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,
I don't think however that this can satisfyingly explain why a dominant acts, well, dominant.
†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.
‡The 330 Hz figure is derived from just intonation, i.e. a Pythagorean fifth. JI is the conceptual underpinning of pretty much all Western harmony prior to atonality. Many instruments approximate JI with the 12-edo tuning system, in which the fifth is defined as 27⁄12 instead of 3⁄2; that comes out to almost exactly the same: the 12-edo E4 is 329.628 Hz, i.e. within 0.5 Hz of 330 Hz. This difference can pretty much only be noticed when superimposing both frequencies over a time of multiple seconds and listening to the beat, but not when listening to the tones in succession. Thus, for the question of whether the fifth lies in the middle of the octave, it is irrelevant whether we're talking about the JI or 12-edo fifth.
I'll just provide some historical context to this question, particularly why the "dominant" note first became prominent in scales. It has to do somewhat with the idea of a "halfway point," but it's a bit more involved.
Let's go back to ancient Greece and the Pythagoreans. They loved math, and the created the mathematics of proportions partly to create music theory. They also came up with several ways of measuring the average (sort of halfway point) between two numbers. The three primary ways are still known as the Pythagorean means.
Suppose you have a string that is 12 units long and divide it in half. If you pluck the 6-unit length and then the whole 12-unit string, they will sound an octave apart, with the longer string having twice the length and thus half the frequency. (The Greeks had no way of measuring frequency directly, so they could only deal in lengths of strings.)
Okay, let's say you want to find a halfway point or a "middle"/mean between these two notes and their string lengths. The Pythagorean means gave three possibilities:
With modern frequency ratios, the means work a bit backwards from what they do with string lengths. Thus, the arithmetic means of two frequencies an octave apart will locate the dominant, while the harmonic mean locates the perfect fourth above the bottom note. (The answer to the question in the title is therefore YES, the dominant is the arithmetic mean of frequencies.) Still, they essentially serve as two types of "means" (i.e., middles) between the end notes in an interval. The geometric mean of frequencies still creates a tritone.
Fast-forward about 1500 years. Medieval music theorists loved classifying scales for chants, and they recognized this structure established by the ancient Greeks building up fifths and fourths using the various means. Thus, octaves were classified using "species" that generally divided them up into fifths and fourths. The note a fifth or fourth above the final (the final being a kind of "home note") in chants was frequently given special importance. In earlier theory, it was sometimes given the name tenor (from the Latin tenere -- "to hold"), as it was a note that tended to be emphasized and held on to in contrast to the final note. Later, these notes eventually came to be known as dominant, and eventually in the 18th century they were split into two categories: the dominant (perfect fifth) and subdominant (perfect fourth above tonic). (I'm skipping over a lot of details here.)
By that point, the dominant note had also acquired harmonic functions too beyond its critical role in scale construction and melodies. In particular, it served as a stable harmony to leading tones, which had come into prominence for resolutions at cadences in medieval times. This is probably why the fifth overcame the fourth and became the "dominant" note in music theory.
However, the ultimate reason why that note was originally placed in scales and was given prominence does date back to the ancient concept of "means" and the various ways to locate a sort of halfway point within an octave when tuning intervals on strings.
(Note about tuning, as there are some comments recently about this: the question doesn't specify JI vs. ET, so I'm not even going to go down that road of discussion here. Historically, the dominant concept emerged by reference to JI. ET fifths are so close that they are practically indistinguishable in most circumstances.)
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.
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.
As for the numerical reasons, the other comments have answered.
As for the function of the dominant, as a triad from the scale degree V, the tones contribute to create a strong "leading" force to the tonic...
Example in the key of C:
Root of the dominant is G, which pulls toward the tonic, not due to the individual harmonic relationship, as the G is contained in the C tonic, and as noted by the above answers, is very consonant due to it's ratio. However, the pull is created by the expectation of music listeners, due to its historic and massively-present frequency in all of Western music. The note is pulled not due to instability, but due to the listener's expectation only.
Third of the dominant is B, which is very unstable, collapses on the tonic as it is the nearest note (B->C is a half step), and strongest dissonant-consonant relationship in proximity (Maj 7th -> Unison). [Note that the root of the vii chord also exhibits this relationship, but it is tempered by the function of the B as root of the vii chord.]
Fifth of the dominant is D, which is a peculiar case. The D is an equal distance between the C and the E, so resolution to either would require the same distance. However, when given the choice between a completely consonant resolution (D->C) and a partially consonant resolution (D->E), it requires less force, and resolves more dissonance in the D->C motion. Also, the symmetric movement to the B->C contributes to a less forceful resolution, adding to more consonance. Along with these functional reasons, the historic reason mentioned in the discussion of the root note also applies.
Given the above voice-leading and historic (expectation) reasons, the dominant chord continues to function (and strengthen) in its role as the precursor to resolution to the tonic.
The dominant is not "halfway" between the tonic and the octave in "equal temperament" tuning. Here, by "not halfway" I mean that the frequency of the dominant is not 3/2 times the frequency of the tonic (in "equal temperament" tuning). If it were truly "halfway" then the ratio of the dominant to the tonic frequency would be 3/2 and the "cycle of fifths" would never repeat (but it does repeat after 12 fifths: C, G, D, A, E, B, F#, C#, G#, D#, A#, F, C).
In modern western music the "equal temperament" choice was made that the "half step" interval was to be obtained from the note below by multiplying the frequency below by a factor of 2 raised to the power (1/12). Thus after 12 half steps the octave is reached. Since the dominant is 7 half steps above the tonic, the ratio of the dominant frequency to the tonic frequency is 2 raised to the power (7/12). Note that 2 raised to the power (7/12) is 1.498307..., which is approximately 3/2, but not exactly.
For example, A3 = 220Hz, E4 = 329.6275561...Hz, A4 = 440Hz.
For more details, see the paper here.
Note that the exactness of the numbers given (e.g., 329.6275561...Hz given to 7 decimal places) does not really apply to any physical instrument, since there will always be tuning errors for any physical instrument. For example, an accredited piano tuner must be able to tune a note to within one hundredth of a half step of the tuning fork (A.K.A. "one cent," which is approximately 0.075 Hz at C below middle C and approximately 0.185Hz at E below concert A). While the difference we are arguing about is fairly small it is still about two cents, so a piano tuner should be able to tell the difference--Unfortunately, I don't know any piano tuners I can ask about this, but would be interested to hear from them.
Furthermore, any physical instrument produces a spectrum of frequencies and the peak at the fundamental frequency has an instrument- and environment-dependent width that further complicates any ultra-precise tuning to "one frequency."
These additional comments I added mostly to point out that I'm not addressing physical instruments, nor realistic tuning scenarios, and I've not said anything about how the ear perceives notes (which since the ears is also a physical analog object will also depend on a further full-spectrum response function of the ear).
Rather, my comments apply to the mathematical theory only. And my point is nothing more than the fact that:
Thus the "circle of fifths" is incompatible with the fifth being 3/2 times the tonic, in general.
This in not really a problem in practice because of the properties of physical objects (some of which are discussed above). But, nevertheless, for a mathematical theory of music this is an annoyance and it is annoying enough for me to prefer to say that the fifth is not 3/2 times the tonic. So, in this context, I would not say that the fifth is "halfway" between the tonic and the octave. Or, to put it another way, I would not say that every fifth is exactly halfway between the tonic and the octave.
Not sure exactly what do you mean by halfway in frequency, but yes you can expect a dominant 7 to have a ~150% frequency of the tonic.
On the other hand, an octave above the tonic is ~200% of the original tonic frequency.
This relationship is also precisely the reason why an octave or dominant 7 sounds so consonant because the sound waves for both notes fit predictably.
As to the tone that is exactly halfway from the tonic to its next octave (in terms of tone not frequency), it would be the tritone. Both are 6 intervals away from each other.
The reason why tritones, although they have the same intervals but don't have the same frequency between the tonic and octave, was due to the non-linearity of growth in wavelengths. Remember the frequency doubles at every octave, the lower you are at the key, the less increase in the frequency of the next.
It would help if you defined what you meant be "half way". In terms of linear distance we can generalize one of the answers.
f_1/2 = (f1 + f2)/2 = (f1 + 2*f1)/2 = 3*f1/2
This matches the definition of the V in Just tuning. This is also commensurate with one of the natural harmonics of a linear vibrating string or resonance tube with fixed node at the ends as boundary conditions. The n = 3 harmonic is an octave and a 5th above the fundamental. It is this fact that may have a stronger influence on the importance of the V in western music. The notes in a "perfect" interval share more harmonics than any other and this is what contributes to the perception of dissonance versus consonance from a physic based analysis. This was determined by Herman Helmholtz in the late 1800s and written about in his text "On the Sensation of Tone" which is still available. Beware that he discusses the variation of the speed of light relative to moving observers (Einstein hadn't produced his work at this point in time). According to the theory (a water down version) when two notes are played together and share overtones these overtones support each other and do not conflict. When the overtones do not match we hear a "beating" between them. Other phenomenon contribute to this. One is the ability to distinguish two pitches versus hearing a combined pitch with beats. Intervals that are judged dissonant have overtones that not only do not match but are close enough to create this interference. It is postulated that this is what causes dissonance. The degree of dissonance is to some extent related to the density of non-matching overtones that fall within a "critical band" for pitch discrimination. You might think that this can be removed by removing harmonics but you cannot to this in theory or practice because the human ear is non-linear and creates harmonics when excited. So, in some sense we have evolved to hear and distinguish these two characters, dissonance and consonance. I cannot say what evolutionary benefit it serves.
In my honest opinion the theory is somewhat flawed. The physics is indisputably true but the attempt to use physics to decide once and for all what is judged as pleasant versus unpleasant is infused with ethnocentric bias. We knew the answer we wanted before the analysis was applied.
On to other definitions of "half way". With the development of 12TET tuning the perfect 5th, 3/2, disappeared and is replaced with a measure of 7 half steps, f = (2)^(7/12)*f1. This is close enough to 3/2 to serve its purpose (1.5 compared to ~1.498...). Whether or not this can be heard depends on the sensitivity of the human system. Some people claim to be able to hear this. As for resonances in musical instruments and beating inside the human ear? Well, all systems have damping and that broadens the response curve. So it is possible for 1.498 to trigger all the same sympathetic resonance as 1.5 and I know very little about the human ear other than a few basic facts so I do not know if it and/or the brain is sensitive enough to hear the diff. However it is not an even division of the octave into steps. 12 1/2 steps make an octave and the 5th is at 7. The flat 5 is at 6, a perfect split of the octave in to equal ratios (we typically describe frequency relations as ratios and not diffs). This has earned the dim 5th the title of the perfectly symmetric interval in some circles. It was also banned by the church for a while.
To get to an answer to your original question perhaps no one is qualified to answer. Whatever music is now has developed over 1000s of years and your question is kind of a chicken versus egg question. Is this relationship what makes the V functionally important? To some extent perhaps it does. Perhaps the fact that we are programmed to hear dissonance and consonance and even create it in our own ears and that vibrating systems have a naturally occurring harmonic at what we now call the V is significant. Perhaps that is why we give so much significance to V. However not all cultures do this. For your question to be fair I think you should investigate eastern music, African, and other cultural music and see if the same choices have been made. It might that from a global perspective the importance of the V is a red herring even if there are physics based reasons to favor it.