Good Math, Bad Math

As long-time readers know, I’m an amateur musician, from a very musical family. My sister is a music teacher, and my brother used to be a professional french horn player and composer. I personally play classical clarinet, a very wide range of folk-flutes, and some bluegrass banjo.

As long as I’ve studied music, my teachers have always talked about how fundamental the pentatonic scale is. For those who don’t know, the pentatonic
scale is a basic scale which has five distinct notes per octave, instead of the 7 of the traditional diatonic scale, or the 12 of the chromatic scale. For example, the
pentatonic scale starting at C is the notes C, D, E, G, A, and back to C.

I’ve never really grasped what’s so fundamental about it. It’s
got a beautiful sound – but just looking at it, it’s hard to see what makes it more
fundamental than any other scale. It’s not an evenly distributed scale – the
steps are second, second, minor third, second, minor third. But there’s something
about it.

This video shows just how fundamental it is. Without being told to, people will
naturally sing the steps of the pentatonic scale. The pentatonic scale is wired into our brains. Watch and be amazed!

World Science Festival 2009: Bobby McFerrin Demonstrates the Power of the Pentatonic Scale from World Science Festival on Vimeo.

Comments

  1. #1 Andrew Jenner
    August 1, 2009

    It’s probably because of the frequency ratios. The ratios of the frequencies of the notes C:D, C:E, C:G and C:A are 9/8, 5/4, 3/2 and 5/3 respectively. These are all ratios of small integers, so almost all the intervals between the notes sound good (harmonic).

  2. #2 Marianne
    August 1, 2009

    I watched and I’m amazed! I’m going to post the link to that one at a couple of message boards. Cool!

  3. #3 Jesse
    August 1, 2009

    Yes, it’s cool we can figure out what next comes next, but what makes the pentatonic scale so special? Can we not do this with a variety of other scales, and even entire songs?

  4. #4 Mark C. Chu-Carroll
    August 1, 2009

    Re #3:

    What makes it so special is that people weren’t told the scale in advance. They weren’t given anything but two notes one step apart.

    If you’d asked me in advance, I would have guessed the common major diatonic scale. But they didn’t. That’s amazing.

    If you do the same experiment with people from different places, where they use different basic scales for their music – chinese, japanese, indian, african, european – it doesn’t matter. They’ll all sing the pentatonic.

  5. #5 Mark C. Chu-Carroll
    August 1, 2009

    Re #1:

    Actually, there’s a simpler mathematical relationship. It’s the circle of fifths. The musical fifth is 1.5 times the frequency of the base.

    If you start at C, the fifth is G. From G, the fifth is D. From D, the fifth is A. From A, the fifth is E. So the first five fifths from the base are the notes of the pentatonic scale.

    There is a mathematical relationship between the notes. But it’s still surprising.

  6. #6 Greg
    August 1, 2009

    I”m not sure I’m that impressed. He gives the audience the C* and the D; then they get the E by themselves, sure. But he himself gives them the A, and then he sings in the pentatonic scale for a while himself, before the audience picks out the G and extends the scale the full two octaves. The demonstration does say something about humans and music: that people will sing in the pentatonic scale without being told to explicitly – they can recognize it and reproduce it. But it sure doesn’t say that the pentatonic is the one scale everybody in the world will start singing spontaneously.

    * They’re actually singing in the key of C sharp; this doesn’t affect the discussion.

  7. #7 Jesse
    August 1, 2009

    Re #4:
    Actually, they were give three of the five notes and without his help might have thought they were going to sing the following at 1:04:
    A# C C# D# F G# A#
    So the audience was given C#, D#, guessed F, was given A#, then guessed G#.
    If you show people C and E, will they not guess G comes next? People can find all sorts of patterns in music. Aren’t they all fundamental? Doesn’t all music follow patterns we can easily recognize and foresee? Why, today, are we pointing out the pentatonic scale?

  8. #8 Jesse
    August 1, 2009

    Re #6:
    What you said.

  9. #9 Scott Hanley
    August 1, 2009

    McFerrin filled in the notes around the tonic for the audience, but the way they identified the boundaries of the scale was pretty much unprompted and that seems significant to me. I would love to see a demonstration like this attempted with the familiar Ionian mode – somehow I suspect the average audience might struggle a bit more, even though most of the music we hear follows that scale.

  10. #10 Chanan Carroll
    August 2, 2009

    Thanks. I’m still a composer, and I still play the horn.

  11. #11 Mark C. Chu-Carroll
    August 2, 2009

    Re #10:

    Yeah, but you’re not really a professional anymore. These days, your main job is being a Rabbi.

  12. #12 porno siteleri
    August 2, 2009

    Actually, they were give three of the five notes and

  13. #13 Brian
    August 2, 2009

    #12, #13: wow, those porn site spambots are getting pretty stylish.

  14. #14 Paul Murray
    August 2, 2009

    Sucessive intervals on the cycle of fifhs (not tempered, of course) are

    F 2/3
    C 1
    G 3/2
    D 9/4
    A 27/8

    correcting these so that they all lie in the same octave and re-ordering them, we get

    C 1
    D 9/8
    F 4/3
    G 3/2
    A 27/16
    C 2/1

  15. #15 nyb
    August 3, 2009

    I don’t think this demo says much about the pentatonic scale and its basic presence in humans; seems to me that it’s very difficult to find an audience that hasn’t been extensively exposed to it, no matter where you are in the world. Even places with non-”Western” musical traditions hear a lot of “Western” pop.

    Right, now I have to go re-watch close encounters. Or perhaps I should say re-listen-to. Let us keep in mind that the pentatonic scale isn’t just on earth! (The pentatonic scale is a virus from outer space? Oh no, William S. Burroughs had it all wrong.)

  16. #16 D
    August 3, 2009

    “If you do the same experiment with people from different places, where they use different basic scales for their music – chinese, japanese, indian, african, european – it doesn’t matter. They’ll all sing the pentatonic.”

    Do you have a citation? That’s fascinating. I’d be a bit surprised if it were true of Indian music actually, because there are so many pentatonic scales. My favorite happens to be C, D, E, G, B and C again. It’s called hamsadhwani, which I think means sound of a swan.

  17. #17 Richard Wein
    August 3, 2009

    I’m pretty sure the audience would have assumed they were hearing a diatonic scale until the evidence indicated otherwise. After C-D-E, if McFerrin had gone up another note without any further clues, they would have sung an F, not a G.

    I think the first indication it wasn’t a diatonic scale was when McFerrin went down from C and sang an A (rather than a B). An interesting question is how many people would find that enough to give them the pentatonic scale. What if McFerrin hadn’t sung a tune over the top?

    What impressed me most was how quickly the audience chimed in with the third note. I know I wouldn’t have been so quick. Was this a particularly musical audience?

  18. #18 Andrzej Stewart
    August 3, 2009

    Howdy Mark!

    As an aerospace engineer that has dabbled in structural dynamics a bit, I can say I’m not all that surprised about the relationships between the notes. As you pointed out, the notes are perfect fifths from each other…that’s pretty significant.

    You see, all the notes we use can be traced back to the frequencies of a vibrating string. Let’s say we have a string tuned to a particular note. The nth fundamental frequency fn of that string can be expressed as fn = n*f1, where f1 is the first fundamental frequency. The first and second frequencies of a string are the note itself and its octave (equivalent notes from a music standpoint), then the third frequency is the octave’s perfect fifth (or pretty close – more on that in a moment). So, the perfect fifth is actually included in the sound coming out of a vibrating string, so if you play another string tuned to the perfect fifth at the same time, it sounds consonant (similarly, if you play two unrelated notes, it sounds dissonant).

    If we keep going, the fourth frequency is another octave, then the fifth frequency is the note’s major third. If we include the major third with the two note chord we were playing earler (note + perfect fifth), then we get a major chord. In C, this would be C-E-G (note that these are all in that C pentatonic scale described in the original article). We’ve just built a major chord using math and science…see folks, there is beauty in math!

    Finally, one other piece of trivia to crack out at your next office party. As I mentioned earlier, there’s a slight difference between the vibrations coming from the string and intended note. For instance, the third frequency of a string tuned at A4 (440 Hz) is 1320 Hz, but musicians will tell you that E6 is 1318.51 Hz. The reason for this is something called “tempered tuning”. The frequencies of a string are eigenvalues coming from a 2nd order PDE. If you were to tune your instrument using these frequencies, it’d sound great in whatever key you’ve tuned to, but not in other keys…each root note would produce a unique set of frequencies for all the notes. Modern instruments instead use tempered tuning. Tempered tuning uses logarithms instead as a compromise, which sacrifices some preciseness of tuning of a particular key to allow an instrument to play all keys equally.

  19. #19 Mark C. Chu-Carroll
    August 3, 2009

    Re #18:

    Yeah, I know about the tempered vs non-tempered scales.

    In fact, in my collection of folk flutes, I’ve got a mixture of tempered and true instruments. The “true” instruments put the pentatonic notes at true, and then temper in between them. You can hear, when you play them, that some of the notes sound a bit off to an ear used to the tempered scale. But if you play harmonies between different true-tuned instruments, it sounds much better than the tempered.

    Of course, the true tuning means that you’re stuck in one key, and everything will sound off if you shift.

  20. #20 Jonathan Vos Post
    August 3, 2009

    Yes, but the Cycle of Fifths works because 5 is relatively prime to 12, so performing the group operation 12 times orbits you through all elements, giving you the Cyclic group of order 12. You take the same operation, inverted, and raise that to the 12th power and you’re running through the same elements in the reverse order, also known as the Cycle of Fourths.

    That is, because 5 is the unique integer less than half of 12, and relatively prime to 12, you have 4 ways to cycle through the 12 notes: chromatically up, chromatically down, Cycle of Fifths up, Cycle of Fifths down.

    If we divided the octave into 30 microtones, we’d have the chromatic up, chromatic down, Cycle of Sevenths up, Cycle of Sevenths down, Cycle of Elevenths up, Cycle of Elevenths down, and there would be too many possibilities for the untutored human brain. The balance between surprising and inevitable in a good melody would be missing.

    Some other cultures have a prime number of microtones in an octave, so that there is only the chromatic scale.

  21. #21 Kapitano
    August 3, 2009

    There’s a problem with the strict circle of fifths. If you follow the maths, you wind up adding about a semitone to the frequency of the octave above.

    If A1 is 110, A2 should be A220, but…

    A1 110
    B 123.75 — (E*1.5)/2
    C 139.218 — (F*1.5)/2
    D 156.618 — (G*1.5)/2
    E 165 — (A*1.5)
    F 185.625 — (B*1.5)
    G 208.827 — (C*1.5)
    A2 234 — (d*1.5)

  22. #22 Jim Plamondon
    August 3, 2009

    The first three notes that McFerrin has the audience sing are each a whole tone apart (Do, Re, Mi), so they could be either the first three notes of the major diatonic scale or the major pentatonic scale.

    When he asks the audience to go “one note lower than the tonic” (almost exactly one minute into the above video), they are confused, and he must sing the note for them — and the note he sings is the “note that’s lower than the tonic” in the pentatonic scale (La), not in the diatonic scale (Ti).

    If the audience had sung “La” without confusion, then that would have meant something. But they WERE confused, so the demonstration meant nothing. It certainly does not prove the universality of the pentatonic scale.

    This is an example of the Clever Hans effect (http://en.wikipedia.org/wiki/Clever_Hans).

  23. #23 Pope Maledict DCLXVI
    August 3, 2009

    #21: Sorry, but the scale you’ve calculated is wrong: the notes pitches corresponding to your frequencies are actually: A, B, C#, D#, E, F#, G#, and A#. (You went wrong assuming that B to F is a perfect fifth, when it’s in fact a diminished fifth.) The note you’re saying is “A2″ is a semitone sharp, because it is A#.

    Generally speaking however, six whole tones (of the 9/8 kind) don’t make an octave, and the difference is the Pythagorean comma, an interval about the size of a quarter of a semitone. (Pythagorean tuning being based strictly on ratios constructed from the octave, perfect fifth, and perfect fourth.)

    The other thing to bear in mind in pentatonic harmonies is that often some tones are sung smaller than others to obtain better tuning: if E is pitched from C as four successive perfect fifths one obtains a ratio of 81/64, whereas the sound of the natural harmonic series is the rather sweeter sounding ratio of 5/4. The size of the step up from the C to D remains at the 9/8 size, but then from D to E is the minor whole tone ratio, 10/9, and similarly the tone from G to A is frequently sung as a minor whole tone, to obtain the ratio of 5/3 compared to the Pythagorean 27/16.

  24. #24 Lowell
    August 4, 2009

    RE #16 – that’s not all that unusual a scale; C major with a 2nd and a 7th. where tones really start getting interesting (and unfamiliar) are when octaves start getting carved up into an unusual number of equal degrees.

    #22 is correct- the audience WAS confused, since ANY three tones signify nothing by themselves. regardless of the notes used, it all depends on how you define the tonic of the scale (the essence of modal theory)- and melodies often do not begin or end on the tonic. the notes C E G are simultaneously C Major (tonic, major 3rd, perfect 5th), E minor (minor 6th, tonic, minor 3rd), and implied G Major (perfect 4th, major 6th, tonic). it’s all relative.

    since the 3rd degree of any scale in Western music determines its overall Major or minor quality, and those three notes omit any information regarding the 3rd in terms of a scale with a G tonic, we can only say G Major is “implied”, as it’s the most common scale that fits the information given. it’s possible for those notes to exist in a minor scale with a G tonic, for instance G A Bb C D E F.

    the reason we “like” or “identify” with the notes used in Western music (the exact resonance frequencies of which continued to change until the mid 19th century) is that they produce the most consistenly consonant harmonics when played in chords, the most consistency in tone when repeated in successive octaves or when transposed to different keys, and through conditioning. music based on scales with, say, 11 equal tones per octave or 13 equal tones per octave (or other exotic, perfectly valid and interesting but comparatively “unmusical-sounding” scales) sounds very strange and dissonant to most people. many of the “exotic” scales used in non-Western cultures utilize quarter-tones to achieve notes between what are possible in Western music, without sacrificing those same consonant harmonics completely.

    Western scales have undergone many changes to find a balance between some mathematically consistent formula and harmonic consonance. there’s a pretty good, succinct explanation of how we’ve arrived at the method we now commonly use (equal temper) here:
    http://www.jimloy.com/physics/scale.htm

  25. #25 Jeff Bowles
    August 4, 2009

    The black keys. They show a perfect cross-section that puts your scale, on one side and the rest of the world, on the other.

    Translating: in the key of C, it’s C/D/E/G/A. If you look at the notes, using a movable-do for names, it is Do-Re-Mi-Sol-La.

    And in the key of F#, which gets us onto the black keys, it’s F#/G#/A#/C#/D#. Again, Do-Re-Mi-Sol-La.

    So, the example you give is a very good one, but if you have a keyboard nearby, “it’s the black keys.”

    I’d bet a dollar, that John Williams composed his “Empire of the Sun” theme on the black notes of his piano.

  26. #26 Sojournposse
    August 4, 2009

    #16, I’ll be interested to know, too. Gamelan is pentatonic. Quite enjoyed the video.

  27. #27 John Miller
    August 4, 2009

    Wow, got to this party late…

    As a background, I work part time for a pipe organ tuner and have a degree in physics.

    First, the interesting thing about this demo is not that this particular audience “got it” it is that any audience will. An American audience would likely have gotten a major scale or major triad, but might have gotten lost in a Doric or true minor, and certainly would have for an equidistant diatonic (unless it was an organ tuner convention). Other cultures may have had trouble with our “normal” major progression. Humans just seem to “get” pentatonic music regardless of culture.

    Second, there are lots of folks messing around with sound ratios. Not a bad thing but there are some psychoacoustic considerations here. Pure tones are boring. Every real instrument makes a sound that has a fundamental and lots of harmonics. It is how one can tell the difference in sound between a violin and a flute. Having two tones that are just slightly out of tune with each other encourages the production of these harmonics. On an organ there is a stop called a celest that is tuned slightly flat and played with a much stronger “string” pipe. It creates a very beautiful and ethereal sound.

    It has been recognized sense the at least Bach’s time (c.f. The Well-tempered Clavichord) that a brighter more enjoyable sound comes from instruments not tuned in perfect fifths. In this case being mathematically precise is not the same as musically enjoyable.

  28. #28 george nagata
    August 5, 2009

    Nice vid & post.
    Just fyi, the first four bars of the Japanese national anthem goes like this:

    | D C D E | G E D | E G A | D B A G |

    Strong sounding pentatonic notes I hear.

  29. #29 tonyC
    August 5, 2009

    very cool – and demonstrates that Jazz & Blues are the only real music!

  30. #30 George Atkinson
    August 5, 2009

    Can you offer a link to a non-vimeo version of the demo?

  31. #31 Jeremy Gibbons
    August 10, 2009

    As has been pointed out, the pentatonic scale is based on the “circle of fifths”. Why “fifths”? That this interval is five notes in the Western diatonic scale is, I think, culturally dependent; but the interval itself (when true) is the harmonious small integer ratio 3:2 in frequencies.

    But why “pentatonic” – why a scale of five notes, and why these five? Well, any interval made out of these five is consonant, and you can’t divide the octave more finely using fifths and avoid dissonance. So it’s a good selection of notes to provide on a simple musical instrument, in order to give an amateur musician the opportunity to make nice sounds without allowing them to make horrible ones. (Wind chimes are often arranged in a pentatonic scale.)

    Interestingly, the fact that the Western even-tempered chromatic scale is divided into 12 semitones is culturally independent: it provides a very close approximation to the consonant ratio 3:2, since 2**7/12 ~= 1.498. To get a closer approximation on an even-tempered scale, you’d need 24 microtones out of a scale of 41 (2**24/41 ~= 1.5004, and no denominator between 12 and 41 is better). So aliens with roughly the same number of fingers as us will likely have the same chromatic scale :-).

  32. #32 Anonymous
    August 12, 2009

    Some have commented here that the step down to A (if we assume he started at C) disproves the theory. It doesn’t. If anything, if this weren’t hardwired into our heads, a step down to A would only confuse us.

    I would have expected everyone in the audience to be able to do a major scale (whole whole half whole whole whole half whole) since anyone who has ever played the white keys on a piano knows how a major key sounds. That’s not what’s happening here. If my assumption of the major scale were correct, the jump down to A would have confused people. Rather, however, they are able to jump all over the scale mostly unprompted.

  33. #33 Steve
    August 12, 2009

    Sing: “I love you, you love _____.”

    I bet most of you said “me”, right?

    People are good at finding patterns anywhere — when given a lead or a hint. (The extra space between hops, for example).

    Having said that, I liked this video!

  34. #34 grahamsw
    August 27, 2009

    Loved the video, loved the performance, and doubtless there are hardwired commonalities in our responses to sound – it can’t be a coincidence that there has never been a human culture that did not have poetry, music, dance and plastic arts – these things are not accidental fluff on top of the serious human stuff, they are fundamental – poetry probably comes before prose.

    But: having an interest in magic, cold reading, fake psychics, propaganda, framing, pedagogy and other forms of misdirection and manipulation, this is not terribly convincing.

    Socrates got the slave boy to show that he knew geometry innately, and the shamwow guy got me to realize that I really, really needed one of these things (then he punched me in the face).

    Showmanship, and the desire not to upset the show, are as fundamentally human as anything else.

    Mark, your skepticism is slipping…

  35. #35 Jonathan Vos Post
    September 1, 2009

    *SIGH*
    http://harvardmagazine.com/2007/01/mapping-music.html
    Okay, let me point you towards the first Music Theory author that “Science” published in 150 years. Tymoczko (pronounced tim-OSS-ko), who spent this past academic year as a composer in residence at the Radcliffe Institute for Advanced Study, has developed a way to represent music spatially. Using non-Euclidean geometry and a complex figure, borrowed from string theory, called an orbifold (which can have from two to an infinite number of dimensions, depending on the number of notes being played at once), Tymoczko’s system shows how chords that are generally pleasing to the ear appear in locations close to one another, clustered close to the orbifold’s center. Sounds that the ear identifies as dissonant appear as outliers, closer to the edges.

  36. #36 Jonathan Vos Post
    September 1, 2009

    About the number of generators of a musical scale
    Authors: Emmanuel Amiot
    Comments: 14 pages, 10 figures
    Subjects: Group Theory (math.GR); Commutative Algebra (math.AC)
    http://arxiv.org/abs/0909.0039

    Several musical scales, like the major scale, can be described as finite arithmetic sequences modulo octave, i.e. chunks of an arithmetic sequence in a cyclic group. Hence the question of how many different arithmetic sequences in a cyclic group will give the same support set. We prove that this number is always a totient number and characterize the different possible cases. In particular, there exists scales with an arbitrarily large number of different generators, but none with 14 generators. Some connex {JVP: convex?} results and extensions are also given, for instance on characterization via a Discrete Fourier Transform, and about finite or infinite arithmetic sequences in the torus R/Z.

  37. #37 bob
    May 25, 2010

    hhh