The Perfection of Bach

This article is part of a series on Baroque Music. See also:

  1. Structure: The Perfection of Bach
  2. Style: Baroque Style Sampler
  3. Performance: Heaven’s Gates

What is music, and how does it assume beauty and structure? The question is difficult, yet awesome – and its answers appear, to me, above all else in the baroque music of Johann Sebastian Bach (1) (2) (3). Bach’s music attains a virtually geometric perfection, dividing time into cascading blocks which each, within themselves, convey struggles, twists, and resolutions – all in perfect harmony. The result evokes a series of late-Renaissance paintings unfolding in real-time.

Music, then, has structure, and it’s of a unique kind. Notes take on meaning in short sequences; these phrases then, in turn, build more complex and nuanced passages. The product is the creation and development of a theme; this theme is introduced and progressively accumulates additional flavor, conflict, and subtlety. In Baroque tradition, it will always end in resolvement.

I will explore a couple themes intimately related to structure in music.


Your humble author has retained, in memory, large segments of almost every piece he has played throughout his lifetime. (At age 7, I played the entirety of Bach’s “Jesu Joy of Man’s Desiring” (4) by memory on the recorder in front of my first grade music class.) How is music memorized? It seems clear that randomized sequences of notes would be much more difficult to memorize than “real music”; this result is indeed confirmed in studies (5). Our answer lies here: the brain, in memorizing music, is not merely memorizing the individual tones in succession. The brain memorizes macro-scale themes and structures. We’ve made progress.

We might draw an analogy to general psychological tests of memory, which demonstrate that “sensory” memory (i.e., segments of purely sensory input such as sound or image) persists only in the short term, while “semantic” memory (i.e. memory of meaning) lasts much longer. It seems that we can conclude that musical memory is semantic in character.

In the same way that the brain finds meaning in places such as literature and language, the brain too identifies meaning in music. Though the nature of this meaning seems unique to music, it’s (apparently) similar enough – at a deep level, that is – for our brains to encode it into semantic memory. Music has strong structure and our brain employs this during memorization.

Music Theory

Why do our brains like music? Music theory describes chords – more precisely, progressions of them and relationships between them – which, in implementation, form music which strikes us as ideal. I claim that classical music, and Baroque music in particular, has “mastered” these chord progressions: composers like Bach create music which is, almost literally, mathematically perfect.

These chords strike our brain as beautiful. Why? Clues might be found in the ratios between pitches: common musical intervals feature mathematically salient ratios between their frequencies. Two notes an “octave” apart are typically said to be the same; more precisely, the muscial scale operates on a “cycle” which repeats every octave. The ratio between octaves? Exactly double. The “perfect fifth”, meanwhile, features a ratio of 3:2. Even more interesting: the ubiquitous major and minor thirds feature ratios of 4:5 and 5:6, respectively. The perfect ratios of these intervals undoubtedly influence our brains’ subjective enjoyment of them.

But why subjective enjoyment – why do we convert these perfect intervals into emotion? This question is even more difficult, and carries significant philosophical depth. I only offer one striking fact: Bharucha (2010) showed that the minor third is replicated in sad speech, mimicking its use in music! (6) The emotional power of music is truly deeply ingrained.

As a closing word, I urge the reader to listen to the links below – and to work to detect the structure. Listen not to each note, but to each phrase, and to each sequence of phrases; broaden your scope, as it were, to the extent that you can. I hope you enjoy.

  1. A playlist containing Bach’s outstanding Brandenburg Concertos.
  2. Bach’s unaccompanied Sonatas and Partitas performed Rachel Podger CD1, CD2
  3. The famous “Concerto for Two Violins” or “Bach Double”.
  4. Jesu Joy of Man’s Desiring
  5. Sloboda, J. A. (1985). The musical mind: The cognitive psychology of music. Oxford, England: Oxford University Press.
  6. Curtis ME, Bharucha JJ (June 2010). “The minor third communicates sadness in speech, mirroring its use in music”. Emotion 10 (3): 335–48.

15 comments on “The Perfection of Bach

  1. Josh says:

    I think it was always clear that the brain likes music for its mathematical perfection; the question is, why is that perfection enjoyable? You acknowledged this question, but stopped just short of answering it. What I think is that music kind of “tickles” the part of the brain that searches for patterns. We do this naturally, without even thinking about it; think about looking up at the clouds and seeing shapes. The brain enjoys making order out of chaos, and music, especially highly “mathematical” music, gives us a great chance to do so. Now, why is 4:5 happy, and 5:6 sad? I don’t have an answer for that one at all.

    Similarly: you mention that music must be encoded as semantic memory. I definitely agree, and this is a good point. However, I think the intrigue goes further. Take language, for example: language is also encoded as semantic memory. It’s much easier for me to remember a logically-constructed English sentence than a Russian sentence, since in English I’m remembering words and ideas, as opposed to the shapes of the characters. The key difference is that, in the case of language, it took many years for me to gain such a grasp of English that a sentence can be easily memorized. With music, the other hand, I’ll find myself whistling along to a song that I just heard for the first time! Sure, you could argue that I might have heard similar songs in the past, thus giving me the necessary training. I argue, however, that even someone completely new to music could memorize music much faster than he could memorize the sounds or shapes of language. The question remains: why is music so immediately learnable, while language is not? It must come back to the brain’s eagerness to seek out and learn geometric patterns. It seems that everyone is born speaking the language of music.

    • Ben says:

      I fully agree with all of your points in the first paragraph.

      In the second paragraph, you claim that music is intrinsically more easily memorized than language. I partially disagree. It seems clear that the ability to memorize music is closely related to musical ability — acquaintance with how music is used to construct emotions and ideas — and this skill is surely not one instantly acquired.

      Contrarily, however, we may note that musical ability itself would surely be slower to develop if “music” were not geometrically perfect in the way that it is. More explicitly: if we imagined an alternate world in which “music” were discordant and random, this musical ability we speak of would be slower in developing. Therefore memorization of music seems influenced by both intrinsic and learned factors.

      Then again, similar arguments have been made regarding language — Chomsky famously argued that language is “hardwired”.

      • Josh says:

        Well, my argument was that, to someone who’s heard neither music nor language, music would be easier to learn. This isn’t to say that his ability to learn music wouldn’t improve with experience–it would. As would his ability to learn language. I’m just saying, at baseline, music seems like it would be more learnable. Of course, I’m just speculating here; I obviously have no data to cite.

        You mention that “amusical” music would be harder to learn. Well, of course! The whole point of my argument is that music is easily learnable BECAUSE of its geometric regularity. Take that away, and it becomes like me trying to memorize an Arabic newspaper. I’m just saying that, given that music IS rich with patterns, my ability to learn it is hardwired–perhaps even more so than my ability to learn language.

  2. susan says:

    bach had a “secret” composition method.
    to find it, apply group theory to his work.
    it appears he knew many of the theorems of permutation groups—long before lagrange.

    makes him even more of a demigod.

    • Ben says:

      Do you know how this might be done? It’s a very curious idea, to be sure.

      Of course, the octahedral group is isomorphic to the symmetric-4 group. Maybe so are the Brandenburg Concertos? Hah.

  3. susan says:

    //At age 7, I played the entirety of Bach’s “Jesu Joy of Man’s Desiring” (4), by memory, on the recorder in front of my first grade music class.//

    did you use the trapp recorder book?

    that, and e.t. bell —i almost feel like i know you.

  4. susan says:

    josh, you might enjoy leonard berstein’s harvard lectures, where he applies chomsky to music.
    try youtube.

    if you haven’t seen them already.

    off to eat, do math….

    stuffed cabbage, and pde.
    washed down with some bach and a small glass of chianti.

    then some poetry.

    anyone for tennyson…….?

  5. Ben says:

    I’m a fan of Petrarch’s poetry.

  6. Richard says:

    I’m deeply sympathetic with your description of Bach as ‘perfect’ and when you say “classical music, and Baroque music in particular, has “mastered” these chord progressions: composers like Bach create music which is, almost literally, mathematically perfect.” I’m also a die-hard Baroque fan, though I like classical music in general too, obviously. A dog’s whines in minor keys too. On the point about music having ‘semantic’ structure, the German philosopher Aurthur Schopenhauer said that music is the purest art form, because it conveys its message or theme without reference to concrete events or topics. No language is used to talk about specifics, no pictures represent familiar scenes; it is a purely abstract art form, beautiful in virtue of structure alone. The connection of key to the tone of voice might threaten this to a point, but there is much more to the structural elements which make fine music so beautiful than just the chords and chord progressions. Melodic structure is also essential, especially with someone like Bach.

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