I was talking with a musically gifted friend recently and this conversation has left a few burning questions on my mind.
In western music, students are taught from a young age that a dominant seventh (as in the V chord of the traditional chordal scale) wants to resolve to the I chord. I’m curious as to whether this has a physiological basis or whether this is something we have just been brought up to expect.
Also, I was wondering if there’s any physiological explanation for why chords are built on 3rds. My friend said it had something to do with more complicated intervals (ie 9ths, 11ths, 13ths) needing to be placed higher in pitch because they have a lower fundamental sympathy or something to that effect.
I realise Plato did some work with 3rds and 5ths and establishing basic chordal structures to some extent, I guess I’m just largely curious as to whether the underlying physiological mechanisms have been explained further in the past few millenia.
The simple answer for your question about thirds is that they are common ratio from the one note to the three note. A major third is the ratio 5/4. The science of acoustics demonstrates that this relationship of two tones will reinforce the tones rather than produce interference. In music this is consonance, as opposed to dissonance.
Even stronger consonances are the fifth (ratio 3/2) and the octave (2/1).
The more complex answer (and these threads always go more complex :)) is that we usually use an Even Tempered tuning, and the major third is something other than 5/4. Why does it still sound ok?.. I bail at this point!
It’s got to do with the ratio of frequencies. If you pluck a stretched string, it plays a note. If you halve the length of the string (doubling the frequency) it plays the same note, but one octave up. If you just move to 3/2 the original frequency, it’s up a fifth. Now, keep going up by fifths, and every time you’re over 2 (up the octave), divide by 2 to come down into the original octave. Also, go down by fifths (multiply by 2/3) and multiply by 2 when needed to come up into the octave.
1024/729 diminished fifth (tritone)
256/243 minor second
128/81 minor sixth
32/27 minor third
16/9 minor seventh
4/3 fourth
1 tonic
3/2 fifth
9/8 major second
27/16 major sixth
81/64 major third
243/128 major seventh
729/512 augmented fourth (tritone)
(apologies if I’m misremembering the terms)
Go look up “circle of fifths” for a better explanation than this.
note that 729/512 = 1.4238…
and that 1024/729 = 1.4046…
So the two are really pretty close, and usually get identified. They’re also pretty close to the square root of two, which is the ratio between the tonic and the true tritone.
Anyhow, obviously fifths will sound very nice since the ratio of frequencies uses such small numbers. Using a fifth and a fourth also keeps numbers small between the various tones, but the tonic-fifth chord is the same as the fourth-tonic chord – not very interesting. Now throw in the minor third instead of the fifth: there’s three different two-note chords, and all of them sound relatively nice on their own.
Of course this is more just about the math than what you asked, which I could now state as “why do low-number ratios of frequencies sound nicer?”
As for why the dominant seventh “wants” to resolve to the I chord (and I think you may already realise this, but just in case not) the V7 consists of the 5-7-9-11 notes from the scale, i.e. 5-7-2-4. The 7 is only a semitone below the root note, and the 4 is only a semitone above the 3. You already have the 5, so it’s only a slight shift from the V to the 1-3-5 of the I (discarding the 2 of course).
So I would say it does have some physical basis. Expectation, familiarity with the scale, they may play a part too.
This subject has cropped up before, such as hereand I’ve disputed the way it’s presented as being purely down to the physics of harmonics. It’s easy for us to refuse to observe how we’re surrounded with western musical systems from the day we’re born, and so absorb many artifices so deeply that the seem natural and obvious.
Dominant sevenths are an excellent example…we probably hear dozens, hundreds, or even thousands of them every day (depending on whether you’ve got the radio on the whole time ) But they’ve not always been present in European harmonic systems, and not all musical traditions use them.
As far as I’m aware, we’re still at the botany stage here. IT has been noted that most listeners find a resolution from the dominant 7th to the tonic to convey a sense of finality, but I don’t think it’s been conclusively demonstrated why. It may simply be a result of cultural conditioning.
Chords are built on 3rds today because over time, we have become used to closer intervals. The earliest written Gregorian Chants were meant to be sung in unison or at the octave. Later, the open fifth and fourth were considered acceptable. Anything closer was considered dissonant.
In later centuries, however, the third and sixth were deemed sufficiently melodious, and today we hardly blink at chords featuring seconds and 7ths.
As far as intervals and sympathetic tones are concerned, those only work in the sort of temperament described by the previous posters. That is not the temperament we use today, which is crafted so that all intervals work out the same regardless of the tonic key chosen. To divide an octave into 12 equal tones, the ratio between any two adjacent notes must be the twelfth root of 2, not the rational numbers described above.
The twelve-hundredth root of 2 (the number when multiplied by itself 1200 times yields 2) is called a cent. When intervals are sounded in equal twelve-tone temperament, the fifth is only one cent off from the ratio 3/2, the rational fifth. This is too close for us to distinguish a difference (IIRC, we need a difference of about 3-7 cents to notice a difference in pitch). The major third, however, is about 14 cents wider than what it would be if rational temperament were used. It does not sound out-of-tune to us, however, because we are used to hearing it.
However, because of these discrepancies, there are those who think that, given advanced technology, we should return to the “perfect temperament” of rational numbers.
I once heard a recording that Wendy Carlos did for Keyboard magazine, where she played a high, dissonant chord in both equal and perfect temperament. The perfect temperament chord yielded a lower note she was not playing, that ws produced by the differences in frequency of the higher tones.
The ancient greeks (not necessarily Plato specifically) layed the groundwork for our understanding of the mathematical relationships in music. However, the interval we now called the third was not sonorous to them, and the fifth may not have been either. I refer you to the section on music in Plato’s Republic.
Also, consider the tritone aspect. In Western harmony, the tritone (either an augmented 4th or a diminished 5th) is the least stable interval.
The Dominant 7th contains a tritone, which feels like it’s got to resolve somewhere. When it takes the form of a dim. 5th (B-F), our ears want it to contract into a Major 3rd (C-E). Conversely, when it’s an Aug. 4th (F-B), it wants to expand to a minor 6th (E-C). Both of these resolutions suggest a V7-I progression.
In fact, in the 4-part voice leading rules one learns in music school, it is expressly forbidden to resolve tritones in any other way.
Here’s a WAG - our pattern-seeking brains crave order, and the cycle of fifths has become (to our western ears) one of the commonest and easily recognizable music sequences there is. If you fail to resolve the V chord to the tonic (or at least come up with a pleasing alternative), it subconsciously offends our sense of order. I’m sure other cultures can recognize the cycle of fifths but may not attach the same significance to it than we do (obviously, failing to resolve the leading tone would have irritated Bach a lot more than it would Miles Davis or Ravi Shankar).
…and conversely, use of the leading note in the way familiar to us from the baroque & classical traditions would have sounded irritating, or just plain wrong, to many earlier generations of western composers.
The end of the second movement of P.D.Q. Bach’s Concerto for Horn and Hardart is a good example of an unresolved dominant seventh.
Doesn’t answer the question (I think it’s mostly universal-mathematical-physical, but reinforced by the direction that Western music has taken in the past millenium), but the laugh that the performance evokes in the audience demonstrates the point of how “funny” this sounds to us (in most contexts).
This doesn’t explain why unresolved conclusions in musics as far apart as Stravinsky and Ives don’t evoke the same response. Something other than the harmony makes dissonant conclusions in PDQ Bach sound funny, while in Stravinsky they sound like conclusive endings, and in Ives they can sound ironic.
Right here is the nugget of the answer to the OP. In other words, the system was designed that way.
It sounds to us like it’s “supposed” to be that way because we’ve all been trained to listen that way. The auditory nerves transmit sounds to the brain, but the brain has to interpret them into meaningful sounds, just as it has to reassemble impulses from the optic nerve into a field of vision. Likewise, I suspect that when we develop a higher-order method for arranging sound, such as Palestrinan harmony
In Euro-American culture we’ve been hearing it used that way since the 16th century when these rules were first set up. I believe Giovanni da Palestrina was the first composer to use the new system of harmony that is still used today, although it continues more in pop song than in art music. The Late Romantic period in European art music had already totally deconstructed Palestrina’s harmonic system by the first decade of the 20th century. Arnold Schönberg’s atonal Pierrot Lunaire was just an open acknowledgement that Palestrinan harmony was dead, within the context of the Romantic evolution toward expanding the rules of harmony every more complex until the whole system just dissolved. Wagner, Mahler, and Debussy are examples of the outer reaches the harmonic system reached before it flamed out.
Palestrinan harmony is now used worldwide. Listen to contemporary Chinese and Japanese pop music. How much of it is built on indigenous Asian music theory, and how much is built on Palestrinan harmony? Almost entirely the latter. Thou hast conquered, O Italian; the world has grown harmonized from thy breath.
Good point. In the particular P.D.Q. Bach piece I cited, the main melody in that part is some well-known motif – perhaps the slow movement of Mozart’s Eine Kleine Nachtmusic ? – so the listener is set up, by the musical context, to expect something that would resolve as Mozart and the like usually does. The examples you cite (Stravinsky and Ives) prompt me to reconsider how much weight I’d give to the physical-mathematical explanation. How close an interval is to a simple fraction tells us much about how we perceive the interval in isolation. “Tension” and “resolution” make sense if you consider, rather, a sequence of intervals, and is in in this that Ives, Stravinsky, and Shikele have come up with different solutions which provoke a response that depends rather more on the larger musical context as filtered by the listener’s previous musical exposure.
Sorry, I needed to finish that sentence. I meant that in this circumstance the brain has to do extra work to further organize the patterns within the auditory impulses into meaningful sounds. So this is learned perception, like learned behavior. It becomes engrained through habit. We’re all on this habit from birth, especially since music composed for very small children tends to restrict itself to the most basic patterns of Palestrinan harmony.
Sorry, but this is inaccurate, to say the least. The way European harmony functions has developed over the best part of a millennium, with Palestrina only being one stage in the process. There’s the first fully-composed vertical harmony, with the Notre Dame school of the 12th century. Complex rhythmic notation of the Ars Nova evolves alongside more complicated-yet-consistent use of melodic lines and of dissonance. The 15th century in particular lays the foundation for functional bass lines. All of these are essential elements of ‘western harmony’, ‘European harmony’, or whatever you want to call it.
I guess we’re actually arguing the same thing here - one person I know happily uses the term ‘indoctrination’ to describe how we acquire our familiarity with western music, and I find it hard to disagree.
) But they’ve not always been present in European harmonic systems, and not all musical traditions use them.
- one person I know happily uses the term ‘indoctrination’ to describe how we acquire our familiarity with western music, and I find it hard to disagree.