7 Scale Systems you might want to know

[IamGroot]

MINOR PENTATONIC
1 1193 MINOR PENTATONIC 1 b3 4 5 b7
2 661 Major Pentatonic 1 2 3 5 6
3 1189 4/b7 Pentatonic Suspended 1 2 4 5 b7
4 1321 Minor b6 Pentatonic Blues Minor 1 b3 4 b6 b7
5 677 4/6 Pentatonic Blues Major Pentatonic 1 2 4 5 6


MAJOR SCALE
1 2741 MAJOR Ionian 1 2 3 4 5 6 7
2 1709 Dorian 1 2 b3 4 5 6 b7
3 1451 Phrygian Spanish 1 b2 b3 4 5 b6 b7
4 2773 Lydian 1 2 3 b5 5 6 7
5 1717 Mixolydian Dominant 1 2 3 4 5 6 b7
6 1453 Aeolian Natural Minor 1 2 b3 4 5 b6 b7
7 1387 Locrian m7b5, Half Dim. 1 b2 b3 4 b5 b6 b7

MELODIC MINOR
1 2733 MELODIC MINOR 1 2 b3 4 5 6 7
2 1707 Dorian Flat 2 Phrygian #6 1 b2 b3 4 5 6 b7
3 2901 Lydian Augmented 1 2 3 b5 b6 6 7
4 1749 Lydian Dominant Overtone 1 2 3 b5 5 6 b7
5 1461 Mixolydian b6 1 2 3 4 5 b6 b7
6 1389 Aeolian b5 Locrian #2 1 2 b3 4 b5 b6 b7
7 1371 Superlocrian Altered Dom. 1 b2 b3 3 b5 b6 b7

HARMONIC MINOR
1 2477 HARMONIC MINOR Aeolian $7 1 2 b3 4 5 b6 7
2 1643 Locrian Natural 6 1 b2 b3 4 b5 6 b7
3 2869 Major Augmented Ionian #5 1 2 3 4 b6 6 7
4 1741 Lydian Diminished Ukranian Dorian 1 2 b3 b5 5 6 b7
5 1459 Phrygian Dominant Freygish 1 b2 3 4 5 b6 b7
6 2777 Aeolian Harmonic Lydian #2 1 b3 3 b5 5 6 7
7 859 Ultralocrian Altered Diminished 1 b2 b3 3 b5 b6 6

WHOLE TONE
1 1365 WHOLE TONE 1 2 3 b5 b6 b7


AUGMENTED
1 2457 AUGMENTED b3 1 b3 3 5 b6 7
2 819 Augmented b2 1 b2 3 4 b6 6

DIMINISHED
1 2925 DIMINISHED WH Whole Step Half Step 1 2 b3 4 b5 b6 6 7
2 2779 DIMINISHED H W Half Step Whole Step 1 b2 b3 3 b5 5 6 7

 

[Malcolm35]

Thanks, now in my study binder.

 

[noeinstein]

Please explain the first four, or so, numbers of each example, e.g., 2741 MAJOR Ionian 1 2 3 4 5 6 7
Thanks!

 

[mambo4]

Please explain the first four, or so, numbers

Those are the "binary equivalents" of the scale's pattern of half and whole steps
Sort of like a finger print of the scale's uniqueness.
They do not contain actionable information and can be ignored.
But if you're curious, they are explained here

 

[noeinstein]

Those are the "binary equivalents" of the scale's pattern of half and whole steps
Sort of like a finger print of the scale's uniqueness.
They do not contain actionable information and can be ignored.
But if you're curious, they are explained here

Thank You

 

[IamGroot]

Please explain the first four, or so, numbers of each example, e.g., 2741 MAJOR Ionian 1 2 3 4 5 6 7
Thanks!

Pretty much what @mambo4 said.
The 5 digit number allows me to look up a scale with 100% certainty I am getting exactly that scale. There are a lot of alternate names for certain scales. Is the Minor Pentatonic the same as the Blues Minor Pentatonic???.Been there , done that. I can also look up scales as a string of notes. For example 1 2 3 4 5 6 b7 is the Mixolydian.

Other than that, unless you are developing software, it wont be too much use. You cant look at a number like 3221 and make any conclusions.

BUT....you will notice that every scale number is odd. WHy?

 

[IamGroot]

Its a 4 digit number , my bad.

 

[noeinstein]

I'm fascinated by the relationship between music and math. I know just enough about both to want to know more.

 

[5StringBlues]

I'm fascinated by the relationship between music and math. I know just enough about both to want to know more.

Name checks out.

 

[noeinstein]

Name checks out.

 

[noeinstein]

Pretty much what @mambo4 said.
The 5 digit number allows me to look up a scale with 100% certainty I am getting exactly that scale. There are a lot of alternate names for certain scales. Is the Minor Pentatonic the same as the Blues Minor Pentatonic???.Been there , done that. I can also look up scales as a string of notes. For example 1 2 3 4 5 6 b7 is the Mixolydian.

Other than that, unless you are developing software, it wont be too much use. You cant look at a number like 3221 and make any conclusions.

BUT....you will notice that every scale number is odd. WHy?

How about a hint?

 

[IamGroot]

How about a hint?

1717 Heptatonic
Binary = 11010110101
Reversed = 101011010110
R b2 2 b3 3 4 b5 5 b6 6 b7 7
1 0 1 0 1 1 0 1 0 1 1 0

So Scale is R 2 3 4 5 6 b7 = Myxolydian. Seven bits = heptatonic scale.

 

[IamGroot]

MINOR PENTATONIC
1 1193 MINOR PENTATONIC 1 b3 4 5 b7
2 661 Major Pentatonic 1 2 3 5 6
3 1189 4/b7 Pentatonic Suspended 1 2 4 5 b7
4 1321 Minor b6 Pentatonic Blues Minor 1 b3 4 b6 b7
5 677 4/6 Pentatonic Blues Major Pentatonic 1 2 4 5 6


MAJOR SCALE
1 2741 MAJOR Ionian 1 2 3 4 5 6 7
2 1709 Dorian 1 2 b3 4 5 6 b7
3 1451 Phrygian Spanish 1 b2 b3 4 5 b6 b7
4 2773 Lydian 1 2 3 b5 5 6 7
5 1717 Mixolydian Dominant 1 2 3 4 5 6 b7
6 1453 Aeolian Natural Minor 1 2 b3 4 5 b6 b7
7 1387 Locrian m7b5, Half Dim. 1 b2 b3 4 b5 b6 b7

MELODIC MINOR
1 2733 MELODIC MINOR 1 2 b3 4 5 6 7
2 1707 Dorian Flat 2 Phrygian #6 1 b2 b3 4 5 6 b7
3 2901 Lydian Augmented 1 2 3 b5 b6 6 7
4 1749 Lydian Dominant Overtone 1 2 3 b5 5 6 b7
5 1461 Mixolydian b6 1 2 3 4 5 b6 b7
6 1389 Aeolian b5 Locrian #2 1 2 b3 4 b5 b6 b7
7 1371 Superlocrian Altered Dom. 1 b2 b3 3 b5 b6 b7

HARMONIC MINOR
1 2477 HARMONIC MINOR Aeolian $7 1 2 b3 4 5 b6 7
2 1643 Locrian Natural 6 1 b2 b3 4 b5 6 b7
3 2869 Major Augmented Ionian #5 1 2 3 4 b6 6 7
4 1741 Lydian Diminished Ukranian Dorian 1 2 b3 b5 5 6 b7
5 1459 Phrygian Dominant Freygish 1 b2 3 4 5 b6 b7
6 2777 Aeolian Harmonic Lydian #2 1 b3 3 b5 5 6 7
7 859 Ultralocrian Altered Diminished 1 b2 b3 3 b5 b6 6

WHOLE TONE
1 1365 WHOLE TONE 1 2 3 b5 b6 b7


AUGMENTED
1 2457 AUGMENTED b3 1 b3 3 5 b6 7
2 819 Augmented b2 1 b2 3 4 b6 6

DIMINISHED
1 2925 DIMINISHED WH Whole Step Half Step 1 2 b3 4 b5 b6 6 7
2 2779 DIMINISHED H W Half Step Whole Step 1 b2 b3 3 b5 5 6 b7

Note the corrected b7 to the Diminished H W scale as pointed out by Bob Ross.

 

[Bob_Ross]

unless you are developing software, it wont be too much use. You cant look at a number like 3221 and make any conclusions.

Perhaps of more value would be treating scales as ordered pitch class collections and then calculating the interval vector of that collection. That way you could have a number associated with each scale that is of use musically.

(Certain sonic characteristics of a collection can be gleaned simply by looking at its interval vector; also, noting that two non-identical pitch collections share the same interval vector -- aka, Z-relation -- can tell you a lot about how those two scales might be similar, and might even be able to co-exist in the same piece of music.)

 

[IamGroot]

Perhaps of more value would be treating scales as ordered pitch class collections and then calculating the interval vector of that collection. That way you could have a number associated with each scale that is of use musically.

(Certain sonic characteristics of a collection can be gleaned simply by looking at its interval vector; also, noting that two non-identical pitch collections share the same interval vector -- aka, Z-relation -- can tell you a lot about how those two scales might be similar, and might even be able to co-exist in the same piece of music.)

Whats a pitch class and whats an interval vector?

My interest in musical scales was to get software in Excel up to do counting in a twelve tone system, identify the major scales against name and number, resolve them down into musical families and get some understanding as to the distribution. Now that that is accomplished, my interest is purely mathematical, ie. counting (Combinatorics).

Fox example, a 7 member scale in the twelve tone system has:

462 scales,
88 families
462 scales with at least 1 half tone or larger,
462 scales with at least 1 whole tone or larger
441 scales with at least a 1.5 tone or larger, implying 21 scales or only three families have the traditional H W pattern
196 scales with at least a interval of 4 half tones or larger,
49 scales (=7*2)with at least one interval of 5 or larger,
7 scales (=7^1) with at least one interval of 6 or larger (The chomatic Unidecamode family)

Do you know why the distribution is 462,462, 441,196, 49, 7 is and what the formula is?

 

[Bob_Ross]

Whats a pitch class and whats an interval vector?

A "pitch class" is all instances of a particular note name, regardless of register. For example, the note C# (with no other qualifications) is by definition of the pitch class C# ...and all C#s on the piano are of that same "pitch class". All Ds are of the pitch class D, all Ebs are of the pitch class Eb, and so on. Contemporary (20th Century and beyond) music theory uses the term "pitch class" to differentiate these All Instances Of A Note Regardless Of Register from the term "pitch"...the term "pitch" refers to a single instance, designated by register (octave). So the note C#0 (lowest octave) is the same "pitch class" as C#3 (three octaves higher) but they are considered two separate, distinct, non-identical "pitches".

"Interval vector" is a numeric expression which identifies the quantity of each interval type contained in a pitch class collection (i.e., a scale) when treated as an ordered set. It's shorthand for an inventory of how many unique interval classes are contained within a musical cell.
For example, the major scale contains two half-steps (designated "1"), five whole-steps (designated "2"), four minor thirds (designated "3"), three major thirds (designated "4"), six perfect fourths (designated "5"), and one tritone (designated "6").*
So the "interval vector" of the major scale would be [254361] ...and that number can convey lots of musical information about how the pitch collection/scale sounds, as well as what types of unique simultaneities (aka, chords) can be built from it.

*intervals larger than tritones are considered inversions of their reciprocal smaller intervals and so don't get counted separately.

 

[IamGroot]

The Interval vector sounds interesting.

If I said a major chord was a [2,5} and a pentatonic minor was a [0,3,2], what name would that have?

 

[Bob_Ross]

If I said a major chord was a [2,5} and a pentatonic minor was a [0,3,2], what name would that have?

Bill?
Fred?
Doogie?

I'm pretty partial to "Fred" myself.

.
.
.



...but seriously, it would depend what you're trying to convey. [2,5] is not the "interval vector" for a major triad so you can't correctly call it that.

 

[IamGroot]

[2,5} is two half tones, 5 whole tones .
[0,3,2] is zero half tones, three whole tones, 2x 1.5 tones

If we wanted to add a whole bunch of happy little trees, that wiuld be [12]

 

[SteveCS]

If I said a major chord was a [2,5}

[2,5} is two half tones, 5 whole tones .
[0,3,2] is zero half tones, three whole tones, 2x 1.5 tones

If we wanted to add a whole bunch of happy little trees, that wiuld be [12]

The major chord contains 3 intervals - R to 3rd (ic4), R to 5th (ic5) and 3rd to 5th (ic4). Because the Root to 5th Interval is larger than a tritone it gets re-evaluated as a P4 (ic5). So the major chord has 1-off M3, 1-off m3 and 1-off P5 (=P4). So the Interval Vector is 0,0,1,1,1,0. The '0' represent interval classes not present in the pitch class set, in this case the notes of the major chord.