A Different Pentatonic Scale

[Avigdor]

Jimmy, thanks for tackling this.
On number of unique pentatonic fanilies, 5 [12] is 792 . Divide by 5 gives 158 families. But a lot of those families are duplicates and i cant think of a way to calculate the number of unique families yet. The reason i asked this question is that i am only aware of a few number of pentatonic systems and there should be more.

A talkbass thread that goes into combinatorics. I think I just died and went to heaven :*)

Any possible pentatonic scale from an instrument in tempered tuning with 12 semitones in an octave will start on
a note, say C. That leaves four more notes to be distributed over the remaining 11 semitones, and we only consider ordered sequences. The number of ways to take 4 from 11 is 11!/(7!*4!)= 11*10*9*8/(4*3*2) = 330.
So we have 330 distinct types of pentatonic scale, and 330*12 (=3960) different pentatonic scales if we count scales built on the same pattern but different root notes as different.

Now if you play many of these they will sound very poor, so the number of USEABLE pentatonic scales is much smaller. However, given my experience with jazz musicians, once someone has said "It can't be done" somewhere a jazz trio is already busy exploring doing just that...

 

[IamGroot]

A talkbass thread that goes into combinatorics. I think I just died and went to heaven :*)

Any possible pentatonic scale from an instrument in tempered tuning with 12 semitones in an octave will start on
a note, say C. That leaves four more notes to be distributed over the remaining 11 semitones, and we only consider ordered sequences. The number of ways to take 4 from 11 is 11!/(7!*4!)= 11*10*9*8/(4*3*2) = 330.
So we have 330 distinct types of pentatonic scale, and 330*12 (=3960) different pentatonic scales if we count scales built on the same pattern but different root notes as different.

Now if you play many of these they will sound very poor, so the number of USEABLE pentatonic scales is much smaller. However, given my experience with jazz musicians, once someone has said "It can't be done" somewhere a jazz trio is already busy exploring doing just that...

Out of those 330, are you sure you arent duplicating? For example, you may find several duplicates of the black key pentatonics using the combination approach.

If you had 6 note symmetrical scales, the number of unique scale patterns is very small, because those scales all use the same interval spacing. .ie, you will have numerous copies of wwwwww.

 

[Avigdor]

Out of those 330, are you sure you arent duplicating? For example, you may find several duplicates of the black key pentatonics using the combination approach.

If you had 6 note symmetrical scales, the number of unique scale patterns is very small, because those scales all use the same interval spacing. .ie, you will have numerous copies of wwwwww.

I *think* I'm not duplicating. Simpler case: 5 notes in an octave (abcde), tritonic scale (three notes).
Then I can have abc, abd, abe, acd, ace and ade, all starting on a. So these 6 all have distinct patterns.
(Note that 6 is (2 from 4) 4!/(2!*2!) = 6, with a fixed root.)

I can now call each pattern by a name and that gives me 6 different types of tritonic scales. I can also start each one on a different note, say on d, then dea has the same notes as my 6th pattern, but the same pattern as my first pattern, just starting on a different root. This is the same with the "standard" major and minor pentatonic scales: CDEGA vs ACDEG, different patterns, but exactly the same notes.

So I think it really is 330 different *patterns*, which can all start on 12 different roots.

 

[SteveCS]

That is a gorgeous sounding scale...

It's a real enigma, isn't it. Restless and energetic, yet light and airy in spite of the propensity towards the minor/diminished.

Interestingly, there is only one triad you can make that has a 'traditional' dominant function, which is the A° (vii°). The natural resolution of A° is Bb (Major I or Minor i), but guess what... there is no way to get anything like a those chords from that scale, with no Bb, no Db (for the minor option) and no F. The other 'normal' Major key resolution, to F7 (V), goes nowhere since the only notes available are A, C and Eb, which is the A°! The last remaining options are the minor key resolutions...
1: Treat A° as vii°, goes to Gb Major (VI in Bb minor or I in Gb), but that is impossible without modulation, given the available notes.
2: Treating A° as ii° in Gm, the closest we can get is D7 with no third or F7 with no root, which, as before, is A°(!)...
There is always A° as an oddball vi° in Cm, but that's something for the real avant-gardists out there.
So we are forced away from pretty much all established norms of functional harmony.

@IamGroot I have to thank you again for bringing this to the table. Coming at it from a 'common practice' background, I've not seen it before and it's the first thing I've been remotely piqued by in years. What I've written above is probably 90% BS and 100% out of context, but it's got me fired up, so I'm cool with that for the time being...

 

[onda'bass]

@IamGroot @JimmyM So I had An assignment in college to take the Slonimski book, specifically penataonics and find out how many different scales there are. The catch is ONLY different scales, not a mode of an existing one. You might be surprised to know the number was quite lower than what is being said here.

 

[Vinny_G]

There may be many, but most people only talk about the 5 modes of the major/minor pentatonic family. The wikipedia article you reference is only talking about ONE pentatonic scale family ( the "black keys") and its 5 modes.
The wiki article is confusing, btw.

It would have been less confusing if it went.
Minor Pentatonic... C Eb F G Bb
Major Pentatonic.... Eb F G Bb C
Etc.

The Coltrane pentatonic is interesting because it comes out of a different family.

Okay, I think I see where you're coming from. I have a classical training, this may explain why I had trouble understanding what seems confusing to you.

 

[Avigdor]

@IamGroot @JimmyM So I had An assignment in college to take the Slonimski book, specifically penataonics and find out how many different scales there are. The catch is ONLY different scales, not a mode of an existing one. You might be surprised to know the number was quite lower than what is being said here.

That's a slightly different question, effectively you're asking how many different combinations of 5 intervalls adding up to one octave there are, while counting rotations of the sequence as a single item, e.g.
the scales that contain two two-whole-tone intervals (T), one whole-tone (W) and three half-tones (H) are TTWHH, TTHWH, TTHHW, TWTHH, THTWH, THTHW, i.e. 6 different scales (with different interval sequences that cannot be obtained through starting on a different note) and their modes. You can do that with all other legal interval combinations to get a pentatonic scale (81111 (1), 72111 (4), 62211 (6), 63111 (4), 52221 (4), 53211 (12), 54111 (4), 44211 (6), 43311 (6), 33321 (4) and 33222 (2).

The number in brackets gives the number of scales with that distinct set of intervals, (e.g. 33321 gives these: 33321 33312 31323 and 32313, the modes of the last two are 31323, 13233, 32331, 23313, 33132 distinct from 32313, 23133, 31332, 13323, 33231)

That gives a grand total of 53 different scales. I may have overlooked a duplicate here or there, but the number should not be far off. @onda'bass, is that closer to the answer you got? Once you rule out a scale that have intervals greater than two-whole tones you come down to 18...

 

[onda'bass]

@Avigdor Yes my findings were in line with your comment. Quite the math your got going there!

 

[nnnnnn]

) How can you tell immediately that the Coltrane Pentatonic is an altered pentatonic?

What's an "altered pentatonic"? Isn't a pentatonic scale any collection of five notes?

 

[vilshofen]

2. How many mathematical combinations of 5 grouped notes out of 12 are there?

The number of pure combinations is 12!/5!x7! = 792. I welcome more math on TalkBass! I know there are tenured profs, polymaths and all-around math geniuses on the TB rolls.

 

[stringtapper]

The “Coltrane Pentatonic” is an example of a hemitonic pentatonic scale, as opposed to the two most common major and minor pentatonic scale forms, which are anhemitonic scales.

hemitonic = contains one or more semitones

anhemitonic = contains no semitones

And it’s that lack of semitones in those two most popular forms that account for their popularity, especially for use in improvisation. Having no semitones takes away the need to treat certain notes of the scale with more care than others. Everything just “sounds right” wherever you play the notes in those “normal” pentatonic scales.

 

[Pacman]

Not technique related in the least. Moved to General Instruction.

 

[vilshofen]

Actually, i just figured out a brute force method to calculate the number of unique pentatonic families that i will program over the weekend .

Groot I believe you have to concoct a permutations problem on the order of "five friends want to sit together among a row of twelve seats in a movie theater, but Igor can't sit next to Malika, and Grace has to sit next to Pascale, and no more than three empty seats between any two friends . . ." Or something.

 

[Groove Master]

Most times, when someone says "Pentatonic", we think of the 5 note scale which contains the major & minor pentatonic modes

But in fact, there are several 5 note pentatonic scales which are not part of the "Major/Minor Pentatonic Scale Family"

Such as the Coltrane Pentatonic:

C D Eb G A

which is cool sounding and worthy of your attention

Yep minor 6 pentatonic.Change the major third for the minor third. Works great on F7 chord and its relative like amin7(b5). But also it can "blues" a C major chord !!
Jerry Bergonzi has a great book on Pentatonic scales with some really great patterns in fourths very difficult to play on bass. It took me couple of months just on the first pattern in all keys on a 4 strings...

 

[IamGroot]

Groot I believe you have to concoct a combinations problem on the order of "five friends want to sit together among a row of twelve seats in a movie theater, but Igor can't sit next to Malika, and Grace has to sit next to Pascale . . ." Or something.

Actually, I think I already have the answer on my computer at my other home, but is not connected to the internet.

I posted a breakdown of scale types and distribution . I think I identified 14 or so heptonic families. I posted the results over in the DB forum, but it wasnt recieved well.

It was part of a bigger project that had nothing to do with music. My collaborator died and i shelved the work. The germans are active in this type of analysis.

 

[IamGroot]

Yep minor 6 pentatonic.Change the major third for the minor third. Works great on F7 chord and its relative like amin7(b5). But also it can "blues" a C major chord !!
Jerry Bergonzi has a great book on Pentatonic scales with some really great patterns in fourths very difficult to play on bass. It took me couple of months just on the first pattern in all keys on a 4 strings...

Doing the same. My stuff is borrowed fro Bergonzi and Campbell.

 

[Carl Hillman]

Most times, when someone says "Pentatonic", we think of the 5 note scale which contains the major & minor pentatonic modes

But in fact, there are several 5 note pentatonic scales which are not part of the "Major/Minor Pentatonic Scale Family"

Such as the Coltrane Pentatonic:

C D Eb G A

which is cool sounding and worthy of your attention

See triad pairs (i.e. hexatonic scales) to go further down the rabbit hole.

 

[shwashwa]

op is just a pentatonic built off the root of a melodic minor scale. another way to think of it is an A-7b5 pentatonic.

 

[mambo4]

Never heard of the "Coltrane pentatonic" by name
But listening to those examples, I hvae heard it before
Branfor Marsallis sprinkles that sound liberally over Sting's stuff.

 

[twelvetrombones]

There are 792 ways to pick 5 notes out of 12. Assuming we then make scales by picking a note to start on and then play the remaining 4 notes in ascending order each one of the 792 groups of 5 notes to start on, so 3960 pentatonic scales in all.