#1
how many unique scales are there if:

1. a scale is a set of intervals (western 12-tone) imposed on a root r

2. a scale must contain at least one element

3. a scale may not exceed one octave

4. a scale may not repeat an interval (taken from r)

5. for a scale S, S(r) indicates the set of notes derived from S with root r.

5. a scale S is congruent (therefore not unique) to a scale T if and only if for some r1, r2 and for every note a in S(r1) there exists a note b in T(r2) for which a = b.

so how many?
i don't know why i feel so dry
#4
Must contain at least 1 element. A one note scale? That would be a note.
A 2 note scale? That would be an interval.
#5
376. Thankyou for your enquiry and I hope that this makes you a better musician.
And no, Guitar Hero will not help. Even on expert. Really.
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#6
Quote by klintala
Must contain at least 1 element. A one note scale? That would be a note.
A 2 note scale? That would be an interval.


very picky you fit right in
i don't know why i feel so dry
#7
No idea what in god's name you just wrote, but there are 2048 possible scales.

Here it is by possible notes in the scale:

1-note scales: 11C0 = 11! / (11! * 0!) = 1
2-note: 11C1 = 11! / (10! * 1!) = 11
3-note: 11C2 = 11! / (9! * 2!) = 11 * 10 / 2 = 55
4-note: 11C3 = 11! / (8! * 3!) = 11 * 10 * 9 / (3 * 2) = 165
5-note: 11C4 = 11! / (7! * 4!) = 11 * 10 * 9 * 8 / (4 * 3 * 2) = 330
6-note: 11C5 = 11! / (6! * 5!) = 11 * 10 * 9 * 8 * 7 / (5 * 4 * 3 * 2) = 462
7-note: 11C6 = 11! / (5! * 6!) = 11C5 = 462
8-note: 11C7 = 11! / (4! * 7!) = 11C4 = 330
9-note: 11C8 = 11! / (3! * 8!) = 11C3 = 165
10-note: 11C9 = 11! / (2! * 9!) = 11C2 = 55
11-note: 11C10 = 11! / (1! * 10!) = 11C1 = 11
12-note: 11C11 = 11! / (0! * 11!) = 11C0 = 1

Found all that I believe in the first thing that came up on google here. It's an interesting read.

Really, only about 36 scales are usable and many of those scales fit into each other. Less of those are pleasant to the human ear.
#8
the math on that page is incorrect. in fact, even what you pasted isn't right. 11 choose 0 = 1? sure bout that

edit: lemme clarify, 11 choose 0 is 1. lol

my complaint is in its inclusion.
i don't know why i feel so dry
Last edited by Eastwinn at Aug 21, 2014,
#12
Quote by MapOfYourHead
There is only the major scale. It, however, has many alterations.

Not true, but even if you want to think of it that way, you still need specific names for specific "alterations".


Quote by Eastwinn
how many unique scales are there if:

1. a scale is a set of intervals (western 12-tone) imposed on a root r

2. a scale must contain at least one element

3. a scale may not exceed one octave

4. a scale may not repeat an interval (taken from r)

5. for a scale S, S(r) indicates the set of notes derived from S with root r.

5. a scale S is congruent (therefore not unique) to a scale T if and only if for some r1, r2 and for every note a in S(r1) there exists a note b in T(r2) for which a = b.

so how many?


more than you'll ever use.
Last edited by GuitarMunky at Aug 21, 2014,
#13
Quote by GuitarMunky
more than you'll ever use.


is that a challenge?
i don't know why i feel so dry
#16
Quote by Eastwinn
the math on that page is incorrect. in fact, even what you pasted isn't right. 11 choose 0 = 1? sure bout that

edit: lemme clarify, 11 choose 0 is 1. lol

my complaint is in its inclusion.


Fair enough, so take out 1 note scale and the 12 note scale. You get 2046 (cause who cares about the chromatic scale, anyway? **** those fusion players. They're too good).

If you don't include the 2 note scale as well because that's simply an interval, you get 2035

EDIT: I have seen the number 2048 multiple times though, especially when I had a little bout studying scale theory (waste of the brain cells that died trying to understand it). I haven't done the maths myself, but it was explained to me once and it made sense. Calculate it yourself and let me know if it checks out. I'm not 100% sure. I had to look it up though, because it's been awhile.
Last edited by mjones1992 at Aug 21, 2014,
#17
1: the scale of your imagination

:3

...modes and scales are still useless.


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#18
Quote by mjones1992
Fair enough, so take out 1 note scale and the 12 note scale. You get 2046 (cause who cares about the chromatic scale, anyway? **** those fusion players. They're too good).

If you don't include the 2 note scale as well because that's simply an interval, you get 2035

EDIT: I have seen the number 2048 multiple times though, especially when I had a little bout studying scale theory (waste of the brain cells that died trying to understand it). I haven't done the maths myself, but it was explained to me once and it made sense. Calculate it yourself and let me know if it checks out. I'm not 100% sure. I had to look it up though, because it's been awhile.


it's 2048 if you follow the criteria in my original post but allow for an 'empty' scale. 2047 if you follow the op exactly. and so on. the approach in the link you posted was not sufficiently rigorous for my liking and suffered because of it. a finite sum of n choose 11 is the method, anyhow.

will consider this challenge
i don't know why i feel so dry
#19
Quote by Eastwinn
it's 2048 if you follow the criteria in my original post but allow for an 'empty' scale. 2047 if you follow the op exactly. and so on. the approach in the link you posted was not sufficiently rigorous for my liking and suffered because of it. a finite sum of n choose 11 is the method, anyhow.

will consider this challenge


So, basically, it came up with the answer you were looking for, but you don't like the way the guy came up with it?

I guess that confuses me a little

I mean, if the solution and the general criteria of what makes a scale a scale is the same, couldn't you just (by the properties of maths) rearrange the equations to where the formula you've come up with is essentially the same (maybe a more streamlined version of) as what this guy did?

Sorry if I have no idea what in the **** I'm talking about (I don't). I really haven't done anything to do with maths (aside from music, but that's pretty much elementary school level shit) in over 4 years.
#20
Quote by mjones1992

Sorry if I have no idea what in the **** I'm talking about (I don't). I really haven't done anything to do with maths (aside from music, but that's pretty much elementary school level shit) in over 4 years.

so, you're saying that music is math?
it's all just coming back
it's all coming back

it's all coming back to me
#21
Quote by Baby Joel
so, you're saying that music is math?


but like math rock tho
There's no such thing; there never was. Where I am going you cannot follow me now.
#22
Quote by Baby Joel
so, you're saying that music is math?

It's an Eastwinn thread. Of course music=math in here.
#23
Quote by Baby Joel
so, you're saying that music is math?


It can be. Sometimes. Practically? Not really. No one's using math consciously while they're performing (aside from counting beats).

I guess math just applies everywhere (that's what I meant, anyway).

But if you wanna get technical (and 'rigorous', apparently) like Eastwinn, math is the science of numbers, quantity, and space, so everything in music is math. Math encompasses patterns and fractions, so notation, time signatures, tempo, scales, modes, intervals, and even the frequencies we derive the 12 notes that we use in western music from, are all maths
Last edited by mjones1992 at Aug 22, 2014,
#24
Quote by MapOfYourHead
There is only the major scale. It, however, has many alterations.

what about the minor scale, brah
Quote by crazysam23_Atax
It's an Eastwinn thread. Of course music=math in here.

cray cray samuel pls
banned
Last edited by deadsmileyface at Aug 22, 2014,
#25
Quote by crazysam23_Atax
It's an Eastwinn thread. Of course music=math in here.

LOL. You beat me to it. I remember how it was last time in the John Cage thread. I wish I could forget.
#26
Since this is mathematical and theoretical. What if the geometry of the actual scale consists of compound intervals? What if intervalic distance plays into the actual scale not just the note names, what if instead of a 4th you had to play an 11th. 3rds stay the same, or become 10ths etc.

(Runs away snckering)


I think all the times I enjoyed blowing the minds of my students with the number of chord possibilities all the way through altered 13ths, using theoretical inversions, inventions and exclusions.

Like:

Ab13/Bb as an Ab13 in 4th inversion! Fun times, it was like counting stars! It had no real world application, but it was cool to see them understand the near limitless potentials of chord voicings.

Eyes round as saucers!

Best,

Sean
Last edited by Sean0913 at Aug 22, 2014,
#27
music is not math anywhere ever thanks. please do not defy my field of interest with your vile sounds.

mjones: i'm particular about how you decide what a "scale" is in this instance. that really matters if you intend to count all of them. presenting the combinatorics without sufficient specification of what you're actually counting is a sin in my book. if that doesn't mean much to you then whatever

sean: eh that's all too difficult to count. counting simple things needs no apology, but i'd need a good reason to want to count contrived scale definitions and possible chords you could define with various traditional naming schemes. props to you though for seeing a definition and thinking "okay but what if it was different" -- that's the spirit really.
i don't know why i feel so dry
#28
Quote by Eastwinn
music is not math anywhere ever thanks.


Then why do you care about the numbers?
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Quote by DisarmGoliath
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#29
Quote by ChemicalFire
Then why do you care about the numbers?


i only care about numbers.

and shapes sometimes too.
i don't know why i feel so dry
#31
Today I played the minor scale and I was like "woah I also played 7 unique one note scales at the same time....I wonder how many other scales I just played". But then I realised that it was really stupid.
And no, Guitar Hero will not help. Even on expert. Really.
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#32
well there's an absurd number of mathematically possible scales, but scales don't exist just because they are technically possible. Most scales are derived from chords/harmonies, and exist to go along with those. They are just re-spellings of a chord.