That’s what I meant? 
edit: what I meant is that equal temperament etc are the compromises.
Hexachord theories allowed for the transposition of the whole system, allowing B/B flat to become E/E flat, etc…and I’m certainly not talking about Greeks, but about 15th and 16th century guys! Also, I have to say that the idea of ‘big gaps’ and an ‘extra note’ does have a whiff of piano-centricity.
Sorry, I’m actually agreeing with you and adding other examples for Bad Astronaut’s edification. I didn’t mean to quote as a contradiction of your post.
edit: I think. Unless now I’m misunderstanding what you were saying. But I think we’re on the same page.
At this point I’m reasonably well convinced that scales are merely cultural constructs, which is interesting in itself.
Also, I was curious about Native American music since we can be pretty confident that there was no cultural contact between the Old and New worlds between the last ice age and the 15th century. Sure enough, their music is vastly different:
Also I’m not Bad Astronomer. If I’d known that was a well-known username on this board I would have picked a different one.
Have you ever fingered a Violin, Guitar, etc?
There are big and small gaps in the diatonic scale. Don’t be worrying 'bout my piano!
I just edited my post when I realized the error, but you beat me to it.
We had a big thread on a similar topic just a couple of weeks ago. My gut reaction to that page is that it’s a big exaggeration to say there’s no known reason for 12 notes in the octave. There’s plenty of reasons on his page alone (some of them ridiculous) - it’s just a matter of argument which one put the 12-note octave “over the line” as far as popularity is concerned.
I think the “circle of fifths” argument provides a pretty good argument for the 12-note octave. If you take the simplest possible ratio of vibrations (3:2) and start stepping up, you get close to a power of two greater (ie, the “same note” as far as our brains process sounds) at steps 5, 7 and 12. And 12 is, of the three, by far the best fit.
Or, to put it another way, if you construct a set of equal-temperament 12, 5 and 7 “octaves”, the 12-note one is far better at replicating the simple rations (3:2, 4:3, 5:3 etc) than either the 5-note or the 7-note (a fifth in the standard octave is only 0.1% off being an exact match to a 3:2 ratio, whereas the corresponding notes in a 5-note or 7-note scale are 1% off)
Also, interestingly, if you look at a keyboard instrument, its 12 notes are split into a 5-note set and a 7-note set - obviously you don’t get an equal temperament scale out of either set exactly, but it’s interesting to see that if you go with a 12-note set you can get the 5 and 7 sort of “thrown in”. I can’t see that any other choice of base would give you so much flexibility.
But that I can’t tell you when a flat B becomes an A would indicate that to me, at least, there is not necessarily any “golden tone.” A golden interval, maybe. But these particular tones, are they anything really?
The meaning of ‘tones’ in this discussion, at least my part of it, would be more accurately described as ‘intervals’ as you say. It seems pretty obvious to me that 440 Hz is a completely arbitrary basis, since a second is a rather arbitrary length of time (1/86400 of the time it takes the earth to rotate once on its axis). Our brains seem to only care about the intervals.
I teach one of the above. (The reference to a viola pupil is a hint!) However, I’m not afraid to acknowledge that it’s a newcomer, embraced long after the issues of scales and intonation were already being debated.
My point is that it’s easy to perceive scales in relation to a piano keyboard, and by extension to perceive other systems in such a way. Far better to try singing along with them, to get a real feel for where the important differences are.
(Guitars? Tuning? Intonation? Is that a joke? 
)
Aspidistra’s description is very much a case of working from the present, that we have a system which functions well with keyboard instruments, and therefore this is where we inevitably arrived. The split in the keyboard directly corresponds to the transposition of hexachords I’ve mentioned earlier.
So I guess if we make contact with extra-terrestrials we should not expect to find any common ground with music, contrary to the theories of Spielberg et al. We should still be ok on primes though!
Cleaning up a few points I wanted to respond to:
I didn’t mean you would perceive it as noise, just that you would know it was different. I know there is microtonal Western music, but it sounds microtonal and is clearly distinguishable from ‘regular’ or what I would consider ‘melodic’ music. It’s hard to describe the difference in meaningful terms.
I think there are some things that can be said about this. A single note from a real instrument (including a singer) consists of a a fundamental plus various harmonics. If we perceived the harmonics as different notes there would be no such thing as a single note. Even with electronic systems you cannot produce a pure tone - all amplifiers and speakers produce some distortion (although with modern technology it can be made small enough not to matter).
So you could theorize that our brains evolved to lump these harmonics together in order to reduce the number of distinct events that needed to be processed.
It’s worth nothign that a major third, tuned to equal temperment, played by a signal generator, sounds BAD. On a piano, it sounds normal and natural (to western ears!)
Actually, that’s also mistaken. Your inner ear is tuned to respond to particular frequencies in such a way that a 440 Hz tone will always excite the same hair cells. People with absolute pitch can tell pitches independently of others. Now, most people are much better at telling intervals apart, but the brain most definitely can perceive absolute frequency. After all, you’ll never mistake a 200 Hz tone for a 5000 Hz tone.
So, while it’s true that the particular value of 440 Hz is arbitrary and far from universal, seconds aren’t really relevant to the discussion.
I knew someone would point this out. I know the ear/brain can detect the difference, but as far as judging the ‘musicality’ of notes everything is relative. Even people with perfect pitch can tell whether one note is a perfect third above the other, regardless of the absolute pitches.
They are relevant in that the standard is 440 Hz, and not 440.2612374 Hz. If the standard was non-arbitrary we would not expect it to be an even multiple of an arbitrary unit (A Hz is defined as 1/second if anyone is not following this).
It’s not so much a question of reducing the number of events as one of providing a likely interpretation. The brain receives low-level signals from the brain that correspond to amplitude levels of various frequency bands – much like a Fourier transform. From this data, the brain has to construct an auditory scene that corresponds as much as possible to the real objects and events that caused those sounds if it is to survive. Since the harmonic series occurs naturally, grouping tones that belong to this series together will, in most cases, provide a “correct” interpretation of the data.
Okay, I see what you mean. (Perceptually, though, 440 Hz and 440.2612374 Hz are almost the same. I thought the question was why 440 and not 480, we do like round numbers.)
Huh? Cycles/second is certainly an arbitrary quantity, because its use in the measurement of the pitch of sounds has no connection with the origination of ‘second’.
They’d be noticably different on violins, though 
Yes. I think this is equivalent to what I was getting at.
Another thing that occurs to me is that since receptors respond to specific frequencies, notes an octave apart will trigger most of the same receptors. The fundamental receptors won’t be activated for the higher note, but the harmonic receptors will be activated in a very similar pattern for both. So it’s not surprising that we perceive them to be so similar.
Nevermind. This point is of no importance. It was pretty much meant as a joke when I mentioned seconds in the first place. As long as we all agree that 440 Hz is an arbitrary choice, it’s not worth discussing further.
When I was in college, I worked out that 19 notes / octave would work well, and built a guitar with 19 frets/octave. It was made from free discard parts from a local guitar store, but it sounded good (well, in-tune).