Music Theory: Set Theory, Part 2

Continuing discussions of set theory. Corresponds with chapter 9, pgs. 173 - 177.
0:00 Introduction
0:51 Equivalences / Inversional Equivalence
5:23 Best Normal Order
11:02 Prime Form

[@kaileegirl1785]

3:23 inversional equivalence 7:47 Best normal order 8:36 careful

[@williamgeiler8866]

gave this comment thumbs up because it is very le epic :-)

[@LemoUtan]

If you start with the four orderings of the notes in the set {3, 4, 8, e} - i.e. 348e, 48e3, 8e34 and e348 and then convert these all to intervals (where each digit is subtracted from its successor) you get 1434, 4341, 3414, 4143 (where the last digit in each of these is provided by the interval jump from the last pitch back to the first). These of course are simply cyclic permutations of the same 'interval string' - in this case, any four successive characters from an endless ...3414341434143414... . One can easily see that the most left compacted of the four sets (by interval) is 1434 (which came from 4-3=1, 8-4=4, e-8=3, 3-e=4), followed by (if it is of interest) 3414 (from 8e34), 4143 (from e348) and (finally, the least compacted) 4341 (from 48e3). It's alphabetical order. To get the inversions, you write those four interval strings in reverse (also in ascending alphabetical order) as 1434, 3414, 4143 and 4341. In this case they are identical (this is not always the case by any means - in fact more often it is not) and you can tell immediately that your original pitch class set was symmetric and also that the 348e form is what you call 'best normal' by default. It seems computationally more effective to work directly with the intervals. I sometimes wonder if it would not be preferable to name PC Sets not from their pitch class content, but from their interval class sequence. The major mode of _any_ diatonic scale, for example, would be (arguably) better 'named' 2212221. The minor turns up as 2122212, the lydian mode as 2221221 and the (best normal locrian) as 1221222. Admittedly one of my favourite octatonic scale names (the Bebop Dominant Flatnine) would end up as 13122111, which isn't anywhere near as much fun. Within it you can also see the normal (and prime) form 11113122 by rotating right three places. Finally, you can see that this set is not symmetric because the reversed interval sequence - with 11112213 as the alphabetically 'earliest' - isn't the same (but happens, in this case, to be 'best normal').

[@DavidEFarrell]

+Paul Sampson It's an interesting thought! The interval-class vector does describe a set by its intervallic content, though obviously it isn't ordered. Whether ordering matters depends a lot on the music we to be analyzed, I suppose...

[@LemoUtan]

+David E. Farrell Right, but an interval class vector (e.g. for a diatonic scale, which covers all 7 modes ionian, dorian, phrygian, lydian, mixolydian, aeolian and locrian) cannot be used to distinguish between equivalent pitch class sets. Whereas the interval sequence can. Indeed the interval sequence exactly describes the scale (or mode). It just forbears to nominate the actual tonic note (if atonal sets can actually be said to have a tonic!). Of course the interval class 'vector' is not actually a vector at all, in any useful mathematical sense, but is a frequency distribution. However, that boat has sailed and there's not much a poor mathematician can do about that misnomer now!

[@jennifertisi8422]

At 12:15, why are you saying 7 8 9 0? It's a C# and you have 7 8 9 1?

[@DavidEFarrell]

No reason - I misspoke there. Good catch!

[@jennifertisi8422]

David E. Farrell thank you for answering. I am studying for a theory placement exam for grad school and it’s been 32 years since I have taken a theory test. I don’t ever remember learning set theory and know it is going to be on the test. Thank you for making these videos. It is extremely helpful.

[@DavidEFarrell]

@@jennifertisi8422 Glad these materials are helpful. Best of luck to you!

[@JacobAdamGarcia]

Hi David, thanks for your videos! Is it possible for a set of 9 pitches to have two different prime forms? One of the arrangements has the set 01245689T, 9-12. But when converted to Best Normal Order, the prime form is 01234578T, 9-7. Is that possible?

[@sanjj_1]

Don't take this the wrong way, but what is the point of studying these mathematical operations on music if they're not tonal?

[@DavidEFarrell]

Sanjay Janardhan These methods are primarily to provide a framework for understanding post tonal music.

[@sanjj_1]

@@DavidEFarrell Wow! thank you for the quick reply! Well, what exactly is the intent or purpose of post tonal music? Most composers and artists nowadays and since the beginning of time seem to create music to evoke emotion, yet this style of music seems bizarre and emotionally not relatable. Forgive me if my comments seem blunt or ignorant, I'm actually a high school student writing a math paper regarding music theory.

[@DavidEFarrell]

@@sanjj_1 I understand - post-tonal music can often seem unusual if we have not encountered it. Artists make their decisions for many reasons - expression is just one of them. If the music seems unusual or unrelatable to you, it might just mean that the musical language is still foreign to you. Imagine trying a new food for the first time - it might taste unusual, but after you get used to it, you might start to enjoy it it new ways. I think for many individuals, this can be an experience they have with post-tonal music. The more they experience with an open mind, the more they find to appreciate.

[@benoitgroulx2248]

Hi David, thank you for your video!, I am trying to obtain the inversion of [5,7,e,0] using the clock diagram but my result is different from yours, does it have to do with the axis I'm choosing? or something else I'm not getting!?, Thank you!

[@DavidEFarrell]

Hi! Hard to know what to say without knowing more about your process. I feel good about my result and how I show it in this video!

[@schmiedenkrieg]

I have a question. What's the use of calling C = 0? Would it not be the same: find the best normal order and then call the lowest note 0. Directly.

[@javier.canseco]

Great job. Thanks for sharing. Best regards.

[@still451]

Thank you for ur videos :)

[@fm42jgtr]

What is the text book that you mention in the videos?

[@DavidEFarrell]

When I made this video, I was teaching with Kostka's Materials and Techniques of Post-Tonal Music.

[@Djvod998]

Thank you very much.