web site that I mught go to have it
explaned to me in good old country boy anglish. Thank e very much.
- moore_r_2000 wrote:
| I have had the circle of fifths explaned to me till I'm blue in the
| face. Can someone tell me a web site that I mught go to have it
| explaned to me in good old country boy anglish. Thank e very much.
Here is one that made the light bulb turn on for me:
It is pretty much plain english, :-)
Murray Phillips <mailto:imphillips@... >
Past-President, Greater Montreal Chapter SPEBSQSA
Editor, The Mini-Pitch, Web Master, Chorus Manager
Quartet Development Chairman
Check us out at: <http://www.nedistrict.org/chapters/montreal >
AHSOW 1998 <http://www.ahsow.org >
Co-Chairman, Montreal International 2003
Bass, Key of Eh? <http://www.nedistrict.org/qtets/keyofeh >
- 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
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
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
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
-- Joel Ellis Rea, Tenor
River Cities Jubilee Chorus
- --- In bbshop@y..., "Joel Ellis Rea" <joelrea@s...> wrote:
> There have been some discussion lately on the Circle of Fifths.One thing
> about it that applies to Barbershop, though, is the fact that theCircle of
> Fifths doesn't actually WORK! It's not a circle! Here's why, andwhat the
> ramifications are:... (several credit hours' stuff snipped)
>... and the practical effect?
> -- Joel Ellis Rea, Tenor
> River Cities Jubilee Chorus
If you are in a quartet with EIGHT REALLY GOOD ears, and you are
singing a song with LOTS of circle-of-fifths progressions, you is a-
gonna end the song SHARPER than the key you started in!
Think about THAT while sipping a little holiday cheer.
Tom in Cincy
Bass - SGC etc. etc.
softly humming "Red Roses For A Blue Lady"
- I forwarded Joel's "not-so-brief" explanation about fifths to our chorus.
There were several positive responses. Here's one...
Al Wolter wrote:
Hey Bruce - this was one of the best explanations I have ever seen (a bit
long to read as e-mail - but excellent). I am going to send this to a few
people I know, and keep a copy for those ever present questions by the
inquisitive. Since Joel's e-dress has been truncated, could you forward this
Hey, Merry Christmas to you and all the GVC'rs. See you soon.
Bruce Sellnow <bsellnow@...> <*)))><
Golden Valley Chorus http://www.gvc.org
Central CA Chapter VP of Music & Performance
TopCats [bari] http://www.topcats.net
Stop the HOAXES! The *real* virus warnings are at:
Symantec Anti-Virus Research Center [SARC]
- --- In bbshop@y..., "Joel Ellis Rea" <joelrea@s...> wrote:
> There have been some discussion lately on the Circle of Fifths.From the scholarly work Joel wrote, I am guessing that Joel is also
> The Circle of Fifths was discovered millennia ago by a really smart
> Greek guy named Pythagoreas.
> [...thousands of words omitted...]
"a really smart guy"! :)
> I'm working behind the scenes with some very talented music softwareNow THIS sounds "REALLY useful" (to quote Thomas the Tank Engine, who
> 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.
is not a really smart guy, but is definitely good-hearted and, of
course, useful). I cannot wait to see what this new notation looks
like. I imagine that the reason we have to wait to see it is for
legal reasons (like patents and copyrights).
Thank you, Joel, for a truly wonderful Christmas present.
MUS Judge (SPEB)
Baritone, Excalibur (SPEB)
Director, City of Lakes (SAI)
brent-a-graham @ exite.com <== use this e-dress for replies
- I suppose that progression around the circle of fifths has always been a part of barbershop.
But is there any info on when barbershoppers realized that is what they were doing? When practice met up with harmonic theory? Was the phrase used in the earliest judging criteria?
- John, you are correct in your suspicion that circle of fifths movement
has been long understood as a vital component of barbershop harmony.
One of the early giants of the Society, Maurice "Molly" Reagan,
published his "Clock System" in The Harmonizer back in 1943 -- and you
can read it today online at
( Also read about Molly Reagan in the Heritage of Harmony History book
--- In firstname.lastname@example.org, "john_cowlishaw" wrote:
> I suppose that progression around the circle of fifths has always been
a part of barbershop.
> But is there any info on when barbershoppers realized that is what
they were doing? When practice met up with harmonic theory? Was the
phrase used in the earliest judging criteria?
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