Saturday, April 12, 2014

The 12th Root of Two

Chances are that if that number means anything to you, you're one of the geeky few who are interested in – or aware of – the intersection of music and mathematics.  Stand by for some observations.  Some of this is going to get weird, I'm sure. 

Music is as old as humanity.  Every society ever found has music, and although the scales may not be the same, there seem to be relationships that follow.  Long before humans could measure frequency (or had a word for it) musicians knew that there are tones that are related to each other.  If you play a stringed instrument, for instance, you can hear beat notes between strings and minimize the differences.  The concept of octaves, two tones that were the same musical tone but twice the frequency, was known for a long time, but not by measuring frequencies.

Western music has evolved to be based what's called the "equal tempered" scale; there's a constant difference between tones.  The scale is based on a series of 12 semitones or half steps, or 7 full tones.  Wait – shouldn't there be 14 semitones if there are 7 full tones?  Welcome to the first oddity: there are two places in a scale where no semitone – no half step – exists.  There are only 7 notes in written music: A through G.  In common notation, a scale has 8 notes - you include the ending note one octave above the starting note.  The easiest scale to show is the C scale, which goes from C to C; C – D – E – F – G – A – B – C.  There are no semitones between E and F or B and C.  The number of half steps between the notes of any scale are 2 2 1 2 2 2 1, and music theory students will laboriously derive scales starting on every semitone in an octave.  If you were to measure the frequency of the two C notes, the second C is exactly twice the frequency of the first one.  For example, middle C on a keyboard, so-named because it is the C closest to the center of a piano keyboard, is produced with a frequency of 261.626 Hz.  It is also called C4  - the fourth C on piano.  The next C (usually called High C or C5) is 523.251 Hz.  Why the oddball numbers?  Why decimals with 3 decimal places?  Because the 12th root of 2 is an irrational number, which means it can't be obtained by dividing any integers.  Its value is approximately 1.05946309435929 although the decimal never ends.

That interval of two gets equally divided among the 12 semitones.  That means each semitone step up is the 12th root of 2 times the frequency of the note you started on, and each step down is divided by the 12th root of 2.  Over the years, the world has more or less settled that the A above middle C (just below high C) is 440 Hz, although some orchestras will vary from that.  Chances are if you have an electronic keyboard in your home, though, that note is 440 Hz. 440 * 12th root of 2 is 466.1638, or A#.  A# times the 12th root of 2 is 493.8833 or B.  Times the 12th root of 2 is 523.2511 Hz or C.  That's how this “equal tempered chromatic scale” system works.

If that reliance on the 12th root of 2 isn't weird enough for you, hang on. There are combinations of notes that naturally sound good to your ears, and other combinations that don't.  These combinations are intervals in musician-speak.  Good combinations are the fundamental, the third and the fifth.  These are 4 and 7 semitones away, or 2^(4/12) and 2^(7/12); the combination of these three tones is called a major triad.  If we drop the third by one semitone so that instead of 4 semitones, we only go 3 (2^(3/12)), this has a tone most people think of as somehow “sadder” than the major triad.  It becomes a minor triad.  Any lessons on the blues will tell you blues music is based on flatting the 3rd and 7th.  In other words, humans perceive tones that vary by (2^(3/12)) or (2^(1/4)) and (2^(5/6)) as sadder sounding than (2^(4/12)) and (2^(11/12)).  Why is that? 

Other forms exist by using other combinations of tones.  Some of them add musical "suspense", somehow sounding stressed or unresolved, as if there's some sort of tension between the tones; replacing the 3rd with a 4th in a Sus4 chord.  Others resolve that tension.  Still others blend in so well with the original major triad that you hardly know they're even present.  Maybe an experienced musician will pick out a C2 being played instead of a plain C major chord; most won't, while everyone can hear the distinctive dissonance of a G6 alternating with a plain G major in the opening of the Eagles' "Tequila Sunrise", for example.  

It gets weirder.  You know that meme that all really famous popular songs come down to a very similar 4 chord progression?  The progression is called I V vi IV ( where the lower case vi means the minor form of 6th chord).  It's a meme because it's absolutely true.   Staying with the key of C (it's easier) C D E F G A B C, that turns into C G Am F.  Not only that, but I vi IV V (C Am F G) accounts for another few thousand songs.  Why do certain progressions work together and others don't?  I have a book of chord progressions, and there are many out there.  This particular one is less than a hundred pages long and yet quite possibly contains the guts of every song ever written.

What this means is that some ratios of multiples of the 12th root of 2 are pleasing to the human ear and others aren't.  Why is that?  Yet another weirdness: the value for A (and, thereby, the entirety of music) has changed since the great composers of the 17 and 1800s were alive.  I've heard speculation that if Mozart were alive today he'd be physically sickened to hear his music.  I don't really know how we could know, but while the individual tones would be different, the ratios of the tones would be the same.  I rather think that music would sound strange to a reanimated Mozart, but that he'd soon adapt to it, since the ratios and harmonies are the same.
I've never seen an attempt at explaining the sensitivity to musical tones as evolutionarily advantageous that seems like a compelling argument to me.  The Wiki entry for octaves says that monkeys experience octaves and that it appears to map to mammalian brains in the thalamus, though. so it seems it has been in the hardware for a long time. 

With no reasonable explanation for why it would offer survival advantage to be sensitive to certain separations of tones characterized by some factors of the 12th root of 2, we're forced to conclude one of two things.  Either it's serendipity: happy accident.  It's just there as the random accident of billions of random matings.  Or we can modify Ben Franklin's statement on wine:  it's "... a constant proof that God loves us, and loves to see us happy."

Now can someone tell me why car horns are in the key of F?


  1. Fascinating post. I wish I had known this when I was trying to start a rock band when I was 17. I would have just started trying chord progressions from that book...

  2. There are people who are "affected" by music and those who are not. And in fact there are those who are musicaholics and must have their "fix" and those who have no interest at all. Why is that? For some people a lot of "music" is discordant noise or at best bearable.

  3. Backwoods Engineer - it's entirely possible to string together good songs that way. Add an intro, a bridge, a few flourishes. Change key once or twice to mix it up. All very commonly done. The basic progression doesn't get you a complete song, but it gets you 90%.

    And anon - that's yet another part of the mystery. Why isn't it universal? Of are some people just more discriminating: that some particular patterns of tone ratios triggers their neural networks, but not all of them.

  4. Because French horns are in the key of F.

    (even if they are double horns, F and B flat)

  5. I do think there is something mystical or entrancing about music at least for some people. In spite my lack of interest in music I have been to music venues. Usually with ear plugs (mostly for the loudness) but it is interesting to watch the effect music has on people. Coincidently I don't drink either and it is also interesting to watch the effect alcohol has on people. Not implying it is the same thing. What I'm saying is I think that when you are "indulging" you can't or don't notice the hypnotic effect of "some" music or "some" venues. I actually went to a concert many years ago and as soon as the lights went down about 1/3 of the people smoked pot. I am implying that listening to music is mind altering (not permenant) and that there seems to be a symbiotic relationship between music and drugs or alcohol. Don't get me wrong I'm not trying to dis it or make something out of it that isn't there. It's just that music seems to be able to touch the mind in a way that words don't and some of that effect is similar to drugs or alcohol. To deep? Too much of a stretch? I am not sure, just a thought.

  6. Anon 0937, I do think it's a repeatable, valid observation.

    I guess my puzzlement is why these relationships of the 12th root of 2 do this in the brain. It's almost like a key fitting a lock: some "thing" is there that the patterns fit into. That's really sloppy talk, but somehow some neural pathways or neural net, or whatever the right word is, get excited by these 2^(1/4) and 2^(7/12) and more patterns.

    It is a mystery.

  7. Out of the park post. I felt like I was reading Asimov or Niven/Pournelle. Amazing.

  8. There's a comedy rock group called Axis of Awesome that has a bit about the "Four Chord Song." Worth a listen. There's also a musician's rant about Pachelbel's Canon floating around out there.

  9. You might want to elaborate in paragraph three a bit more.

    Your example scale is a major scale, which has the 2 2 1 2 2 2 1 step pattern. A melodic minor scale, for example, would have a different step pattern with a flattened third.

    C (2) D (2) E (1) F (2) G (2) A (2) B (1) C

    Melodic Minor:
    C (2) D (1) E♭ (2) F (2) G (2) A (2) B (1) C

    Note that there are three minor scales. Because one is never enough, I guess.

    Natural Minor Scale = C - D - E♭ - F - G - A♭ - B♭ - C
    Harmonic Minor Scale = C - D - E♭ - F - G - A♭ - B - C
    Melodic Minor Scale = C - D - E♭ - F - G - A - B - C

    Also, your discussion of semitones as they relate to the scale is a bit off. There is one semitone between the third (E) and fourth (F), and seventh (B) and eighth (C) notes of the scale. There are two between the rest of the notes. The C major scale is nice in that all of the notes are playable with the white keys on a piano.

    Now if you want to have fun, look up the modes and come up with the step patterns for, say, E mixolydian, or D phrygian.

  10. And you didn't even broach the "church" modes! Nowadays we have minor and major, and forget there were once 12 or more. Careful though....Satan was worship leader before his pride got the better of him.

  11. The 12th root of 2 also determines the fret spacing and a visualization of compound interest. Take the open string length and multiply it by .059463 percent and get the fret spacing from the guitar nut to the first fret. Now take the remaining string length from the first fret to the bridge and multiply this length by .059463 and you will get the distance to the next fret. If we do this 12 times you will get at a note one octave higher or at twice the open string frequency, thus the 12th root of two. You will also be at a string length almost one half of the open string.

    Because strings have different diameters and stiffness, strings have a physical length and a speaking length. Thicker strings have more mass and start vibrating just a little forward of their resting place. That is why bridges on guitars seem slanted and longer on the bass strings side.

    Most investor advisers use the rule of 72 to estimate how long it takes to double your money based on the compound interest rate. Divide the interest rate into 72 and it takes 10 years to double your money when getting 7.2 percent. 6 percent would take 12 years. As you can see with the 12th root of 2 being very close to 6 percent, the guitar fret spacing getting closer with each increment visually represents compound interest otherwise each fret increment would be the same spacing and would take you longer to double your money.

    Now you can appreciate music and money in the same mental model.