I'm a self-taught composer currently trying to make my way through Arnold Schoenberg's "Theory of Harmony." I'm unfamiliar with what the numbers super-scripting certain notes means- is it the octave?

Schoenberg "Theory of Harmony" excerpt

  • This seems to refer to the natural overtones of a vibrating system, but some seem to be off. The exponent might refer to the number of octaves above the lowest overtone in the sequence. – ggcg Dec 27 '18 at 2:17
  • What does the "Klang" refer to? The sound of a bell or cymbal? – ggcg Dec 27 '18 at 2:17
  • Klang just means "sound" in German; Schoenberg's use here is nondescript. – Richard Dec 27 '18 at 4:42
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    In the linked duplicate, see especially the discussion of Helmholtz notation, which is what Schoenberg is using here. – Richard Dec 27 '18 at 4:45

Schoenberg is (as mentioned above) writing down the harmonics of a tone. "Klang" is German for "tone" in English (but I will resist the obvious jokes).

The harmonics of an ideal string (as opposed to a real string which has weight and tension) have the ratios, 1,2,3,4,5.... times the fundamental. For a fundamental of "c" the next note would have a ratio, of 2/1 (c1)[I don't know how to make sub and superscript on this board] the string vibrates a half-length. Then next is g1 (vibrating at 3x the fundmental) then c2 (4) then e2 (5) then g3 (6) and now b3 (at 7:1, this is really out of tune according the the "just scale."

It seems that Schoenberg isn't exactly correct in his list. (As noted above).

  • The true harmonics do not perfectly fit the just tuning ratios. I've never seen a music theory book get that correct. None of them fit the equal tempered scale by definition. Perhaps damping brings them close enough for a response. – ggcg Dec 27 '18 at 11:25
  • The Just Scale is the best one can do with numbers divisible by 2,3,or 5 only. Natural harmonics (in a simple harmonic vibrator) go through all the numbers. A type of 7-limit tuning is used in (Highland) bagpipes and by barbershop quartets. – ttw Dec 27 '18 at 14:28
  • Will you ever get 4/3? Nope. My comment was simply that the "harmonics" listed in terms of intervals are not entirely correct. The 1, 3, 5, and one of the sevenths and seconds are in there, but not 4th and a couple others. Since the major triad is present in the natural harmonics and the I is the 5th of the IV building the scale from arpeggios does make sense. But to say that string x will generate the fourth in its spectrum is simply false. – ggcg Dec 27 '18 at 14:44
  • True that 4/3 (and other) ratios do not occur in the harmonic series. However, the "speculative" music theorists also allow in inversion by the octave so the fourth is the inverse of the fifth, etc. Going a fourth up is octave equivalent to being a fifth down. – ttw Dec 27 '18 at 21:06
  • That has no bearing on the harmonics present. You are misunderstanding something or beating a dead horse. The statement made is that the natural vibration will include the 4th and this is false. It will not be true in theory or practice. – ggcg Dec 27 '18 at 21:17

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