Bela Bartok
Music for Strings, Percussion and CelestA
The Golden Ratio and Polar Opposite Keys
There is much to say about this brilliantly creative work, Music for Strings Percussion and Celesta. If you find it strange or difficult to listen to at first, please give it a chance and a good listen all the way through. It's one of those pieces that grows on you and becomes a lifelong friend. Here's a good performance of it by The Orchestre Philharmonique de Radio France conducted by Alan Gilbert.
Just two of the many interesting things about this piece and Bartok's music in general:
1) The Golden Ratio
Bartok was fascinated by the Golden Ratio. (The Golden Ratio is about 0.618033. It's a mathematically magical number with a lot of beautiful properties. Greek architects recognized it; the ratio was used in rectangles in some Greek buildings; it also has a tight relationship to the Fibonacci numbers: 1, 2, 3, 5, 8, 13, ... (each number is the sum of the two previous ones). If you extend this series infinitely, the ratio of two adjacent numbers approaches the golden ratio))...
One nifty property (or maybe definition) of the golden ratio is clear from this diagram; the ratio of b to a is the same as a to (a+b). In other words, the ratio of the small part to large part is equal to the ratio of the large part to whole. i.e.
b/a = a/(a+b) =~ 0.618033.
Bartok proportioned a lot of his music this way. This is a waveform (using Audacity software) of the first movement of this piece.
You can see the biggest peak in sound just after 4:30. This is the dramatic climax of the movement. The proportions are the same as in the a-b diagram above. The whole length of the 1st movement (in this recording) is 440 seconds. The big climax is exactly at 272 seconds. 272 / 440 = 0.618. So he placed the climax according to the proportions of the golden ratio.
Listen to the big climax here ...
1st mvt climax.mp32) Polar Opposite Keys
Bartok had a unique twist on tonal relationships. You may know closely related keys (keys that are 1 sharp or flat away from one another) are a 4th or 5th away (adjacent on the circle of 5ths). Early classical music would typically modulate from the home key to one of these, most often the dominant (5th above the home key = adding 1 sharp (or subtracting 1 flat)). For example, if Mozart started a piece in C he would typically modulate first to G.
Bartok's twist on this was to do EXACTLY THE OPPOSITE! instead of centering a piece around two adjacent keys, he would center it around two opposite keys (or tonal centers). These two keys are as "distant" as you can get.
If you're not familiar with the Circle of Fifths, read more about it here.
Pairs of keys that are opposite each other on the circle of fifths are what I'm calling polar opposites. For example, E-flat major and A major (red line). Two interesting things to note about these pairs:
1) The sum of the number of flats and sharps in the two opposite key signatures always add to 6. For example, E-flat has 3 flats, A has 3 sharps. B-flat has 2 flats, and is opposite E which has 4 sharps, and so on.
2) The interval between two notes that name the keys is always a tritone. (A tritone is a diminished 5th, or an augmented 4th). For example, C and G-flat is a tritone. F and B is a tritone, and E-flat to A is a tritone.
Bartok highlighted this key relationship throughout this piece.
Listen to the eerie fugue at the beginning of the first movement...
Note that it begins on an A.
1st mvt beginning.mp3Now listen again to the climax...
What note do the strings play on the climax with the big smack of the bass drum?
E-flat - the polar opposite of A.
1st mvt climax.mp3And at the end of the movement... back to A. Note that this phrase begins on A, the two voices move in opposite directions to the upper and lower E-flat, then meander back down to the central A, again playing with that distant E-flat to A relationship.
1st mvt end.mp3
One last point - this isn't about either of the two ideas above; it's just a very cool thematic transformation. This is a cyclic piece too! (See the Theory Corner about cyclic music). Listen again to the opening theme of the first movement...
1st mvt beginning.mp3and compare to this spastic passage in the 2nd movement. What do you notice? It's the same melody starting upside-down (inverted), then rightside-up. Also with a completely different texture and mood...
2nd mvt them trans.mp3And it pops up again - more explicitly this time - in the 4th movement...
The texture is similar to the eerie opening of the 1st mvt, but in a slightly more major-ish key giving a feeling of reconciliation toward the end of the piece.
Like a recurring theme I keep coming back to cyclic music!
4th mvt cyclic.mp3