why do our scales have seven notes?? and not 8, 9, 10 or 11?

[aciddose]

"Why does wood burn?"

Wood burns because you lit it on fire.

Vs.

The carbon content of wood is oxidized by atmospheric oxygen at a certain temperature which raises and maintains the temperature at that required to maintain the reaction given certain conditions are met.

We don't care whether someone has lit it on fire. It still burns regardless of whether anyone ever did or ever will do so!

Does that make sense? It doesn't matter if the wood was lit or not, we still know it burns because we've been able to explain how. We can make predictions about it without needing to actually ever see it burn.

We don't just understand "how" it burns, we haven't merely observed it, but we know "why".

How: by what means? (history, what actually happened, experiment, observation)
Why: for what reason? (facts and causes independent from history, regardless whether it happened or not, theory)

Unfortunately how/why are very nearly interchangeable in English.

[stringtapper]

So you're happy to settle on a subjective explanation for things?

It's not subjective. It's history, i.e. facts.

You don't think there is a "best fit" system and you don't see history as an approximation toward that?

Well, no of course not. To think that would be terribly ethnocentric of me. When you say "history" you must understand you are talking about "western music history" which is not the only game in town.

Instead, you see history as both the "why" and "how" and answer to everything?

What I'm talking about is only "history" because it already happened. I'm talking about specific examples of the reasons that a certain musical system was employed in a certain musical culture.

This is far more musical religion than science.

No, again they are just the facts, i.e. what happened.

Just like you can pick apart the whole of religious systems by pointing out fundamental flaws with their foundations it is easy to point out the slew of problems encountered with any system other than 12TET.

The limitations of any system other than 12-notes, or of scales other than 7-notes.

And yet somehow music was being made for all of those years that 12tet didn't exist. Most of the history of musical practice in fact. How did they do it with all of those "limitations"?

When you look at the bigger picture here it should be obvious that 12TET and 7-note scales are indeed best-fit systems that line up with the mean of "subjective" human preference.

And how do you go about proving this? If you're going to try to say that because 12tet 7-note scales are widely used in western music then they represent some musical universal then I'm afraid you may have a narrow view of the musical world. It's basically like saying "it's popular so it must be 'good.'"

[aciddose]

Are we suddenly talking about some other musical system? Why bring non-western music into this?

Again, we care about the why of western music and it's 12TET and preference for 7-note scales.

I'm sure you can find all sorts of reasons for some other system but that is totally out of context.

If you want to create an all encompassing theory it is definitely a good idea to ensure that it fits given all the variations available. If you wanted to prove that my theory / explanation for 12TET and 7-note scales is incorrect you're very welcome to point out how it is completely incompatible with some other musical system, other than "western".

Note though that I already pointed out that the same theory does apply perfectly fine to other systems and I've already looked into this myself. The numbers are different in other systems because the preferences are slightly different, but the other systems still are selected due to the fact they are a best-fit.

Of course even in western music 12TET isn't used exclusively. Some composers are happy to work with the limitation that they can not freely transpose and easily predict the results. That doesn't however take away from the fact that 12TET is preferred because you can freely transpose and predict the results.

[aciddose]

And yet somehow music was being made for all of those years that 12tet didn't exist. Most of the history of musical practice in fact. How did they do it with all of those "limitations"?

In my opinion, not very well.

We're subjective now, but if you want to go there I think the complexity and feeling of music composed in some of the more ancient systems outright sucks.

While I feel that certain just intonations are great for certain compositions, I'm also perfectly aware of the additional difficulty associated with composing with these systems.

So, not very well and only with great difficulty.

[IncarnateX]

@aciddose. If you really want to put all this in the numbers themselves as if musical scales consisting of 7 notes somehow are mathematically given and universal, you owe Weber and Fechner to take their psychophysiological law into consideration. You need the math of the brain, not only the stimulation. Here you go:

http://en.wikipedia.org/wiki/Weber-Fechner_law

And a quote to work on

The total number of perceptible pitch steps in the range of human hearing is about 1,400; the total number of notes in the equal-tempered scale, from 16 to 16,000 Hz, is 120.[3]

Good luck making the numbers fit.

[aciddose]

What does that have to do with anything?

Also the idea that we're only capable of perceiving 1400 steps between 20hz and 20khz is completely ludicrous. It only makes sense when you take into account all the little specifics they've employed in their tests.

2048 steps is not enough. 4096 is barely beginning to become enough, you just start to lose your ability to identify the steps in a glide between notes at this resolution (12 bits) in the range 1k - 4k, our most sensitive range.

Given 4096 steps total and a limited range of 128 notes (10 + 2/3 octaves) we have only 32 steps per semi-tone, 3 + 1/8th cents. This is just barely, barely perceptible if you're modulating the pitch slowly back and forth. Barely perceptible enough to make it perfectly acceptable to use for general-purpose frequency control - although not accurate enough to use for fine-tuning as you want far more accuracy, usually at least 1/10th of a cent.

(More notes on this - you should test with a harmonic rich ramp waveform. A single sine is not an accurate test, does not reflect real-world sounds and so doesn't accurately reflect our perceptive capability in the real world. Having more of the harmonics available allows us to more quickly and more accurately pin-point the frequency we're hearing as the frequency measurement is made with 1000s of sensors rather than only a handful.)

That said that is entirely a different topic. I've explained exactly why there are more 6-note scales than others and also speculated about the reason there are more 7-note scales in common use than others, even 6-note.

[IncarnateX]

What does that have to do with anything?

That you are only speaking of a well tempered system that contains 120 out of 1400 possible pitch steps (or more if you like but that does not do your argument any favor and note that pitch steps do not equal total freq. range but the range of perceivable note steps you can make within total hearing range)
Thus it is hard to see that somehow the tempered scales of 7 notes appear from the math in itself beyond any culture or historical consensus.

Also see about the use of quarter tone scales:

http://en.wikipedia.org/wiki/Quarter_tone

I'm counting 13 out of 35 with 7, more than other numbers.

That leaves you with 22 exceptions to the rule to explain. Numbers are not on your side here.

I've explained exactly why there are more 6-note scales than others and also speculated about the reason there are more 7-note scales in common use than others, even 6-note .

I am sorry, in that case I did not get it then for some of the same reasons already noted by others. So put shortly as possible: what is the main proposition of your theory and how does it answer the OPs question?

[aciddose]

I'll just be repeating exactly what I've already said, but here, it's simple:

Scales are only unique by the number and position of gaps in them.

For example, C-Maj can be represented by:

1101110

Here, each number represents the number of steps to skip before the next one which is a component of the scale.

An octave scale would then be:

11

Although, 11 would be confused with two 1s, so I'd recommend using commas or alphabetical characters to represent steps larger than 9. (so, B = 11)

If we can agree on this (I don't see why not?) then we should be able to see that there are a limited number of unique scales possible if you ignore transpositions.

For one note, the only possible scale is: 0 (or nothing at all, it's any single note)
For eleven notes, the only possible scale is: 000000000001 (any eleven consecutive notes with a gap anywhere)

All "rotations" of the pattern are the same pattern. So 1000 = 0100 = 0010 = 0001.

If we can also agree on this (again don't see why not)...

We're also assuming this applies specifically to a 12-tone system, although it applies similarly to any number of tones just with different results.

http://en.wikipedia.org/wiki/Combinations

We can then look at our 12 tones and see that we can produce the most unique combinations of 6 tones. Slightly less for 5 and 7, again less 4 and 8 and so on until we have only one possible combination for 1 and 11.

So now... if we can agree on that (again, no idea why anyone would refuse to)...

The question is why do we see 7-note scales show up so often in modern "western" music?

The simplest explanation for this is going to be that 7 is only one step away from 6. Now we need to identify the reason that scales of 7 show up more often than 6, and more importantly why more often than 5. We should also show why other numbers like 8-4, 9-3, 10-2 are only rarely seen.

In the case of other ranges excluding 5, 6 and 7, the answer is simple. There are merely less combinations available to start with. Unless the preference was very strong for scales with more or less notes, it is simply impossible to see more scales in use with these number of tones.

Now comes the only part of what I've said related to this that is theory: My theory is that 7 note scales provide greater melodic flexibility and are the best trade-off between the total number of scales available vs. the number of notes within them. There are too few 8-note scales available to start with to out-number 7-note scales and there are too few notes available in 5-note scales.

Apparently, it also seems that the bias toward "more notes" is not strong enough to overcome the limited number of scales available for greater numbers, 8, 9 and 10, or that having more than 7 notes doesn't in fact provide any significantly greater melodic flexibility.

The reason for that is likely that there are not enough notes in 12TET closely approximating "just" (as in just intonation) fractions to allow you to pick a set of 8 notes that provides an advantage over a similar set of 7 - that the 8-note scale will be a 7-note scale with "one extra note", and that it will likely be dissonant.

The same idea applies to why there are not an equal number of 6-note and 7-note scales in common use - likely that a majority of those 6-note scales are 7-note scales with "one missing note".

That sums it up.

[IncarnateX]

The simplest explanation for this is going to be that 7 is only one step away from 6. Now we need to identify the reason that scales of 7 show up more often than 6, and more importantly why more often than 5. We should also show why other numbers like 8-4, 9-3, 10-2 are only rarely seen.

In the case of other ranges excluding 5, 6 and 7, the answer is simple. There are merely less combinations available to start with. Unless the preference was very strong for scales with more or less notes, it is simply impossible to see more scales in use with these number of tones.

Now comes the only part of what I've said related to this that is theory: My theory is that 7 note scales provide greater melodic flexibility and are the best trade-off between the total number of scales available vs. the number of notes within them. There are too few 8-note scales available to start with to out-number 7-note scales and there are too few notes available in 5-note scales.

Apparently, it also seems that the bias toward "more notes" is not strong enough to overcome the limited number of scales available for greater numbers, 8, 9 and 10, or that having more than 7 notes doesn't in fact provide any significantly greater melodic flexibility.

The reason for that is likely that there are not enough notes in 12TET closely approximating "just" (as in just intonation) fractions to allow you to pick a set of 8 notes that provides an advantage over a similar set of 7 - that the 8-note scale will be a 7-note scale with "one extra note", and that it will likely be dissonant.

The same idea applies to why there are not an equal number of 6-note and 7-note scales in common use - likely that a majority of those 6-note scales are 7-note scales with "one missing note".

That sums it up.

I see a lot of assumptions and propositions based on an already established western scale, so all this really begs the question to me instead of answering it. Now in addition answers like "The simplest explanation for this is going to be that 7 is only one step away from 6" seem like pure cabalism to me, where you can make any number mean anything. I have to chime in with the rest I am afraid: Not convinced.

Thanks for your time, anyway.

[vurt]

venus and uranus are the same.

I love quoting out of context. Apart from they're roughly spherical, how are they the same to anything, especially each other?

who can say?

Probably the people that have looked at them.
Don't forget that Venus has been visited (well, landed upon). Uranus has had a close flyby, and it's weird.
Steve

i hope now, reading what has passed since this post, why i dont bother with being serious around here too often...

[stringtapper]

I'll just be repeating exactly what I've already said, but here, it's simple:

Scales are only unique by the number and position of gaps in them.

For example, C-Maj can be represented by:

1101110

Here, each number represents the number of steps to skip before the next one which is a component of the scale.

An octave scale would then be:

11

Although, 11 would be confused with two 1s, so I'd recommend using commas or alphabetical characters to represent steps larger than 9. (so, B = 11)

If we can agree on this (I don't see why not?) then we should be able to see that there are a limited number of unique scales possible if you ignore transpositions.

For one note, the only possible scale is: 0 (or nothing at all, it's any single note)
For eleven notes, the only possible scale is: 000000000001 (any eleven consecutive notes with a gap anywhere)

All "rotations" of the pattern are the same pattern. So 1000 = 0100 = 0010 = 0001.

If we can also agree on this (again don't see why not)...

We're also assuming this applies specifically to a 12-tone system, although it applies similarly to any number of tones just with different results.

http://en.wikipedia.org/wiki/Combinations

We can then look at our 12 tones and see that we can produce the most unique combinations of 6 tones. Slightly less for 5 and 7, again less 4 and 8 and so on until we have only one possible combination for 1 and 11.

So now... if we can agree on that (again, no idea why anyone would refuse to)...

FYI, you're reinventing the wheel with a lot of what you've written here. Pitch-class set theory already exists. All available combinations of pitches in the 12tet system have been catalogued for at least forty years now. I gave you links to the lists. Set classes and their interval vectors give you all the information you need. Also, the use of characters in place of 10 and 11 is already commonplace. I use "T" and "E", some theorists use "A" and "B."

The question is why do we see 7-note scales show up so often in modern "western" music?

The simplest explanation for this is going to be that 7 is only one step away from 6. Now we need to identify the reason that scales of 7 show up more often than 6, and more importantly why more often than 5. We should also show why other numbers like 8-4, 9-3, 10-2 are only rarely seen.

In the case of other ranges excluding 5, 6 and 7, the answer is simple. There are merely less combinations available to start with. Unless the preference was very strong for scales with more or less notes, it is simply impossible to see more scales in use with these number of tones.

Now comes the only part of what I've said related to this that is theory: My theory is that 7 note scales provide greater melodic flexibility and are the best trade-off between the total number of scales available vs. the number of notes within them. There are too few 8-note scales available to start with to out-number 7-note scales and there are too few notes available in 5-note scales.

Apparently, it also seems that the bias toward "more notes" is not strong enough to overcome the limited number of scales available for greater numbers, 8, 9 and 10, or that having more than 7 notes doesn't in fact provide any significantly greater melodic flexibility.

The reason for that is likely that there are not enough notes in 12TET closely approximating "just" (as in just intonation) fractions to allow you to pick a set of 8 notes that provides an advantage over a similar set of 7 - that the 8-note scale will be a 7-note scale with "one extra note", and that it will likely be dissonant.

The same idea applies to why there are not an equal number of 6-note and 7-note scales in common use - likely that a majority of those 6-note scales are 7-note scales with "one missing note".

That sums it up.

Sorry, also not convinced. It appears you are going a long way to discuss mathematical properties of pitch collections to then merely say "7 is one more than 6." "Melodic flexibility" seems to be your final argument but you don't really define what you mean by this term.

The problem is that you are attempting to place an objective measure on something that is inherently subjective: musical practice. You're trying to get the "one true answer" by only dealing with one aspect of something that has a lot of moving parts. To ignore the cultural and anthropological underpinnings of the issue is to ignore the issue altogether, IMO.

In fact if you want to get scientific then I think you aren't going far enough with it. I don't see how you could do it without getting into the psychoacoustics of pitch perception, i.e. critical bandwidth and the discernment between consonance and dissonance. See Plomp 1965 for some research into this:

http://pubman.mpdl.mpg.de/pubman/item/e ... l_1965.pdf

I appreciate a rigid, quasi-scientific approach, I really do. But I don't think we can get the entire picture by only looking at numbers when we are ultimately talking about a subjective human activity.

[robogone]

its because bach had only 7 fingers

The other 3 were busy working out his first movement.

Insert constipation joke here

[IncarnateX]

To ignore the cultural and anthropological underpinnings of the issue is to ignore the issue altogether, IMO.

+1

[VariKusBrainZ]

Because of "quantised" scaled isntruments with keys and frets

[VariKusBrainZ]

Also, musical scales in various cultures are related to the pitch structure of their spoken language

http://discovermagazine.com/2008/jan/mu ... oFHhiefnlY