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Circle of Fifths

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  • moore_r_2000
    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
    Message 1 of 8 , Dec 18, 2001
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      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.
    • I. Murray Phillips
      ... Here is one that made the light bulb turn on for me: http://www.mikemurphy.net/GuitarLessons/lessons/lesson18.htm It is pretty much plain english, :-)
      Message 2 of 8 , Dec 19, 2001
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        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:

        http://www.mikemurphy.net/GuitarLessons/lessons/lesson18.htm

        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 >
      • Joel Ellis Rea
        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
        Message 3 of 8 , Dec 20, 2001
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          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). :-)

          Happy Holidays!


          -- Joel Ellis Rea, Tenor
          River Cities Jubilee Chorus
        • sgc_tom
          ... One thing ... Circle of ... what the ... ... and the practical effect? If you are in a quartet with EIGHT REALLY GOOD ears, and you are singing a song with
          Message 4 of 8 , Dec 21, 2001
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            --- 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 the
            Circle of
            > Fifths doesn't actually WORK! It's not a circle! Here's why, and
            what the
            > ramifications are:
            >
            >
            ... (several credit hours' stuff snipped)
            >
            > -- Joel Ellis Rea, Tenor
            > River Cities Jubilee Chorus

            ... and the practical effect?

            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
            barit1@...
            Bass - SGC etc. etc.
            softly humming "Red Roses For A Blue Lady"
          • Bruce Sellnow
            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 -
            Message 5 of 8 , Dec 21, 2001
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              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
              him?

              Hey, Merry Christmas to you and all the GVC'rs. See you soon.

              Al


              *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*
              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]
              http://www.symantec.com/avcenter/
            • thebl00per
              ... From the scholarly work Joel wrote, I am guessing that Joel is also a really smart guy ! :) ... Now THIS sounds REALLY useful (to quote Thomas the Tank
              Message 6 of 8 , Dec 21, 2001
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                --- In bbshop@y..., "Joel Ellis Rea" <joelrea@s...> wrote:
                > There have been some discussion lately on the Circle of Fifths.
                > [...]
                > The Circle of Fifths was discovered millennia ago by a really smart
                > Greek guy named Pythagoreas.
                > [...thousands of words omitted...]

                From the scholarly work Joel wrote, I am guessing that Joel is also
                "a really smart guy"! :)

                > 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.

                Now THIS sounds "REALLY useful" (to quote Thomas the Tank Engine, who
                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.

                In Harmony,
                Brent Graham
                MUS Judge (SPEB)
                Baritone, Excalibur (SPEB)
                Director, City of Lakes (SAI)
                brent-a-graham @ exite.com <== use this e-dress for replies
              • john_cowlishaw
                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
                Message 7 of 8 , May 7, 2013
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                  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?
                • Brian
                  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
                  Message 8 of 8 , May 7, 2013
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                    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
                    http://barbershop.org/harmonizer/Harmonizer_vol2_no4_may1943.pdf
                    <http://barbershop.org/harmonizer/Harmonizer_vol2_no4_may1943.pdf>

                    ( Also read about Molly Reagan in the Heritage of Harmony History book
                    -- http://barbershop.org/history/HeritageofHarmony_50Years.pdf
                    <http://barbershop.org/history/HeritageofHarmony_50Years.pdf> )



                    --- In bbshop@yahoogroups.com, "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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