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

[aciddose]

Yes I know 12 tones are chromatic and atonal. But every other amount is not chromatic, why 7 notes? And why two half steps?

Premise is wrong. There are a good deal of scales that do not consist of 7 notes. Count them yourself:

Ain't it simple as that?

I'm counting 13 out of 35 with 7, more than other numbers. I like my explanation which seems to fit perfectly when you look at the distribution presented here.

12 = 1
11 = 0
10 = 0
9 = 0
8 = 3
7 = 13
6 = 3
5 = 8
4 = 1
3 = 3,
2 = 1,
1 = 1,

We don't have all possible scales listed, just a small selection of them but already we can clearly see a normal distribution centered around 7. (Actually, would be 6.5 wouldn't it?)

It's also possible some of the scales in that chart are identical transpositions, it would take me far too long to check for that so I haven't.

If they were in a format like I proposed where each number represents the number of steps to take before the next note in the scale, that would make it far easier to identify duplicates by rotating the resulting pattern.

[stringtapper]

We don't have all possible scales listed, just a small selection of them but already we can clearly see a normal distribution centered around 7. (Actually, would be 6.5 wouldn't it?)

It's also possible some of the scales in that chart are identical transpositions, it would take me far too long to check for that so I haven't.

If they were in a format like I proposed where each number represents the number of steps to take before the next note in the scale, that would make it far easier to identify duplicates by rotating the resulting pattern.

We're pussyfooting around here talking about "scales." Your predilection for 7 is easily proved as unfounded by delving into pitch-class set theory.

Check out these lists of pitch-class sets:

http://lulu.esm.rochester.edu/rdm/pdfli ... .table.pdf

http://www.courses.unt.edu/josephklein/ ... _CLASS.pdf


These sets represent all possible orderings of pitches within the 12tet system accounting for transposition and inversion. The second link contains Allen Forte's original list from The Structure of Atonal Music.

As you can see the number of septachords totals 38 while hexachords win out at 50. Also notice that because of the relationship between complimentary sets there are just as many pentachords as septa chords (5+7=12).

So much for the "math" proving anything significant about 7-note pitch collections.

[aciddose]

You might want to go back and read my post about where I specifically pointed out that 6 = 1/2 of 12, and that 7 is very near to 6.

I talked about regression toward the mean and the number of available scales being highest at 6. I mentioned that I suspect 7 is favored due to it's improved flexibility relative to 6.

I then said "Although, it's also possible I suppose that there may be more available at 7."

6.5 would make sense, perhaps, meaning there would be equal numbers of 6 and 7 note scales. 6 though seemed the initial intuitive "center" value.

The list of scales presented earlier in this thread is an example of the preference for 7-note scales vs. 6-note scales. That is the result of preference, nothing mathematical.

Unless you want to get into a mathematical definition of "flexibility" of 6 vs. 7 note scales, which I suspect would also have a perfectly valid mathematical explanation

[fmr]

Let's look how Mr. Pythagoras, our old Greek friend, has founded the Western tuning system:

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

Considering that only in the seventeenth century musicians started to use what is now mentioned as the "scales" (basically, major and minor modes), and only in the eighteenth century did the equal temperament estabilshed (still with some variants), and that even the greeks were not using Pithagoras tuning (as far as History can go), it's not bad.

I think many wrong questions and answers have been written around here. People seem to consider today's western tonal system (which actually is not in use anymore by many composers) as a "standard"...

Well, it isn't. It is just a system among many others that existed and still exist. And it only started around 300 years ago. In our western civilization, modal system, by comparison, existed for much longer, and was based on the hexachord (six notes - go figure).

And what can we say about the "progressive tonality" in late romatic period?

When music is predominantly chromatic, can we still say the composer is just using seven noyes, or the whole twelve notes?

[stringtapper]

You might want to go back and read my post about where I specifically pointed out that 6 = 1/2 of 12, and that 7 is very near to 6.

I talked about regression toward the mean and the number of available scales being highest at 6. I mentioned that I suspect 7 is favored due to it's improved flexibility relative to 6.

I then said "Although, it's also possible I suppose that there may be more available at 7."

6.5 would make sense, perhaps, meaning there would be equal numbers of 6 and 7 note scales. 6 though seemed the initial intuitive "center" value.

The list of scales presented earlier in this thread is an example of the preference for 7-note scales vs. 6-note scales. That is the result of preference, nothing mathematical.

Unless you want to get into a mathematical definition of "flexibility" of 6 vs. 7 note scales, which I suspect would also have a perfectly valid mathematical explanation

Doesn't answer the issue of complementation between sets whose combined cardinality equal 12 (i.e. the fact that all complementary sets are equal in number and therefore there are the same amount of five-note sets as there are seven-note sets).

Using your logic you would have to concede that pentachords are just as "favored" as septachords.

[aciddose]

Doesn't answer the issue of complementation between sets whose combined cardinality equal 12 (i.e. the fact that all complementary sets are equal in number and therefore there are the same amount of five-note sets as there are seven-note sets).

You are probably just missing out on the probability aspect of this. Obviously there would be equal numbers of 5 and 7 because they're equal distance from the center. There are likewise equal numbers of 1 and 11 note scales... 1 of each.

The only possible 1-note scale is a single note. Doesn't matter which one, they're all the same, just transpositions.

Likewise for 11, the only possible 11-note scale is a scale with a single gap in it, doesn't matter where, they're again all just transpositions.

Using your logic you would have to concede that pentachords are just as "favored" as septachords.

What would motivate you to come to this conclusion?

My logic was that the extra note available in a 7-note scale yields greater melodic flexibility and so it is favored by musicians.

The opposite should apply to 5-note scales!

With 8, 9, 10 note scales this can't happen anymore because although they may be more flexible there are simply less of them available.

[stringtapper]

Using your logic you would have to concede that pentachords are just as "favored" as septachords.

What would motivate you to come to this conclusion?

My logic was that the extra note available in a 7-note scale yields greater melodic flexibility and so it is favored by musicians.

Well what would cause me to come to it is that you've seemingly been using two streams of logic. One based on "it's all in the math" and one based on a notion of "melodic flexibility."

[Musicologo]

Because seven is the maximum amount of fifths, that maximize the harmonic series (have the lowest numbers), you can fit within the same octave?...

Let's arbitrarily start in C.

-C (harmonic 1,2,4,8,16...)
-D (harmonic 9)
-E (harmonic 5,10)
-F* (harmonic 11)
-G (harmonic 3,6,12)
-A (harmonic 13)
-B (harmonic 15)
------
-F*# (harmonic too high, not "natural", etc...)




So basically for me it's the best compromise you can get between having fifths superimposed on each other AND the lowest ratios of harmonic series.

Of course that led to some "decisions" on not using harmonic 7/14 (that fifht doesn't show up soon, even though it has a low harmonic) at all and using a "stable" version of harmonic 11 (in fact it should be a quarter tone above F). I believe this second issue was compromised because F is also a fifth below our initial note. So it can make "sense" to include an "F" (based on the fifth superimposition) than a "quarter tone above F" (based on harmonic series).



One could argue based on this that if you want to focus on harmonic series, perhaps our scale should be an octatonic C-D-E-F*-G-A-Bb-B. IT isn't so because then this doesn't reflect a pure stacking of fifhts.

That's why I say 7 is the best compromise between the two criteria.

[stringtapper]

The actual answer to this question lies in a study of the history of musical practice—at least in Western music—from the middle ages to the 17th century. The move from the hexachordal system of mutation to the tonal system employing a leading tone was key to the shift from hexatonic to heptatonic scales in musical practice.

[aciddose]

Of course, because my argument is that the reason for the 12TET system is entirely mathematical. The reason for the selection of the structure of piano keys is again, entirely mathematical.

The reason there are more 5, 6 and 7 is again, entirely mathematical.

This is where we depart from that stance - The reason 7-note scales are favored is due to the increase of "melodic flexibility" offered by them vs. any lesser number of notes.

...Combined with the fact that there are more chords available in the 5, 6, 7 ranges than anywhere else. The majority of scales exist here and so even if there were some significant preference for 8-note scales for example, there are too few of them available in the first place and this would cancel out any preference based upon flexibility.

If we were simply concerned with flexibility and not the total number of available scales (uniqueness PLUS attractive intervals), we'd all be using all twelve notes and wouldn't be bothered by scales at all.

So it turns out much like the reason for 12TET's existence and the common preference for it, 7-note scales are favored for the same reason. They're the best possible trade-off available.

[aciddose]

The actual answer to this question lies in a study of the history of musical practice—at least in Western music—from the middle ages to the 17th century. The move from the hexachordal system of mutation to the tonal system employing a leading tone was key to the shift from hexatonic to heptatonic scales in musical practice.

Now you're doing the same thing jancivil did... "cart before the horse" indeed.

The mathematical facts didn't come about due to history.

History came about due to the mathematical facts!

You're describing exactly what I've just said about favoritism for 7 over 6 due to it's increased flexibility. You're just putting it in the context of what already happened rather than why.

My explanation offers the answer to why it happened.

I don't care whether it actually did happen or not. The fact it did work that way only provides evidence in favor of my explanation!

Any other explanation would need to fit these existing facts better than mine does before it could replace mine.

Best way to disprove what I've said would be to find an example of a fact that does not fit according to my explanation.

[stringtapper]

Of course, because my argument is that the reason for the 12TET system is entirely mathematical. The reason for the selection of the structure of piano keys is again, entirely mathematical.

The reason there are more penta, hexa and septa chords is again, entirely mathematical.

Eh no. The reason was to accommodate a greater number of keys for modulation in the tonal system. This is just history and well documented. The reason it works is entirely mathematical, but the reason it was employed is not. It was always a practical matter and was borne of the expansion of tonal practice in Western music.

[aciddose]

Of course, because my argument is that the reason for the 12TET system is entirely mathematical. The reason for the selection of the structure of piano keys is again, entirely mathematical.

The reason there are more penta, hexa and septa chords is again, entirely mathematical.

Eh no. The reason was to accommodate a greater number of keys for modulation in the tonal system. This is just history and well documented. The reason it works is entirely mathematical, but the reason it was employed is not. It was always a practical matter and was borne of the expansion of tonal practice in Western music.

I don't see what you're arguing here. You're saying "It wasn't X, it was X."

Reason it was employed? Are you saying it was employed for some other reason than that it works?

That doesn't make sense at all.

I would expect people to employ the most practical system available for the purpose required.

The fact that our use of notes and scales has changed over time does not change the fact that the current systems we employ for the purposes we employ them are the most practical systems available.

It makes absolutely zero sense to argue that the "most appropriate fit" is invalid and instead go with an argument that the reason we've come to where we are is due to the "natural evolution" of that which we're attempting to fit to.

Your argument requires my explanation to make sense. Arguing against my explanation is therefore arguing against yourself.

This is what I've been saying: You can argue against my explanation all you want, but if you can't provide a better one you've accomplished nothing.

[stringtapper]

You're describing exactly what I've just said about favoritism for 7 over 6 due to it's increased flexibility. You're just putting it in the context of what already happened rather than why.

My explanation offers the answer to why it happened.

This seems a bit embarrassing but I'm quite sure you are confusing the words "why" and "how." Why means "for what reason" or "for what purpose." As I've demonstrated the purpose of the seven-note scale was to accommodate the tonal system and the shift away from the cumbersome nature of the hexachordal system in the face of a changing musical practice.

How the seven-note diatonic scale works to fulfill that purpose can be described with mathematics if one wants to go there, but math is not answering why it was implemented, which is ultimately what the OP's question was(!!).

Ultimately it's a musical system and barring the earliest work of the Pythagoreans, musical practice itself is the impetus for the development of musical systems, whether those systems' purpose is to explain existing practice or to accommodate evolving practice.

[aciddose]

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

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

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

This is far more musical religion than science.

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.

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.