The Physics of Music: Subdivision

Part I: Subdivision

This is the first in a series exploring the physics of music. They are geared toward musicians seeking a better understanding of the basics of music. If nothing else, they're food for thought.

I'll address subdivision in front two angles: first by establishing it's importance in keeping accurate time and secondly, why it works.

First Experiment

A bell tower strikes 12 times indicating the time is noon. Suppose the bell rings every 4 seconds. The experiment is simple: clap at the same time the bell rings.

Interpreting the music above: the bell rings on downbeats, subdivide according to the notation (1 beat per ring, then 2, then 4, then 8). Be careful to not subconciously subdive the 1 or 2 beats -- try to "feel" the time.

When I have tried this I found that it is difficult to match the ring with 1 or 2 beats. With 4 beats per bar I do a pretty good job and with 8, even better. The takeaway is simple: subdividing beats is an effective way of keeping steady time.

Second Experiment

Imagine yourself with a monk (high production values on this blog...).


The rules of this are the same as the first: clap at the same time the monk hits the gong. What makes this difficult is that you are not allowed to use any external or internal "clock" -- don't look at a watch, count numbers, or even notice the rhythm of your breating. Focus on "feeling" the time. The idea is that in real life we're very good at syncronizing with external clocks, but how good is our internal, natural, clock?

 

Here is what we might expect if our target is 15 minutes. We'll probably do a poor job but, on average, we might hope to be on target. What's important is the relative error, for example, if you were off and clapped after 14 minutes (an error of 1 minute) your relative error is 7%. 

Now imagine you do this again for a baseline of 1 minute. To get that same relative error you need to be within 4 seconds. Finally, imagine the baseline is our 4 seconds from the 1st experiment. To achieve the same relative error as before you need to be within 0.3seconds!

Hypothesis

Our ability to divide intervals of time does not improve as the intervals get shorter. In other words, whether one is counting in minutes, seconds or milliseconds the relative error is the same.

Why subdividing works

If smaller subdivided intervals are still error-prone, then how can it be more accurate? The answer is that as we add up many small intervals the errors cancel out.

As a more concrete example imagine we have two options rushing or dragging, each count can be ahead or behind by some random amount. Whether we are ahead or behind on any given count amounts to a coin flip -- pure chance. As we add up many coin flips the result is 50% heads (ahead) and 50% tails (behind) -- right on time!  [Fun math note: there are an infinite ways to be behind or ahead of the beat but only one way to be exactly on time -- in practice you'll always be either ahead or behind, if only very slightly!]

Conclusion

Scientists create phonemically accurate atomic clocks using the same principle -- adding up many finely spaced "ticks" -- the only difference that their clocks finely divide nanoseconds (0.000000001sec) instead of minutes.

It is imperative that long held notes are subdivided to ensure subsequent entrances are played in time. For example, rushing a 4 beat whole note by 7% leads to an early entrance by a over an sixteenth note [at M.M. =60: 4s*.07= 0.28s while a sixteenth note is 4s/(4*4) = 0.25s].

We all have natural limit to how quickly we can count. For myself, in moderate tempi, I find subdividing eighth notes to be satisfactory; beyond that I get diminishing returns for my efforts. After all, there's more to music than keeping precise time. 

Appendix: Some math

A fair coin is equally likely to come up heads or tails, and is an example of the more general Bernoulli_trial in probability theory. There is a proof than may physicists encounter while studying Condensed Matter Physics or Thermodynamics showing that for a large number of coin flips, the probability distribution is a Gaussian (Bernoulli Process).

The takeaway is that flipping a coin many times produces a "sharply peaked" Gaussian centered on a probability of 1/2 (heads comes up half the time). What's interesting is that there are no other options, in other words, for a large enough number of coin flips you will always get 50% heads!

Driving me crazy

If playing the clarinet is like driving an automatic (give it gas and it'll go), then playing the bassoon is like driving a manual, with 8 clutches, that sometimes it won't go into gear no matter how gently you press the clutch or how much gas you give it!

The Bassoon

After about 10 hours of practice I can play the Bassoon!
Dynamics: on/off
Intonation: +- semitone
Rhythm: note speaks in the same bar
Vibrato: none

What's the point of a conductor? Part II

Part I talked about the ubiquity of quality classical music in the modern era. That should be good right? I'm not so sure.

Take a look at this video of Liberace:

http://conductorland.tumblr.com/post/58939768438
I find it depressing, and not just because it's a lousy performance. What's disturbing is to hear a lack of commitment in the music. Liberace isn't interested in Tchaikovsky's music here and that cheapens the music. What's worse is this opened his show night after night!

Shostakovich 5 

This summer I listened to an academy orchestra as they prepared, in the span of a week: Shostakovich's Symphony No 5, Beethoven's 8th and a modern piece I've already forgotten (oh, it's Fandangos). The conductor was a high-profile music director who is prominent on the conducting circuit. (A post on pedestrian conductors will come later). His main purpose here is to introduce the next quote by Furtwangler about his task as a conductor.

Furtwangler

Nowadays people are apt to praise a clean and accurate reproduction of what's written in the score above everything else....Broadly, one can say, that no matter how high the technical capacity of an orchestra, the conductor has one archenemy to fight: routine. Routine is something very human, very understandable. It's the line of least resistance and there's no denying that in daily life it has its advantages. But all the more must we insist that it plays the most deadly role in music, especially in the performance of old and familiar works. In fact routine, with its loveless mediocrity and its treacherous perfection, lies like a rust on the performance of the most beautiful and best know works.
- Wilhelm Furtwangler

From an interview with H. Brailsford in London November 2nd 1948
Can be heard on Spotify: Furtwangler's Beethoven (1954-1954)
Transcript: http://www2.ee.ntu.edu.tw/~d85010/furtboard/speech/bbc1.html

Conclusion

The symphonies of Beethoven, Brahms, Mozart, Mahler have been recorded hundreds of times -- there's no shortage. While every concert is branded an marketed as an "event," the truth is that it's the same repertoire conducted by the same conductors who travel the conducting circuit year after year. Who are these conductors? They're the folks who get along so as to be re-invited next year -- they don't break the routine of the orchestra! 

The problem I have with summer festivals, and so many performances in general, is that there is a lack of commitment behind them, they're routine run-throughs of repertoire and they aren't special.

What's the point of a conductor? Part I

Kennedy's Complaint

I read an article about the violinist Nigel Kennedy a few days ago.
http://www.theguardian.com/music/2013/aug/30/nigel-kennedy-interview-conductors-overrated

In it he gives the complaint that has been around as long as conductors have existed (I assume) -- what do they do? Even most conductors would acknowledge that any decent orchestra can put on a good performance of an "warhorse" without a conductor.

I'd add a joke along the following lines: "Those who can't perform, teach -- and those who can't teach, conduct!" 

let them believe is was their magic hands

Marcel Tabuteau had a devilish comment at the end of his lessons: "My dear little friends, between us! Don't give my secrets to anyone! Especially to conductors! Let them believe is was the magic hands that did the trick!"

Most musicians know the repertoire better than ever

Summer festivals are everywhere these days -- pick a scenic setting, recruit faculty and the students will come. The purpose of these festivals is straightforward: they provide performance opportunities for young musicians. Therefore, a successful festival programs many concerts into a the span of a few weeks. For the most part, the performances are good -- the music is played technically accurately and in the proper style. And this happens not just all across the United States, across the globe even! As musicians will tell you, "standards are always rising," and that may well be attributed to these festivals.

So if musicians know the repertoire better than ever and can get along without a conductor, what's the point of a conductor these days?

Ups and Downs: Tabuteau on Brahms 3

Tabuteau demonstrates on the oboe three different ways of playing the solo in the first movement of Brahms' 3rd symphony. Below is the entire audio clip from "Marcel Tabuteau: How do you expect to play the oboe if you can't peel a mushroom?"

Audio:

Version 1

1st C# up, 2nd C# down 

Audio:

Version 2

G# up, C# down

Audio:

Version 3

Audio:

Final Thoughts

I've listened to this recording more times than I care to admit and I haven't come up with the third version; Tabuteau describes only two of the three different versions. Any help identifying the third version and what he's demonstrating would be appreciated.

1st C#	2nd C#	Tabuteau's Version
Up	Up	?
Up	Down	1
Down	Up	?
Down	Down	2

I love that classical station but they play too much Bach

1. Classical Guitar
What is with classical guitar? 

2. A piece by a composer and conductor AND ensemble I've never heard of.
On the other hand, entering a live performance with no expectations or preconceptions is thrilling! Imagine entering a venue where you don't get a program and don't look up the performers but simply listen to the music and judge it on its own merits then and there.

3. Mendelssohn Violin Concerto

4. Handel
I'll say it...Handel is an overrated composer. 

 5. When I hear three melodic notes played identically or when a fragment is repeated without any variation.

Have you ever heard someone say "I love that classical station but they play too much Bach"? I rest my case.

The Clarinetist's Checklist

I've been catching some bad habits lately: excessive body movement, slouching, and a collapsing embouchure. A framework I use is not to fix bad habits but to continually remind myself of and to reinforce good habits (1).

The following checklist will be mentally run through before I play the first note of whatever I'm playing.

Checklist

1. Feel the 'sit bones' (2)
2. Sit up straight with my head on top of a the spinal column 
3. Form my embouchure (3)
4. Breath well (4)
5. Insert clarinet, feeling the lip pressure all around the mouthpiece

References:
(1) - Arnold Jacob
(2) - See Embouchure Building
(3) - See Embouchure Building
(4) - Maria Callas at her masterclasses at Julliard often said "Take the time to breath well." There are, of course, hundreds of ways thinking about and accomplishing this -- again, I refer to Embouchure Building and Arnold Jacobs for guidance.

Don't let anyone get bored

Glenn Dicterow, the concertmaster of the New York Philharmonic, is in residence at the Music Academy of the West for 3 weeks this year. In his first week, he gave a masterclass and in the second he performed the Debussy Violin Sonata. 

I took away the following idea from the masterclass: the audience has a short attention span, they want to check their email, play a game on their phone, or leaf through the program and it's your job as a performer to not let that happen. He said, "Don't let anyone get bored with anything you play." 

His performance of the Debussy Sonata tonight embodied the belief that it's the job of the artist to hold the audience's attention. Many performers want to paint with a thousand colors (I have at my disposal a couple dozen) but he actually does! Some people make a reputation with a beautiful tone, others with virtuosic technique, but what makes Mr. Dicterow so great is his creativity in finding a color for every nuance in every phrase.

John de Lancie

"If you are intellectualizing, and the goal is perfection, the result cannot help but to be inspiring."
- John de Lancie

June 8, 2013

heavy and pear shaped
beauty in eye of beholder
sound of clarinet

Thursday, May 30

reeds all terrible
try another hopefully
should have played the flute

Reed troubles...

I'm unashamedly stealing an idea from Bass Blog.

 
working on my reeds
improvement hard to come by
nice tone is fleeting
 

"Last Chair"

I was nicknamed "last chair" in high school by a good friend because in band I was, well, last chair. For most of my life I've played clarinet in the last chair: in band I'd be the last person on 3rd clarinet, 2nd clarinet in the orchestra, 5th reed in a pit orchestra and so on.

That being said, I love music! Third chair, second chair, principal -- it doesn't matter to me any more. Music is music, no matter what part you play.