3 ms·
Some comments on tones. Pythagorean tuning is based on repeated ³⁄₂ increases in frequency with occasional halving to stay in the octave, so we have, e.g., A =
by dhosek 4y ago
Some comments on tones. Pythagorean tuning is based on repeated ³⁄₂ increases in frequency with occasional halving to stay in the octave, so we have, e.g.,
A = 1
E = 3/2
B = 9/8 (here we halved to get back into our 1–2 range)
F# = 27/16
C# = 81/64 (another halving)
etc.
Another approach is to use harmonic overtones. When a string (or a column of air) vibrates, it vibrates not just at its fundamental, but in a series of integer divisions of the string.¹
Fundamental: A
Octave (½): A' (up an octave)
Twelfth (⅓): E (up a fifth from the octave)
Double Octave (¼): A''
⅕: C#²
⅙: E
⅟₇: G (but a bit flat from most tunings).
We invert wavelength to get frequencies and halve to get into the 1–2 range and our E matches up at 3/2 and C# comes out as 5/4 which is pretty close to the 81/64 of Pythagorean
I would also note that 24-tone music does occur with some moderate frequency in avant-garde music where half-flats and half-sharps have their own notation, although these notes are not easily accessible from many standard instruments, but the sound of a quarter-tone difference in pitch is definitely distinct. Many non-Western musics apply various micro-tonalities, such as Indonesian scales which are closest to a subset of a 9-tone equal temperament.
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1. In some cases, e.g., overtones of a cylindrical pipe vs conical pipe, or open at both ends vs open at one end, you won’t get all of these tones, so a flute, which is cylindrical and open at both ends can hit the fundamental and the octave, while a clarinet, which is cylindrical but closed at one ends hits the fundamental and then the third partial (the twelfth) but not the octave.
2. The place where you hear notes produced to these pitches most commonly is in bugle calls: Taps, for example, would be ⅓ ⅓ ¼, ⅓ ¼ ⅕, etc.
- ajross 4y agoI've never liked explaining the scale as a Pythagorean derivation. It's not really correct historically (Greek music didn't have anything approximating a full major scale) or mathematically (it doesn't understand the idea of a "third" interval the way tonal music does, so playing triads with pythagorean tuning sounds awful!). Here's my take: late medieval singers discovered The Major Chord. That's the combination of three (!) notes that is "most consonant" (mathematically: beats in the shortest period). This combines two notes a major fifth apart (ratio 3:2), with a third note that is 5:4 with the low note. You can write some code to prove this if you like. So now take that "best" chord with its three notes, and start moving it around. If you go up a fifth (i.e. by "the most consonent interval", that is the "closest best chord to your first best chord") you can play the same chord, adding two needed notes that weren't in the scale before. You can likewise go down a fifth to add two new notes. Then you compress these seven notes into a single octave, and you get... the major scale! It's just there. All you need is that one "best" three-note chord and an obvious metric for "nearest" (i.e. transpose by a fifth) and you have almost all of modern tonal music. Play the same tunes starting on different notes and you get "modalities", etc... You can transpose up and down to nearby keys and keep playing by "cheating" with your tunings to move a note half way up or down. And the practice of formalizing those transpositional cheats because what we now know as the equitempered scale. But they're still just cheats. And the fact that pow(2, 1.0/12) happens to work is, basically, just dumb luck.