Why are a piano’s black keys arranged in groups of three and two?
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ELI5: That’s how music is shaped.
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ELI10: So you might know that sound is made of vibrations in the air. Western cultures (America, Europe) ended up liking sounds with certain math ratios of each other (like 2x, 1.5x etc.) After hundreds of years this settled into known patterns in music theory, forming things like “octaves” (2x frequency) and “scales” (series of notes in order). There are 8 notes in a scale including 1 repeating note before a pattern repeats, with the 8th one being one “octave” up from where the scale started. It just so happens that, again because of how culture and math worked out, there are actually 12 “half steps” between octaves because some notes sound better going the next half step and some sound better going the next whole step (2 half steps). The white keys are then organized so they sound best in a “default” setting, but the other half steps are still there for funky things. Try it yourself! Start on a low (left) white key and play only white keys in sequence up (move to right). It sounds kind of happy, right? That’s C Major scale. Or, start on any key and go the nearest note in sequence up (right), including black keys when they’re there. It sounds kind of mystical, right? That’s the chromatic scale.
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ELI18: So, although the pitch of a note is often represented by a single frequency, there are these things called overtones and harmonics that underlie practically every sound. The waveform of a given sound often looks far wigglier than a sine wave, and those wiggles within wiggles can be expressed as additions of multiple waves of certain mathematical ratios, with the most prominent overall frequency called the fundamental. The wavelengths are related by the harmonic series: 1, 1/2, 1/3, 1/4… The harmonics contained in a sound define its “timbre,” the texture of a note, and the reason a flute and a violin playing the same note sound noticeably different than each other, and why different people’s voices sound different. (If this waveform analysis interests you, you might Google “fast Fourier transform,” a common way of breaking down a sound into its fundamental and harmonics).
Despite differing timbre, the prominence of the first few overtones tend to be quite common, and thus causes notes with fundamentals that hit overtones of other fundamentals to resonate (being at the same frequency) with those overtones, which ends up defining quite a few of the notes found in music. In both Western traditions and the stereotypical Asian pentatonic scale, for example, you find what western music theory calls the “fifth”, which is the relation of the 1st and 2nd harmonics above the fundamental, or 1/3 vs 1/2 = 3/2. It forms something like the halfway point between octaves – if X hz is the fundamental, and 2*X is the next octave up, then 3/2*X is the fifth.
Western music ended up taking advantage of a few more partials than that, which by the Baroque period (1600 to 1750), in large part thanks to the prolific composer Johann Sebastian Bach, settled more or less into 2 archetypes, the major scale and minor scale. Within major and minor scales, 7 repeating notes can be defined, with an 8th corresponding to the 1st note (called the tonic), hence “octave.” These notes could roughly be defined as being a “whole step” or “half step” away from each other, with a whole step being roughly the distance between 7th and 8th partials. So despite 7 mixed whole- and half-steps (+ 8th to restart cycle) for scales, if measured only in half-steps, it would be 11 half-steps (+12th to restart cycle). The half-steps only scale is called the chromatic scale.
Note that I mentioned “roughly.” The ratios between notes end up “drifting” a little when key changes come into play; that is, if one splits an octave into 12 multiplicatively equal parts to form the half-steps of the chromatic scale, notes don’t quite match up with the harmonics noted above. For example, tonic X hz * 2^(7/12) represents the “fifth” on the scale so it sounds very close to X hz * 3/2, but it doesn’t line up exactly. If you’re interested in this difference, you can Google “equal temperament” (the system of 2^(n/12)) and/or “just temperament” (the system following harmonics like 3/2). In fact because of the challenge of tuning (temperament), JS Bach wrote the “well tempered clavier,” a set of pieces meant to be played based on each note of the chromatic scale (and thus hitting all the white and black keys as the tonics of their respective scales).
Further reading:.
* https://en.m.wikipedia.org/wiki/Harmonic_series_(music)
* https://dsp.stackexchange.com/questions/61908/i-dont-understand-harmonics-why-do-they-happen (waveform analysis)
* https://en.m.wikipedia.org/wiki/Equal_temperament
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