The Math Behind Beethoven’s Music

Almost all the biggest math enthu­si­asts I’ve known have also loved clas­si­cal music, espe­cial­ly the work of Bach, Mozart, and Beethoven. Of course, as San Fran­cis­co Sym­pho­ny music direc­tor Michael Tilson Thomas once put it, you can’t have those three as your favorite com­posers, because “they sim­ply define what music is.” But don’t tell that to the math­e­mat­i­cal­ly mind­ed, on whom all of them, espe­cial­ly Bach and Beethoven, have always exert­ed a strong pull.

But why? Do their musi­cal com­po­si­tions have some under­ly­ing quan­ti­ta­tive appeal? And by the way, “how is it that Beethoven, who is cel­e­brat­ed as one of the most sig­nif­i­cant com­posers of all time, wrote many of his most beloved songs while going deaf?” The ques­tion comes from a TED-Ed seg­ment and its accom­pa­ny­ing blog post by Natalya St. Clair which explains, using the exam­ple of the “Moon­light Sonata,” what the for­mi­da­ble com­pos­er did it using math. (You might also want to see St. Clair’s oth­er vides: The Unex­pect­ed Math Behind Van Gogh’s “Star­ry Night.”)

beethoven music gif

“The stan­dard piano octave con­sists of 13 keys, each sep­a­rat­ed by a half step,” St. Clair writes. “A stan­dard major or minor scale uses 8 of these keys with 5 whole step inter­vals and 2 half step ones.” So far, so good. “The first half of mea­sure 50 of ‘Moon­light Sonata’ con­sists of three notes in D major, sep­a­rat­ed by inter­vals called thirds that skip over the next note in the scale. By stack­ing the first, third, and fifth notes — D, F sharp, and A — we get a har­mon­ic pat­tern known as a tri­ad.” These three fre­quen­cies togeth­er cre­ate “ ‘con­so­nance,’ which sounds nat­u­ral­ly pleas­ant to our ears. Exam­in­ing Beethoven’s use of both con­so­nance and dis­so­nance can help us begin to under­stand how he added the unquan­tifi­able ele­ments of emo­tion and cre­ativ­i­ty to the cer­tain­ty of math­e­mat­ics.”

Explained in words, Beethoven’s use of math­e­mat­ics in his music may or may not seem easy to under­stand. But it all gets clear­er and much more vivid when you watch the TED-Ed video about it, which brings togeth­er visu­als of the piano key­board, the musi­cal score, and even the rel­e­vant geo­met­ric dia­grams and sine waves. Nor does it miss the oppor­tu­ni­ty to use music itself, break­ing it down into its con­stituent sounds and build­ing it back up again into the “Moon­light Sonata” we know and love — and can now, hav­ing learned a lit­tle more about what math­e­mati­cian James Sylvester called the “music of the rea­son” under­ly­ing the “math­e­mat­ics of the sense,” appre­ci­ate a lit­tle more deeply.

Relat­ed Con­tent:

Stream the Com­plete Works of Bach & Beethoven: 250 Free Hours of Music

Beethoven’s 5th: The Ani­mat­ed Score

Leonard Bern­stein Con­ducts Beethoven’s 9th in a Clas­sic 1979 Per­for­mance

Beethoven’s Ode to Joy Played With 167 Theremins Placed Inside Matryosh­ka Dolls in Japan

Man Hauls a Piano Up a Moun­tain in Thai­land and Plays Beethoven for Injured Ele­phants

Slavoj Žižek Exam­ines the Per­verse Ide­ol­o­gy of Beethoven’s Ode to Joy

Oliv­er Sacks’ Last Tweet Shows Beethoven’s “Ode to Joy” Mov­ing­ly Flash­mobbed in Spain

Does Math Objec­tive­ly Exist, or Is It a Human Cre­ation? A New PBS Video Explores a Time­less Ques­tion

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities, lan­guage, and style. He’s at work on a book about Los Ange­les, A Los Ange­les Primer, the video series The City in Cin­e­ma, the crowd­fund­ed jour­nal­ism project Where Is the City of the Future?, and the Los Ange­les Review of Books’ Korea Blog. Fol­low him on Twit­ter at @colinmarshall or on Face­book.

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Comments (6)
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  • MTB says:

    That was a ter­ri­ble video that explained noth­ing of the genius of Beethoven and only loose­ly explained har­mon­ic prin­ci­ples that apply to EVERY com­pos­er. Also, an octave con­sists of 12 chro­mat­ic notes and major/minor scales use 7 notes. The out­er notes they use in the C–C exam­ples are the same.

  • Alan Drabke says:

    Physics teach­ers and physics researchers befud­dle their stu­dents with the cliché: “Some state are for­bid­den.” Or words to that effect. As Sir James Jeans explains in the pages of ‘Sci­ence and Music’ (1937) Pages: 65–69

    Vibrat­ing strings, strings in res­o­nance can only assume whole num­ber mul­ti­ples of their fun­da­men­tal fre­quen­cy.

    And so it is with atoms and mol­e­cules. A fact of nature unknown to every physics pro­fes­sor in the world. Jean’s expla­na­tion could be a part of every high school and col­lege physics text book. It could be. …

  • Slartibartfarst says:

    Many thanks for this post and its links.
    Huge­ly infor­ma­tive.
    To com­menter @Alan Drabke: thanky­ou so much for the link to the “Sci­ence and Music” by Sir James Jeans — is a real bonus.

  • Chuan Chang says:

    I explain how the begin­ning of LVB’s 5th sym­pho­ny, when trans­lat­ed into math, reads like the first chap­ter of a book on group the­o­ry; the sym­pho­ny was writ­ten before math­e­mati­cians woke up to the con­cept of group the­o­ry; see sec­tion (67) of my book:

    The Moon­light is an inven­tion by LVB in which he super­posed har­mo­ny with dis­so­nance to bring out the beau­ty of har­mo­ny, and the pain of dis­so­nance, sec­tion (49)

    The Appas­sion­a­ta is a piano ver­sion of his 5th sym­pho­ny, sec­tion (51)

  • Nitay Arbel says:

    The first com­pos­er to real­ize that musi­cal con­so­nance derives from shared over­tone fre­quen­cies was Jean-Philippe Rameau, in his “Trea­tise on Har­mo­ny” (Traité de l’Har­monie).

    Jeans is good but not per­fect. His state­ment that dif­fer­ent keys can­not pos­si­bly have dif­fer­ent tonal char­ac­ter­is­tics, for exam­ple, clear­ly does not apply to those of us with absolute pitch.

    Anoth­er (old­er) book, which is heavy going at times because of all the math, is Helmholtz’s clas­sic: “On the sen­sa­tions of tone”.

    Physics (and, for that mat­ter, applied math) and music have been inter­twined for a long time. In fact, the first per­son in the West to actu­al­ly apply a ver­sion of equal tem­pera­ment in instru­ment build­ing was the lute mak­er Vin­cen­zo Galilei — the father of Galileo!

  • BW Acuff says:

    This is a com­mon mis­take of — for want of a bet­ter word — a type of “pre­sen­tism”. Only instead of judg­ing his­to­ry by mod­ern stan­dards, peo­ple make the mis­take of think­ing com­posers build music math­e­mat­i­cal­ly.

    The mis­take comes from how their music can be decon­struct­ed math­e­mat­i­cal­ly — which can be done with most every­thing human. The rea­son it’s done so fre­quent­ly with music is because of our love for music and how sim­ple the lan­guage actu­al­ly is: the love of music pro­vides the moti­va­tion and the sim­plic­i­ty makes the decon­struc­tion log­i­cal, neat and tidy.

    But it’s still a mis­take. I’ve nev­er used any math, ever, while writ­ing a piece. Nor have I ever known some­one who did although there are a few ‘per­fo­mance art’ com­posers who have done it on pur­pose. That work how­ev­er, tends to be unlis­ten­able from a pure­ly enjoy­ment stand­point.

    And being deaf was­n’t the hand­i­cap non-com­posers think it is. It was a chal­lenge but most­ly for rea­sons the aver­age per­son would­n’t think of: tem­pos and orches­tra­tion. Most com­posers can write with­out using an instru­ment — they’re experts on the sub­ject, after all: that’s why they’re com­posers — and they know the notes they write will sound how they intend­ed them to, but dial­ing in the cor­rect tem­pos and tim­bres can be a chal­lenge.

    It’s because when you’re writ­ing in your head, you’re using a per­fect orches­tra and you under­stand the emo­tions you’re attempt­ing to con­vay and where in the score you’re attempt­ing to con­vey them. So the tem­po is irrel­e­vant to you: it works in your head per­fect­ly, at any tem­po.

    Think of a nov­el­ist who’s try­ing to explain a sto­ry they already know back­wards and for­wards in their head. Have you ever read a book where you feel lost at some point? The author was­n’t. He or she just for­got to give you enough infor­ma­tion to keep you informed of what they were think­ing.

    And tim­bre is very much like tem­po: much eas­i­er to mess up than the actu­al notes.

    That’s not an attempt to dis­miss the bril­liant accom­plishe­ments of Beethoven’s work but mere­ly an attempt to put it in per­spec­tive.

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