Saturday 23 November 2013

A Sense Of Scale

When I was a kid, growing up in a rough area of Glasgow, we were all taught music at school - even at elementary school.  I have a memory going back to about age eight, sitting in a classroom that was right next to the school gym.  I recall it containing gym equipment.  And I recall the teacher writing two very strange words on the blackboard - “Beethoven” and “Mozart”.  Frankly, I don’t remember much else about it.  I do know that we were taught the so-called “Tonic Solfa”, - Do, Re, Mi, Fa, So, La, Ti, Do, which is in musical parlance the major scale.  On a piano keyboard this is easily played as C, D, E, F, G, A, B, C.  I think it is sad that this sort of thing is no longer taught in most schools as part of the core syllabus.

I think we also all know that those notes I mentioned form only the white keys on the piano keyboard, and that there are also black keys that sit between them, set back slightly from the front of the keyboard.  Every pair of white notes has a black note between them, save for E/F and B/C.  This gives the piano keyboard its characteristic pattern of black keys, which alternate up and down the keyboard in groups of two and three.  It is this breakup of the symmetry that allows us to immediately identify which note is which.  For instance, the C is the white key immediately to the left of the group of two black keys.  The other thing most of us know is that every black note has two names - the black note between C and D can be called either C-sharp (written C#) or D-flat (written D♭).  And if you didn’t know that before, well you do now!

Any performing musician will tell you that it is critically important to get your instruments in tune before you start playing.  And if you are in a band, it is important that all instruments are in tune with each other.  Some instruments (most notably stringed instruments) have a propensity to go out of tune easily and need frequent tune-ups, some even during the course of a performance.  Even the very slightest detuning will affect how the performance sounds.  Let’s take a close look at what this tuning is all about, and in the process we will learn some very interesting things.

Something else that I think you all understand is that the pitch of a note is determined by its frequency.  The higher the frequency, the higher the note.  And as we play the scale from C to the next C above it (I could denote those notes as C0 and C1 respectively), we find that the frequency of C1 is precisely double the frequency of C0.  In fact, each time we double the frequency of any note, what we get is the same note an octave higher.  This means, mathematically, that the individual notes appear to be linearly spaced on a logarithmic scale.  If we arbitrarily assign a frequency to a specific note by way of a standard (the musical world now defines the frequency 400Hz as being the note A), we can therefore attempt to define the musical scale by defining each of the adjacent 12 notes on the scale (7 white notes and 5 black notes) as having frequencies which are separated by a ratio given by the 12th root of 2.  If you don’t understand that, or can’t follow it, don’t worry - it is not mission-critical here.  What I have described is called the “Even-Tempered Scale”.  With this tuning, any piece can be played in any key and will sound absolutely the same, apart from the shift in pitch.  Sounds sensible, no?

As I mentioned earlier, if you double the frequency of a note you get the same note an octave higher.  If you triple it, you get the note which is the musical interval of “one fifth” above that.  In other words, if doubling the frequency of A0 gives us A1, then tripling it gives is E1.  By the same logic, we can halve the frequency of E1 and get E0.  So, multiplying a frequency by one-and-a-half times, we get the note which is a musical fifth above it.  Qualitatively, the interval of one-fifth plays very harmoniously on the ear, so it makes great sense to use this simple frequency relationship to provide an absolute definition for these notes.  So now we can have A0=400Hz and E0=600Hz.

The fourth harmonic of 400Hz is another A at 1600kHz, so let's look at the fifth harmonic.  This gives us the musical interval of “one third” above the fourth harmonic.  This turns out to be the note C#2.  So we can halve that frequency to get C#1, and halve it again to get C#0.  The notes A, C#, and E together make the triad chord of A-Major, which is very harmonious on the ear, so we could use this relationship to additionally define C#0=500Hz.

We have established that we go up in pitch by an interval of one-fifth each time we multiply the frequency by one-and-a-half times.  Bear with me now - this is what makes it interesting.  Starting with A0 we can keep doing this, dividing the answer by two where necessary to bring the resultant tone down into the range of pitches between A0 and A1.  If we  keep on doing this, it turns out we can map out every last note between A0 and A1.  The first fifth gives us the note E.  The next one B.  Then F#.  Then C#.  Let’s pause here and do the math.  This calculation ends up defining C# as 506.25Hz.  However, we previously worked out, by calculating the fifth harmonic, that C# should be 500Hz!  Why is there a discrepancy?  In fact, the discrepancy only gets worse.  Once we extend this analysis all the way until we re-define A, instead of getting 400Hz again we end up with 405.46Hz.  And what about the “Equal-Tempered Scale” I mentioned earlier - where does that fit in?  That calculation defines a frequency for C# of 503.97Hz.

The problem lies in the definition of the interval of one-fifth.  On one hand we have a qualitative definition that we get by observing that a note will play very harmoniously with another note that has a frequency exactly one-and-one half times higher.  On the other, we have a more elaborate structural definition that says we can divide an octave into twelve equally-spaced tones, assign each tone with the names A through G, plus some black notes (sharps/flats), and define one-fifth as the interval between any seven adjacent tones.  I have just shown that that the two are mathematically incompatible.  Our structural approach gives us a structure where we can play any tune, in any key, and defines an “Equal-Tempered” scale, but our harmonic-based approach is based on specific intervals that “sound” better.  How do we solve this conundrum?

This was a question faced by the early masters of keyboard-based instruments, where each individual note can be precisely tuned at will to a degree of precision that was not previously attainable by other instruments.  All this took place in the early part of the 18th Century, back in the time of our old friend Johann Sebastian Bach.  It turns out they were very attuned to this issue (no pun intended).  The problem was, if you tuned a keyboard to the “Equal-Tempered” tuning, then pieces of real music played on it did not sound at all satisfactory.  So if the “Equal-Tempered” tunings sounded wrong, what basis could you use to establish something better?  There isn’t a simple answer for that.  Every alternative will, by pure definition, have the property that a piece played in one key will sound slightly different played in another key.  What you want is that the different keys have the property of each having a sound which we accept may be different in character, but such that none of them sound “bad” in the way that the “Equal-Tempered” tuning does.

This problem shares many aspects with the debate between advocates of tube vs solid-state amplifiers, of horn-loaded vs conventionally dispersive loudspeaker, even of digital vs analog.  If the solution is to be found in a consensus opinion of a qualitative nature, there is always going to be a divergence of opinion at some point.  In Bach’s time, there was a consensus which emerged in favour of what is termed “Well-Tempered” tuning.  I won’t go into the specifics regarding how that particular tuning is derived, but in short this is now the basis of all modern Western music.  Bach wrote a well-known collection of keyboard pieces titled “The Well-Tempered Klavier” whose function is to illustrate the different tonal character of the different musical keys which arise from this tuning.

One thing which emerges as a result of all this is that the tonal palette of a composition is determined, to a certain degree, by the key in which it is written.  This is what is behind the habit of classical composers to name and identify their major works by the key in which they are written.  You may have wondered why Beethoven’s ninth symphony was written in D-Minor, or, given that it had to have been written in some key, why the key always gets a mention.  If so, well now you know.

Here is a web site that explores the “character” of each of the different keys.  Of course, since this is a purely qualitative assessment, YMMV.  Enjoy!…