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Is the dominant tone of a major scale halfway in frequency between the tonic and octave?

If so is this why it functions as it does? ... because it's sounds less dissonant to the tonic?

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Yes, that is true. For instance,

  • A3: 220 Hz
  • A4: 440 Hz
  • E4: 330 Hz ≡ (220 Hz + 440 Hz)/2

I don't think however that this can satisfyingly explain why a dominant acts, well, dominant.

  • The Ⅴ tone itself is actually contained in the tonic triad as well, and there it clearly does not evoke any urge to resolve it to the Ⅰ or somewhere else. In fact I daresay because it is so consonant, it does not act as a dominant...
  • ...unless you add extra leading tones. A dominant chord contains at least a (high-intonated) third, which leads up to the tonic's octave. It very often also contains the 7 (sometimes “dominant chord” and “dominant-seventh chord” are considered synonyms!), and that's a dissonance which strongly suggests resolving it.
  • That effect still works when the Ⅴ tone is not even part of the chord, as exemplified by diminished seventh chords.

At least in an analytic sense. As Dave remarks, our ears actually perceive pitch rather logarithmically than linearly, and on a logarithmic scale the Ⅴ is further that half-way between the Ⅰ and its octave.

  • It just seems to be that when I strike a I and then V they are simply not dissonant whereas the I & II are far more. WWWH – Randy Zeitman Nov 14 '17 at 1:17
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    @RandyZeitman That is because a major second is a much more complicated relationship than the simple fifth. Because of (I'm assuming) wave cancellations, two notes (say C and D) generate a beat ( especially audible for a minor second), which is usually interpreted as dissonance. – Ye Dawg Nov 14 '17 at 15:04
  • @leftaroundabout- what do you mean by a "high-intonated" third? I find the resolution of a just dominant chord to its tonic- say 12/15/18 to 8/16/20, to be just as convincing, and with less cringeworthy dissonance. But maybe that's just me.... – Scott Wallace Nov 14 '17 at 15:09
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In one way you could say that a perfect fifth is halfway between root and octave. The relationship in frequency is 3:2, which is equivalent to 1.5:1. At first glance this looks like halfway. This is a very simple and consonant interval. But it doesn't quite hold. Two perfect fifths is not an octave, but a ninth! The true halfway interval is the tritone/augmented fourth/diminished fifth as two of them stacked equal an octave. Mathematically, in equal temperament the relationship of a tritone is 2^(1/2):1, or approximately 1.415:1 . Here you already see that by taking the square of it you get 2, which is the frequency relationship of an octave. No, it's not that beautiful sounding, at least not in western culture. It is what it is.

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    The OP asked specifically about the interval "halfway in frequency," not the "halfway interval". – Alex Basson Nov 14 '17 at 22:09
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There is a sense in which the equal tempered (ET) augmented 4th or diminished 5th is the "middle" of the octave. Note how on a piano keyboard if you move up 6 semitones, and then move up 6 more, you'll get to the key that is an octave above your starting point. This has to do with the fact that we perceive pitch differences logarithmically. In a perceptual sense, the relationship between A2 and D#2 is the same as the relationship between D#2 and A3 (in equal temperament), while the the interval from A2 to E2 is perceptually distinct from the interval between E2 and A3.

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