Scale Possibilities

[Unanything]

Here's a cool octatonic for you all.

1 - m2 - m3 - M3 - b5 - 5 - M6 - dim7

Oh, I'll bet no one can work out how many heptatonics there are with:

a 1st and perf. 4th,
and one of each of the 2nd, 3rd, 5th, 6th, and 7th notes.

[Unanything]

I'll tell you now, then.

There are 32, because 32 = 2 to the power of 5.

If 1 and 4 are constant, then you only need to worry about the 2nds, 3rds, 5ths, 6ths and 7ths. As each has two notes inherent, one and one only are included in these scales. Therefore you can count using base two like you do in gate logic diagrams in technological studies, or, as it is popularly known, BINARY. Using this, you can find all the combinations of figures and find all the scales possible in the stated condition.

0 - minor/flattened
1 - major/perfect

2nd 3rd 5th 6th 7th
0 0 0 0 0 - Locrian mode
0 1 1 0 0 - Phrygian Dominant scale
1 0 1 1 0 - Dorian mode

That means there would be 16 possible scales made up of two adjoined tetrachords.

Clever?

[davie_gravy]

Damn...

That's heavy. I'm a computer science graduate, so that makes killer fuckin sense. I was always told there's a correlation between programming and music. You just told me...it's all about base 2. Nice.

[Unanything]

Damn...

That's heavy. I'm a computer science graduate, so that makes killer fuckin sense. I was always told there's a correlation between programming and music. You just told me...it's all about base 2. Nice.

Glad I could be of service.

That means there are 2 to the 11 one octave scales of all possible number of degrees and intervals. Use eleven digits (ROOT is constant therefore excluded) with 0 - excluded, 1- included.

This means 11111111111 is the chromatic scale.

Which is... 4096/2
Which equals... 2048.

WHOA...

I'm not sure how many heptatonics there are. Mr. Gravy, can you figure a binary system? You could look through all 2048 posibilities and filter out all possibilities with more than six 1's, but that'd be way too time-consuming. Although I did make a point of writing out all the 32. And I have tabbed a lot of Meshuggah's I...

Like my octatonic by the way?

You could have this as wll for a similar sound.
1 - m2 - M3 - 4 - m6 - M6
A hexatonic, like the blues scale. Made up of adjoined Gypsy tetrachords.

[vivaldi]

wtf?
didnt understand a thing.
mate could ya explain in simple words
cheers

[davie_gravy]

He's just showing you the relationship that scales and music take in relation to binary.


0 - minor/flattened
1 - major/perfect

2nd 3rd 5th 6th 7th
0 0 0 0 0 - Locrian mode
0 1 1 0 0 - Phrygian Dominant scale
1 0 1 1 0 - Dorian mode


See, the root and perfect 4 are constant, the 1's and 0's represent whether it's a minor tone or major tone for that particular degree (2nd, 3rd, 5th, etc).

1 1 1 1 1 - Major Scale (Ionian Mode) because the major scale contains all major scale tones. Therefore, they would all be 1's which represent major/perfect scale tones

1 0 1 0 0 - Minor Scale (Aeolian Mode) because the minor scale contains a flattened 3rd, 6th, and 7th scale tone.

See how all scale possibilities are in relation to binary now?

[vivaldi]

thnaks dude

[johnmansley]

I'm not sure how many heptatonics there are. Mr. Gravy, can you figure a binary system? You could look through all 2048 posibilities and filter out all possibilities with more than six 1's, but that'd be way too time-consuming.

To arrive at a figure for the number of included notes you would permutate the number of notes to be included among the total number of notes available according to the following formula.

C(m,n) = m!/(n!)(m-n)!

where n is the number of notes to be included and m is the total number of notes for the permutation to be applied to. In all cases m =11 but in the case of heptonics, n = 7.

Now:

C(11,7) = (1x2x3x4x5x6x7x8x9x10x11)/[(1x2x3x4x6x7)x(1x2x3x4)]

C(11,7) = 8x9x10x11/1x2x3x4

C(11,7) = 7,920/24

C(11,7) = 330

Similarly, the permutations for all scale lengths are as follows:

C(11,0) = 1
C(11,1) = 11
C(11,2) = 55
C(11,3) = 165
C(11,4) = 330
C(11,5) = 462
C(11,6) = 462
C(11,7) = 330
C(11,8) = 165
C(11,9) = 55
C(11,10) = 11
C(11,11) = 1

Which gives a total of 2,048 scale combinations. However, the true number of scale combinations is 2,036 if a scale is to include at least two notes.

PS: I knew the Combinatorics module I opted for in my final year of university would come in handy!

[davie_gravy]

Very Impressive mansley!

[johnmansley]

Cheers!

[Unanything]

To arrive at a figure for the number of included notes you would permutate the number of notes to be included among the total number of notes available according to the following formula.

C(m,n) = m!/(n!)(m-n)!

where n is the number of notes to be included and m is the total number of notes for the permutation to be applied to. In all cases m =11 but in the case of heptonics, n = 7.

Now:

C(11,7) = (1x2x3x4x5x6x7x8x9x10x11)/[(1x2x3x4x6x7)x(1x2x3x4)]

C(11,7) = 8x9x10x11/1x2x3x4

C(11,7) = 7,920/24

C(11,7) = 330

Similarly, the permutations for all scale lengths are as follows:

C(11,0) = 1
C(11,1) = 11
C(11,2) = 55
C(11,3) = 165
C(11,4) = 330
C(11,5) = 462
C(11,6) = 462
C(11,7) = 330
C(11,8) = 165
C(11,9) = 55
C(11,10) = 11
C(11,11) = 1

Which gives a total of 2,048 scale combinations. However, the true number of scale combinations is 2,036 if a scale is to include at least two notes.

PS: I knew the Combinatorics module I opted for in my final year of university would come in handy!

I know what the ! does in Maths but we haven't reached that yet in it. There was a freak question on it in my exam last year and no one but me had a shitting clue. I lost a couple of marks though because I forgot the quadraric formula. I think I'll open a thread on it in Chit chat. I was breaking up algebraically while the teacher rattled on about circle and tangent equations because everone (but me) is failing at it. i seem to be the only one who knows what a negative reciprocal is. In fact I got called a smartarse by a co-Higher Maths student for mentioning it when we did the gradients of perpendicular lines.

I'm gonna be a-noting that formula. :beer:

[davie_gravy]

Yea,


I always hated fuckin factorials. Thank god for powerful calculators.

[lizardly]

Are we writing an analysis program? What's the point of this?

Manx

[johnmansley]

It's of interest to me and therefore has a point.

[powersofterror]

Uhhh, sorry to burst your bubble Unanything, but there are only two real octatonic scales--the major octatoc (0-2) and the minor octatonic (0-1). In some weird cases, there may be a third, and that is 1-2.

Major:
D-E-F-G-Ab-Bb-B-C#-D

minor:
D-Eb-F-F#-G#-A-B-C-D

You gave only the minor one, and the last note is a half step up, making it a m7 from the root, not a dim7. If it was dim, it wouldv'e been Cb.

[Rattlehead]

Uhhh, sorry to burst your bubble Unanything, but there are only two real octatonic scales--the major octatoc (0-2) and the minor octatonic (0-1). In some weird cases, there may be a third, and that is 1-2.

Major:
D-E-F-G-Ab-Bb-B-C#-D

minor:
D-Eb-F-F#-G#-A-B-C-D

You gave only the minor one, and the last note is a half step up, making it a m7 from the root, not a dim7. If it was dim, it wouldv'e been Cb.
Yeah. Isn't a dim7 just the same thing as an M6?

[davie_gravy]

Yeah. Isn't a dim7 just the same thing as an M6?

It is in terms of the actual note your playing, but it's all about the chord type that determines if it's a dim7 over major6.

[powersofterror]

That's another way of saying it's "enharmonic."

[Unanything]

Music theory can extend beyond the established you know. You should know that. I called it an octatonic because there are 8 notes in it.

You should some of the chords I'm using... I've been getting Mr. oRg to name them and he's been using powertab to name some. In fact cred' to the man.

[Rattlehead]

Music theory can extend beyond the established you know. You should know that. I called it an octatonic because there are 8 notes in it.
Yeah but the reason I was asking is because if there are enharmonic notes in an octatonic scale, like in
1 - m2 - m3 - M3 - b5 - 5 - M6 - dim7
Then it technically should be called septatonic, on account that there are only 7 actual notes.
I'm confused. Is there something I'm missing here?

You should some of the chords I'm using... I've been getting Mr. oRg to name them and he's been using powertab to name some. In fact cred' to the man.I didn't understand this passage at all.

:beer:

Edit: Cool, I just found out that you can set Calculator to binary and octal number systems! Then I thought I'd share this useless information with you kind folks, so I desecrated my post by adding this. Enjoy!

[powersofterror]

Music theory can extend beyond the established you know. You should know that. I called it an octatonic because there are 8 notes in it....
:p

[lizardly]

Catatonic Scale? :flame:

[Unanything]

OOOOH!!!
What's that? That's the coolest name for a scale! Quite, creepy! Is that the octatonic I gave? Because if it's not, I wanted to call the octa' the Macdonaldian scale. That's my last name. Has a ring if you place the stresses right.

[Unanything]

Glad I could be of service.

That means there are 2 to the 11 one octave scales of all possible number of degrees and intervals. Use eleven digits (ROOT is constant therefore excluded) with 0 - excluded, 1- included.

This means 11111111111 is the chromatic scale.

Which is... 4096/2
Which equals... 2048.

WHOA...

I'm not sure how many heptatonics there are. Mr. Gravy, can you figure a binary system? You could look through all 2048 posibilities and filter out all possibilities with more than six 1's, but that'd be way too time-consuming. Although I did make a point of writing out all the 32. And I have tabbed a lot of Meshuggah's I...

Like my octatonic by the way?

You could have this as wll for a similar sound.
1 - m2 - M3 - 4 - m6 - M6
A hexatonic, like the blues scale. Made up of adjoined Gypsy tetrachords.

I've just realised that many of these scales would in fact be chords. Haha!
That proves some of Zen's philosophies, and indeed a few of primitive Buddhism's.