Beatsong wrote:

Mr.Samsa wrote:The only other alternative is to have lots of kids and raise them how I want...

Don't bother. They'll just end up listening to Beyonce anyway.

If that ever happens then I'm taking my wife down to the abortion clinic..

"So Mr.Samsa, how far along is the baby?"
"Oh.. approximately 168 months. We thought we wanted to keep it, but we just discovered a horrible genetic defect that causes it to enjoy Beyonce songs. Just put it out of it's misery now, please..."

Sorry about the very late bump, but I wanted to read through the whole conversation and it has developed too fast for me to do it.

Beatsong wrote:Absolutely. The tierce de Picardie was a very common device throughout the Renaissance and baroque periods, because composers intuitively felt that the minor chord I of a minor key, while OK for most purposes, just didn't sound sonorous or stable enough to finish a movement or a main section. The opposite (minor chord I closing in a major key) NEVER happens, because there's simply no reason why it would.

But what I actually meant by "final movement" was in piano sonatas, string quartets, symphonies etc where the piece as a whole has several (usually four) "movements". These being played one after the other with a few seconds' break between, but still considered part of the same "piece". Often titled by their tempo markings: "1. Allegro Moderato; 2. Adagio" etc.

There are many cases of such pieces where the overall piece is called "Quartet in D minor" or "Symphony in C minor" or whatever, but the last movement is actually in the tonic major key. The most famous example off the top of my head is Beethoven's Fifth Symphony, with the famous C minor opening, the second and third movements also in minor, but then that huge triumphant finale in C major. This pattern is like the tierce de Picardie idea on a larger scale: instead of just the stability of a final major chord, the composer is increasing the stability of the piece as a whole, via a final major movement.

OK, now that is clear, I know what a movement is and I should have known that you knew it too. I couldn't find an example off the top of my head, so I wondered whether you had meant something else.
By the way, I thought of another instance of the same pattern : there can be a major coda to a minor movement (e.g. Brahms' 1st cello sonata, 1st movement, Schubert's piano sonata D960, 2nd movement, probably many other examples), but hardly the other way round. Just underlining the fractal properties of music.

Once again, I can't think of a single example where the opposite happens. It's the kind of thing you'd imagine one of those deranged Romantics would have thought of - a final movement changing from major to minor just to make sure you go home and slit your wrists. But even among them I can't think of an example.

Interesting challenge, I tried but didn't succeed either. I would expect some cases to exists though.

You've got exactly the right idea but actually it does go further than you thought. leaving aside transitional modulations and looking at the main overall, structural key centres, baroque and classical pieces that start in a major key almost always modulate to the dominant - also major - at the main articulated section point about a third or half way through, and then return to the tonic. Similar pieces in minor keys however, usually modulate to the relative major at the same point, before returning to the tonic.

So the basic tonal "architecture" of major key music is through the fifth above (the dominant). This is because of the relationship of the fifth being so fundamental to the harmonic series, which is another story. But in minor keys, the latent tonicity of the relative major is so strong that it overrides this architecture. It's like the minor key is so inherently unstable that right from the beginning of the piece, it simply has to "fall into" its relative major, which is waiting with its superior harmonic stability to step in to its inevitable role. Talking of large and small scale examples of the same phenomena, this is Rameau's theory of "co-generation" - the sense that the third of a minor chord (the tonic of its relative major) is actually a kind of "secondary tonic" latent in the chord - in practice.

Like I said I simply can't accept that such a consistent a far-reaching inequality in the relationship between the two chords and keys is only "symbolic", only about people having "learnt" to associate them with particular feelings. It tallies so perfectly with the difference between their acoustical propoerties vis a vis the harmonic series, and runs with remarkable consistency through the works of every baroque and classical composer. (And I only don't automatically include Romantic music in most of this because the huge increase in chromaticism and use of far away keys renders everything much less prone to valid generalisation). The major key CAN modulate to its relative minor for expressive purposes if it wants to, but doesn't have to. The minor OTOH has to modulate to its relative major, because the source of tonal stability that actually resides there, not in the minor tonic, inevitably pulls it that way.

These are the kinds of factors I was referring to in my post above to Mr Samsa, but I wasn't sure how far to go into the technical stuff as I realise most people here probably won't be into it to that level. But since you asked...

Very interesting, and no problem with following you here. I play the piano, know tons of classical music more or less by heart, but have skipped the theoretical cursus as much as I could because I thought it sucked.

As a general comment to the whole of your posts, I basically agree with you, but would be a bit more cautious in moving from the structural/physical level of analysis to the functional/perceptual level. All the examples you are giving make sense, but it doesn't necessarily mean that they are true. That the functional aspects of music are inherent to it is almost self-evident to me, but experiments are necessary in order to infer that a specific structural property translates into the expected psychological effect.

Some such experiments are discussed in David Huron's Sweet Anticipation - I didn't want to sound like the author's sockpuppet and quote again a book that I quoted already twice or thrice on music discussions on RDF, but here we are, it is really the only great book I read about music.

Mr.Samsa wrote:We're discussing the same thing, just coming at it from different angles so there's nothing wrong with your wording

Like I said above, all learning is statistical learning. When we form "associations", all we are doing is forming conditional probabilities; i.e. if event X happens, then how likely is consequence Y? So when we say that a certain stimuli "elicits" a certain response from an organism, we simply mean to say that the organism has learnt that consequence Y reliably follows event X. And then we get into the more complicated territory of continuations and discontinuations, but basically if the same behavior is rewarded over and over again then the control of the stimuli will increase which each subsequent reinforcer, however, if you put in a discontinuation (reinforcement for a different behavior) then you will see a large shift away from the previous behavior toward the new one. I still need to figure out how exactly this works into what you're saying and with music, but something in the back of my brain is telling me it's relevant

Great. And you should trust the back of your brain more. Oh wait, no, that is a psychological fallacy.

If you want some hard data to support that claim, the book I mentioned in my above post (yes, always the same one ) is the way to go.

It depends what you mean by "external" cues - if you simply mean that someone needs to be crying, or dying nearby when you hear the song then definitely not. You'd need a hell of a lot of crying and/or dying at the exact right times for such an association to occur. Realistically, I think Beatsong is right about there being numerous elements to what makes a song sound happy or sad, so I think there is some coincidental factor going on (such as minor chords sounding better when played with a slower tempo, or something) which makes them more likely to be played in a sad song.

Well, the controversy was all about the information being "intrinsic" - contained in the music, even if not contained only in the specific music one is listening to at the moment - vs "symbolic" - arbitrary meaning that has to be related to external cues (=non-musical stimuli) in order for it to be learned. I think that it is the point where your analogy with language is very misleading. Yes, music is a language is some way, but a self-contained one, without dictionary. Or did I miss something and is that distinction an illusory one?

Apart from that, your sanity checks regarding us romantic musicians is always helpful and appreciated.

Does anyone know if there is a name for this subject; of understanding reasons for effects of different musical elements on the brain? Or the study of people's reactions to different types of music? I find this kind of thing really interesting.

n8banks wrote:Does anyone know if there is a name for this subject; of understanding reasons for effects of different musical elements on the brain? Or the study of people's reactions to different types of music? I find this kind of thing really interesting.

Psychoacoustics

I did a little sound engineering back in the day. It's absolutely fascinating!

thanks!

Beatsong wrote:That is the essence of it.

The major triad sounds inherently more stable or maybe even natural than the minor triad, because the major triad consists of a root plus the two notes that correspond to its two lowest overtones. The lowest overtones above any given fundamental are by far the strongest, so these overtones harmonise clearly with the root and contribute to its sense of stability.

The major triad does not consist of two notes corresponding to the two lowest overtones: the lowest overtones are, in order, octave, twelfth, double octave, and only then do you get the major third - it's the fourth overtone/fifth harmonic.

It's easy to compare it directly with the major triad on the same root, and find that the major 3rd (the fifth harmonic) has been replaced by the minor 3rd (the nineteenth harmonic). So clearly this will negatively affect the stability, the sense of self-affirming consonance, and the happy "ring" of the chord.

Historically though, and oftentimes in a cappella renditions and so on, you get the minor third closer to a much simpler interval than 19/16: 6/5. If we borrow Harry Partch's terminology, the major chord is an otonal chord. It forms a series of the form 4/4, 5/4, 6/4. (Sometimes notated as 4:5:6.) The minor chord is (notionally) an inverse of this: 6/6, 6/5, 6/4. (Sometimes notated as 1/(4:5:6) in the kind of theory dicussions that deal with this.) Now, alas, it turns out that equal temperament detuned every interval - and the 5/4 and 6/5 are rather bad victims of this, both suffering about a sixth of a semitone of a detuning. (And this is a rather audible detuning if you compare a just 5/4 = 1.25 with a 12-tone equal temperament 2^(1/3) = 1.259921... or a 6/5 = 1.2 with a 2^(1/4) = 1.1892...

If the minor third were designed as a 19/16 interval, the major third would have to be designed as (3/2) / (19/16) = 48/38 = 24/19 - an even less natural interval to come up with when designing a scale for something with relatively normal overtones.

What I find interesting about Partch's otonal and utonal chords is that you can extend the idea to new kinds of chords; you can build an otonal septimal diminished chord (septimal because the highest prime factor present is 7) like so: 5:6:7, and it'll sound rather dim-like. But contrast it to its utonal septimal diminished version 7/7, 7/6, 7/5 - and they form a similar albeit slightly less distinct pair analogous to major and minor. You can go and do the same to its octave inverse 3:5:7, and you get a more audible minor/major-like split. You can do it to 7:9:11-sonorities, and you get similar results, although your mileage may vary to some extent. (Of course, these different things are not possible in 12-tet, the difference between 6/5 and 7/6 is tempered out in 12-tet, so 5/5, 6/5, 7/5 and 7/7, 7/6, 7/5 are identical. A chord of the type 4/4, 7/4, 10/4 would also have a minor analogue at 10/10, 10/7, 10/4, etc.) Of course we can come up with chords that don't really sound distinct. Take something along the lines of 18/18, 22/18, 27/18, the utonal 27/27, 27/18, 27/22 will be almost audible indistinguishable. (They're a major/minor pair of neutral chords ... the difference between the neutral thirds is something like a 16th of a semitone.)

To some extent Partch's utonal/otonal idea fits with Rameau's explanation. However, Partch provides not only an explanation of things as they are, he provides a hint at things we also could do outside of what we already have.

Are you sure you're not just making this stuff up?

John

z8000783 wrote:Are you sure you're not just making this stuff up?

John

Nope. The twelve tone scale of western music can be built in a number of ways:

you can stack twelve fifths on top of each other, corresponding to (3/2)^12. This is fairly close to 2^7. 2 corresponds to an octave - the doubling of a tone. The difference is (3/2)^12/2^7 ≃ 1.01. In western music, we distribute this error equally over the twelve 3/2s, so instead of 3/2 being 3/2, it's represented by ((2^7)^1/12) = 2^(7/12).

Due to a kind of awkward mathematical fact known as the fundamental theorem of arithmetic, no two different primes to any non-zero power will ever be equal, so 3^n/2^n will never give an integer, as 3 will cycle through the final digit being 3, 9, 7, 1, 3, ... and 2^n will cycle through the last digit being 2, 4, 8, 6, 2, ...
We'll get near misses, though, and that's where we attack the problem.

However, this only really gives us good approximations of ratios with low powers of three (and any power of 2, as 2 is represented by a perfect 2) in them (and some weird random ratios of other primes - 19/16 happens to get a very good approximation.

We could just as well approach it by noticing that 2^(n/12), for different n gives good approximations of 3/2 and 4/3, less good approximations of 5/4 and 6/5 (and for an obvious reason 5/3 and 8/5), about as good approximations to 15/8, 16/15 and 9/5 and 10/9 as those of 5/4. 9/8 is very well approximated. Notice how the very good approximations all have factors no higher than three. 5 is badly approximated, 7 even worse- so seven-limit harmony isn't used in western, tempered music. We could use temperaments that approximate 7 and 5 better than currently - in fact, the most prevalent scale in the 16th and 17th centuries in the west, quarter-comma meantone, if remade so that all keys are playable, and it's somewhat tempered, comes very close to 31-tone equal temperament. A system where chords of the form 4:5:6:7, 5:6:7:8, (as well as the 7/7, 7/6, 7/5 inversion) all work. (When it was in use, alas, quarter-comma meantone only was used in a range of about 12 tones, so only a few keys worked in it, but in some of the further away keys, you got these alien yet overtone-like intervals.)

Anyways, uh, music theory is how I learned to calculate approximate logarithms in my head (to three decimals accuracy for very many numbers, numbers with higher prime factors might get a bit worse as far as accuracy goes), so ...trust me, I know this shit.

Zwaarddijk wrote:

Beatsong wrote:That is the essence of it.

The major triad sounds inherently more stable or maybe even natural than the minor triad, because the major triad consists of a root plus the two notes that correspond to its two lowest overtones. The lowest overtones above any given fundamental are by far the strongest, so these overtones harmonise clearly with the root and contribute to its sense of stability.

The major triad does not consist of two notes corresponding to the two lowest overtones: the lowest overtones are, in order, octave, twelfth, double octave, and only then do you get the major third - it's the fourth overtone/fifth harmonic.

I was discounting the octave and double octave because they are simply doublings of the root in a higher octave. I meant the two lowest overtones apart from doublings of the root itself.

As for the rest: WHOAAAA!

Beatsong wrote:

Zwaarddijk wrote:

Beatsong wrote:That is the essence of it.

The major triad sounds inherently more stable or maybe even natural than the minor triad, because the major triad consists of a root plus the two notes that correspond to its two lowest overtones. The lowest overtones above any given fundamental are by far the strongest, so these overtones harmonise clearly with the root and contribute to its sense of stability.

The major triad does not consist of two notes corresponding to the two lowest overtones: the lowest overtones are, in order, octave, twelfth, double octave, and only then do you get the major third - it's the fourth overtone/fifth harmonic.

I was discounting the octave and double octave because they are simply doublings of the root in a higher octave. I meant the two lowest overtones apart from doublings of the root itself.

As for the rest: WHOAAAA!

The doublings of the root do contribute to dissonance and consonance, though, so they're kind of relevant. In modern theories of consonance, dissonance is caused by overtones that are more than a few hertz, but less than ~6/5 apart. (The ~6/5 span grows as you descend, but 6/5 is rather constant for most of our hearing span above something like 200hz. Towards the lower end of the musical gamuts in use, it is almost fixed at 50 hz, to the point that even octaves can be perceived as dissonant without very odd timbres).

Of course, some instruments lack octave overtones - conical bore wind instruments only have odd harmonics/even numbered overtones, so you get something like 1, 3, 5, 7, 9, 11, ... x the fundamental. (Which has been used in some experimental music for tuning scales that lack octaves altogether and use 3:5:7 and 5:7:9 as the main chords instead of 4:5:6). Turns out thirteen equal steps to the twelfth (often called the tritave by musicians involved with this, due to its 3:1 ratio) give a way better approximation of those chords than 12 steps to the octave gives for 4:5:6.

Zwaarddijk wrote:Of course, some instruments lack octave overtones - conical bore wind instruments only have odd harmonics/even numbered overtones, so you get something like 1, 3, 5, 7, 9, 11, ... x the fundamental.

What sort of instruments would be able to generate them?

John

z8000783 wrote:

Zwaarddijk wrote:Of course, some instruments lack octave overtones - conical bore wind instruments only have odd harmonics/even numbered overtones, so you get something like 1, 3, 5, 7, 9, 11, ... x the fundamental.

What sort of instruments would be able to generate them?

John

Clarinets, tubas, etc. I seem to recall some instruments have only the even harmonics, but don't remember which.

The 13-steps-to-the-perfect-twelfth scale is called Bohlen-Pierce after its two inventors (both microwave engineers, both inventing it independently). A bunch of Bohlen-Pierce clarinets have been constructed.

[youtube]http://www.youtube.com/watch?v=y8Dimhs_GX8[/youtube]
Alas, no harmonies in this video

IMHO, it is a result of conditioning, plus a little nature. The vast majority of music we are used to hearing is in a seven note scale of some kind or other. My theory is that 3.43 semitones above the root is the center of the vicinity in which we expect a 3rd scale degree to be, because it is precisely two sevenths of the way from the tonic to its 8ve. Naturally, most music ends up rounding this up or down to 3 OR 4 semitones above the tonic to create minor or major 3rds, which even in equal temperament are close to the consonant simple ratios of 5:6 and 4:5, respectively. The result is that most music ends up with two options when it comes to a 3rd scale degree, one being high within the vicinity of 3rds (57 cents above average) and therefore perhaps conveying a sense of uplift, and one being low within the vicinity of 3rds (43 cents below average) and therefore perhaps being an emotional downer. If you add to this the fact that the major triad is more natural than its inversion (as proven in the naturally occurring harmonic series), one can easily see why western composers started to use major chords with positive texts and minor chords with negative texts, thus adding a "nurture" factor to the very possible, though tricky to prove "nature" factor.

ewanclark wrote:IMHO, it is a result of conditioning, plus a little nature. The vast majority of music we are used to hearing is in a seven note scale of some kind or other. My theory is that 3.43 semitones above the root is the center of the vicinity in which we expect a 3rd scale degree to be, because it is precisely two sevenths of the way from the tonic to its 8ve.

Why do talented singers find singing a neutral third so difficult then?

Naturally, most music ends up rounding this up or down to 3 OR 4 semitones above the tonic to create minor or major 3rds, which even in equal temperament are close to the consonant simple ratios of 5:6 and 4:5, respectively. The result is that most music ends up with two options when it comes to a 3rd scale degree, one being high within the vicinity of 3rds (57 cents above average) and therefore perhaps conveying a sense of uplift, and one being low within the vicinity of 3rds (43 cents below average) and therefore perhaps being an emotional downer.

Why should we in the first place assume chords built on every other tone in a seven tone scale? There actually exist good chords in 7-tet, none of which have that structure. If major and minor chords occur as accidental approximations to ~7tet chords, you should explain why they're not approximations of more consonant 7tet chords.

(If we look at the history of how major chords came into use, we also find that the 3:4:5 inversion predates the 4:5:6 inversion in use; it's quite likely the major chord has come into being not as a result of some arbitrary stacking of intervals, but rather as a result of an attempt to line simultaneous tones up with each other in a consonant way, which due to harmonics best is achieved by simply having them correspond to some overtones (relatively low overtones at that).

Why do talented singers find singing a neutral third so difficult then?

Because it's a nasty interval that sits between two nice ones and doesn't occur in western music. I'm not saying that 3.43 semitones above the root is the golden standard of the ideal 3rd, or even the most natural 3rd. I'm saying that IF we're talking about music built with a seven notes scale, then the more the scale attempts to divide the 8ve evenly, the more likely the 3rd is to be in the *ball park* around 3.43. As it happens, 3.43 is a hard to sing dissonance, but in its ball park are two excellent options, the minor and major 3rd, almost equidistantly on either side of 3.43. These two kinds of 3rds have therefore become important in relatively equal measure, and it makes sense that they have come to be compared to each other, and that the higher one has come to have positive associations attached to it. Your second point is harder to respond to, and I will do so tomorrow. Thanks for the discussion!

ewanclark wrote:

Why do talented singers find singing a neutral third so difficult then?

Because it's a nasty interval that sits between two nice ones and doesn't occur in western music. I'm not saying that 3.43 semitones above the root is the golden standard of the ideal 3rd, or even the most natural 3rd. I'm saying that IF we're talking about music built with a seven notes scale, then the more the scale attempts to divide the 8ve evenly, the more likely the 3rd is to be in the *ball park* around 3.43. As it happens, 3.43 is a hard to sing dissonance, but in its ball park are two excellent options, the minor and major 3rd, almost equidistantly on either side of 3.43. These two kinds of 3rds have therefore become important in relatively equal measure, and it makes sense that they have come to be compared to each other, and that the higher one has come to have positive associations attached to it. Your second point is harder to respond to, and I will do so tomorrow. Thanks for the discussion!

The Blues is all about playing around with these intervals. It plays around with the 7th and (since the 1950s) the 5th, but the 3d is where it gets most of its evocative power.

The Blues is all about playing around with these intervals. It plays around with the 7th and (since the 1950s) the 5th, but the 3d is where it gets most of its evocative power.

Ok, fair point. When I said "doesn't occur in western music" I knew I should have instead said "the vast majority western art music" to clarify that I wasn't including blues in the statement. All dissonances are powerful and evocative when used well, so I shouldn't perhaps have called it nasty. But my point to Zwaardik was that the neutral 3rd (more precisely 3.43 semitones above tonic) was the center of the vicinity in which 3rds occur, rather than necessarily everyone's ideal 3rd. The fact that neutral 3rds also occur in some musics is consistent with this theory about the vicinity of 3rds.

Zwaarddijk wrote:Why should we in the first place assume chords built on every other tone in a seven tone scale?

Good question. The answer of course is that we shouldn't assume that a major chord *must* exist in the context of a seven note scale. But the question at hand is about the perception of major and minor chords; presumably the perception of the average person alive today. Now, regardless of the way in which the major chord historically came into use (though that is certainly very good to know) I would argue that our perception of major and minor chords is more a result of a) the natural properties of those chords and b) the way in which those chords are used in music that we are accustomed to.

For most people alive today in westernised countries, we are used to hearing triads within the context of a seven note scale, far more than we are used to hearing them in any other context, although I would be fascinated if someone can convince me otherwise.

Why the seven note scale is so prevalent is a harder question to answer, other than to say that the first 14 partials in the harmonic series suggest a seven note scale. I know, I know, partials 15 and higher complicate things substantially, but the point still holds some weight, just as the harmonic series can be used to explain the prevalence of the perfect 5th and the major triad in various kinds of music.

Zwaarddijk wrote:There actually exist good chords in 7-tet, none of which have that structure. If major and minor chords occur as accidental approximations to ~7tet chords, you should explain why they're not approximations of more consonant 7tet chords.

(If we look at the history of how major chords came into use, we also find that the 3:4:5 inversion predates the 4:5:6 inversion in use; it's quite likely the major chord has come into being not as a result of some arbitrary stacking of intervals, but rather as a result of an attempt to line simultaneous tones up with each other in a consonant way, which due to harmonics best is achieved by simply having them correspond to some overtones (relatively low overtones at that).

I agree that the major chords came about as you say, through its consonance and through its occurrence in relatively low overtones in the harmonic series. And I agree that major and minor triads did not come about as a result of being approximations of 7-tet, and that 7-tet contains some closer approximations to chords found in 12-tet, such as C-D-F and C-D-G.

But none of this negates my central point. 7-tet does not need to be the historical origin of diatonic pitch relationships in order for my point to hold true. Let me try and make more point in a different way. If we wanted to find the perfect seven note scale, and evenness of octave division was our only priority, we would come up with 7-tet. If we wanted to find the perfect seven note scale, but made acoustic consonance (simple ratios) a higher priority than evenness, we would come up with a diatonic scale of some kind; maybe a major scale in just intonation. Yet evenness is still a priority that has been very slightly violated in the forming of this scale, which perhaps, just perhaps, causes our subconscious mind to hear a major 3rd as being a bit high and a minor 3rd as being a bit being low, compared to the 7-tet 3rd that places evenness as highest priority.