How many music scales are there?

How many scales in music are there?

There are 154,440 scales in music, depending on certain parameters.

I got this answer from a friend of mine, Lobert O Reary, but be careful: he has red hair.

Here is what Rob said :

To get the number of possible scales by these standards you want to calculate n!/(n - r)! where N is the total number of symbols in the system (13), R is the minimum sequence length (5), symbols must remain in sequence (only ascending scales) and repetition is excluded. The function is called Pascals Triangle - it's very useful in combinatorics. Unless I'm missing some factor, 154440 should be the number you're looking for.

Here is how I replied:

thanks for that! nice one. So I better get practicing then! and I trust your answer far more than any nonsense I try and calculate!. Combinatorix? a friend of * and obelix?!

Anyway, here is the original essay that prompted the answer above:

How many scales in music are there?

I don't know the answer. Please, if you are good at maths, tell me the correct answer using the 7 parameters below.

Children in particular ask this brilliant question so I like giving them some kind of answer which doesn't involve getting too philosophical with regards to infinity, Neptune, gravity and cats etc.

Being a teacher of music, it's a question I have been asked so many times. And I ask it to myself every second of every day. There are an infinite number of scales for an infinite number of reasons, but parameters have to be put in place to give a child (or anyone) some kind of answer one can get their head around.

Keep it simple: stick to piano. It's easy to visualize and most musicians can relate to it in some kind of way. Let's stay with scales starting on middle C and ending on C exactly one octave above. All these restrictions, I know, I know. But if you don't chuck them in, well, your away to the moon. Daydreaming. Scales turn into light years, and start becoming elusive. I met a scale, Jimola Mac Bunty, near Neptune yesterday, and she was made of well over eight billion notes, spanning millions of octatatataves. We argued: its was confusing. (I left the keys on the table).

Ending an octave above gives a great sense of finality to the sense of a scale, the feeling of beginning and ending, although of course a scale is whatever sequence, permutation or combination of notes you want it to be.

I emphasize this fact simply because of the reams of nonsense I've read about how many scales people pretend there to be, or not to be.

So back to earth, to now and here, as they might say.

Here are our Beacons so far:

1. Were doing this on the piano.

2. Start on middle C and finish on C an octave above.

(If we didn't do it on piano, and did it on violin, we'll have the "F sharp is flatter than G flat !" conversation, and once again, Jimmiloa Mac Bunty is crying tears of enharmonic non-equivalents. Piano it is. Non-fretted string players, shuffle aside.

Good stuff. So, there is a maximum of 12 notes in a scale (13 if you include the octave one note above). What about the minimum? Hmm…well, I'm going to stick to a minimum of 5 notes. The pentatonic scale (pent is a great way of making anything to do with 5 sound relevant. I really am interested in the pentarelavance of hedgehogs) is extremely common across the world and once you reduce a scale to just four notes it just feels like were in quadra-cluster-quartet-quadrant- territory. It just annoys me. I think of groups of 4 notes as arpeggios or just parts of scales. That's how my brain works right now. (Of course catch me tomorrow and I'll be ranting on about the infinite variety of two note scales. Yup. Jimolola is a-calling). 5 is the minimum, ok!? Take your 3 note scales and your predator drones and sod off. Go on, go. Get out.

Great. So a minimum of 5 different notes, maximum 12.

(or put more simply, or more complexly, the full c chromatic scale is the longest scale. You wouldn't believe how many drafts I've done of this entire essay, trying to hone in and refine all these numbers for the sake …yawn, pardon me. Pass the salt, Gertrude).

Oh and another parameter: All scales ascending. Remember, keeping the parameters simple. Thats because if you go back down again, we will have the melodic minor argument, and philosophize about the other trillion a.w.o.l. scales that want to change going back down. As far as I know jugglers don't juggle with balls which change their…um….dimensions? on the way back down. (:

Keep It Simple, always the key.

I'm tired: I need food.

Here are the seven very simple parameters.

1. Piano

2. Each scale must start on middle C, and end on the C an octave above.

3. A maximum of 13 notes (the chromatic scale is obviously the Maximum number of notes) per scale.

4. A minimum of 5 different notes (this number includes the starting note C, excludes the C an octave above )

5. Only continually ascending, no descending.

6. No repetition of notes (the C an octave above is not considered a repetition. It's only a necessary function of the scale)

7. No notes played at the same time

6. No "ambiguities". This is a standard, normally tuned piano, with no gimmicks/prepared strings/etc.

7. Each scale is the same dynamic (playing loud or soft has no relevance to the type of scale)

So above we have 7 parameters I have plucked out of the blue. Of course you yourself can play games with this. Bear in mind there is an element of sheer nonsense (a.k.a. Fun) because so many of these scales will not sound like scales as you think about them normally. Just imagine, or sing, or play a C,Csharp D, Dsharp and then jump up to the B above those guys. Most music academics or scholars would not consider this a scale, but most music academics or music scholars often forget what smiling is, and have no concept of what music is. Let's not have this conversation now. Do you like hedgehogs?

So when you do the maths, or get someone to do it for you, the answer is that there are 1246234527579842385345684548960452ß85069 scales. I'm lying.

Shannon airport. Cash. Middle East. Trump-Brexicide…. Oh dear.

ok, sorry. It's late. The answer is…..I don't know the answer. Please, if you are good at maths, tell me the correct answer using the 7 parameters above.

If a child, or anyone, asks, make sure you bring up infinity, otherwise Jimola will end up thinking Barack Obama is somehow peaceful. (I can hear Obama's drawl: Hey, I mean its just collateral damage. Shedding tears over high school shootings in his own country, and just shrugging off his victims in Whereisthatistan, like dusting off cobwebs. What? Who? Whereiszan? What's that? Exactly.

Lack of education is lethal.

And don't even pretend you did all you could to prevent the Iraq war. Did you actually climb over the fencing and lie down on the runway? No. Me neither. I watched television instead. Radical, huh? Awesome. In fact, how many anti-war marches have you gone on in your lifetime? Wait a second…

Thanks for reading. Wicked. Wicked. How many scales are there really? Please tell me.

In fact I was so strange recently I started to try and write out all the scales:

C C# D D# E F F# G G# A A# B C

But then I got tired. How many scales are there, anyway?

Simon, 13th December 2016

P.S. Happy Solstice.