# Music Theory Approaches: 2. Atonality



## millionrainbows

It's all been said before, and this is not the end of the series, so let's try not to get bogged down on this particular subject.

The use of graphic aids can help in conceptualizing a tone row. Graphs and grids have been used, using the up/down axis for the chromatic note scale, and the horizontal axis for the notes of the row. In this way, you can see what symmetries are present in the row by drawing a diagonal. The symmetries are found by their distance from "center" or the diagonal, which also reveals an "axis of symmetry" which can be found for any row.

Franck Jedrzejewski, in his _Mathematical Theory of Music_ has shown how "knot theory" can be used to uncover these symmetries. The notes of the row are written around a circle, and lines are drawn connecting any tritones. This shows the symmetries in a different way than the method above.

With Forte's book _The Structure of Atonal Music,_ the chromatic scale was completely analyzed and presented in all its possibilities. This book even standardized a way for theorists to speak about the various sets, as Forte 4-19 (0,1,4 8).

There is more to this, as we can go on to look at Forte's "Z-related pairs," etc.


----------



## millionrainbows

So Forte identified and catalogued all of the possible aspects of the "12" chromatic, into all possible 2-note, 3-note, up to 12-note sets, and also identified those sets which "behaved" similarly under different conditions of inversion, direction, and other transformations, named them, and put them into "indexes" of notes.

Of course, this is all dependent on looking at the 12 notes as a _totality,_ and as a list of possibilities which are mathematical in nature, being unique as individual members, not as a "tonal gestalt" in which notes are "graded" into an hierarchy of importance because of their relation to a tonic drone of tonality.

Thus, the real nature of the "chromatic" approach is inherently atonal; it has nothing to do with establishing an hierarchy of notes which reduce down to one note, or tonality. That's not its nature.

Schoenberg, who can be credited with starting this way of thinking, is unfortunately a poor example to use in explaining this new chromatic world, due to his ties to tradition and tonality. Mentioning his music as an example or an exception regarding any of this only leads to confusion on the part of those who do not have a total overview of the chromatic world.


----------



## millionrainbows

These Forte sets can also be drawn on a chromatic circle, and reveal their symmetries and intervals.


----------



## Vox Gabrieli

Psychoacoustics would play a major role as a form of Gestalt in atonality. In terms, of musical structure, I would say that the music would shy away from melodies, but attempt to invoke certain emotions in the audience. 

I have seen different reasons for writing atonal pieces - Atonality is sometimes excused to write without a key signature, accidentals galore. This is used a lot in solo ensemble in contemporary music. I've seen Shostakovich do this a few times in his piano works. Paul Creston, a writer for saxophone, writes technically challenging music that is atonal.

These threads are a good challenge for me! Excited to see future posts from OP. :tiphat:


----------



## millionrainbows

Thanks for the positive vibes, Macduff. There is a book called "Ear Training…" for atonal music.


----------



## Vox Gabrieli

Where has my milionrainbows gone! I'd like to keep seeing these threads, i'm closest to the music theory message board than the others.


----------



## millionrainbows

Thankee, McDuff, I was hoping this would turn into a discussion, but since you are still interested, I'll keep going. BTW, the skeleton for this survey thread is inspired by the book "Other Harmony" by Tom Johnson.


----------



## Vox Gabrieli

Here is a fantastic internet article that is relevant to this discussion:

http://www.sscc.edu/home/jdavidso/Music/MusicNotes/MusicNotes.html

It seems that we've gone full circle our talk of sine waves has once again come to the foreground, while association psycho acoustics and intervals together. Certain notes, like for instance, C and F#, produce tension. While this is common knowledge, Leonard Bernstein took this, and made it something more with the addition of Musical Linguistics, in his Norton lectures. The creation of tension between C and F# will also create a mood, and the contour of the piece will be tense.

Mathematically, one note is tuned to the geometric mean of an octave. These notes are perfectly spaced out keyboard do not blend in a stable fashion. The graph I provided indicates a more random pattern, than an interval of a fifth.









My thoughts aren't finished. I'll edit this post later.


----------



## millionrainbows

Interesting. Although the tritone C-F# creates a tension, it really has nothing to do with "halving" the octave, if that's what you are thinking. The octave is strictly a 1:2 or 2:1 relationship. If 1:2, it means A=440 becomes A=880, the octave above. If expressed as 2:1, A=440 becomes A=220, the octave below.

As you can see, this is strictly mathematical. The tritone, although it lies in the "center" of the octave and is the midway point, is not the same sort of relationship. 

This is because with the factors 1 and 2, there are no divisors which can be anything other than a multiple or factor of "1".

That's why Pythagoras was never able to close the octave circle. The fifths he was stacking are 3:2 or 2:3, and 1 cannot be divided in such a way to "close" the circle and come out as an whole number.


----------



## BabyGiraffe

Some different tritones in the musical scales are 
25/18 568.717426 classic augmented fourth
36/25 631.282574 classic diminished fifth
45/32 590.223716 diatonic tritone
64/45 609.776284 2nd tritone
7/5 582.512193 septimal or Huygens' tritone, BP fourth
10/7 617.487807 Euler's tritone

The 12 ET tritone is 131836323/93222358 - 600.
It's easy to see why it sounds bad - A=440 x 1.41421356237 = 622.253967444
The other tritones are usually way better for melodical and harmonical purposes.


----------



## pkoi

Here's prof. Olli Väisälä's doctoral dissertation concerning prolongation in early post-tonal music. Especially interesting are his ideas on broadening the Strausian set-theory by incorporating registral aspects, thus omitting the octave equivalence built in the Strausian theory. Also the concept of prolongation, a Schenkerian point of view, discussed in post-tonal enviroment IMO is interesting and something I haven't come across that often.

http://ethesis.siba.fi/ethesis/files/nbnfife200910222275.pdf


----------

