Doh! A Deer!

Written by Richard Murison

When I was a kid, growing up in a rough area of Glasgow, we were all taught music 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 clearly recall the teacher writing a very strange word on the blackboard – “Beethoven”, which in my mind I pronounced to rhyme with Teeth Oven.  Frankly, I don’t remember much else about it.  I do know that we were also 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.

We probably 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 except 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.  If we play our Do, Re, Mi,… scale on the piano keyboard beginning with the C, then we only need play the white notes.  But if we start with any of the other keys then we have to incorporate one or more of the black notes to make it work properly.  So a Do, Re, Mi,… scale that starts with a D will use a different pattern of notes than one which starts with a C.  In fact the required note pattern is unique for each of the different notes that you can start your scale on.  We call those patterns “key signatures”, and each one is named for its starting note on the Do, Re, Mi,… scale.

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 the 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 rather 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 denominate 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 suggests, mathematically, that the individual notes would 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 generally prefers to define the note A as having a frequency of 440Hz), 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 which is given by the 12th root of 2, or 1.0595:1.  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”.  Using even-tempered tuning, music has the property that any piece can be played in any key and will sound absolutely the same, apart from the shift in pitch.  Sounds simple and sensible, right?

Of course it does.  But unfortunately, it’s not the whole story.  With music, it never is.  We need to dig a little deeper into it.  As I mentioned earlier, if you double the frequency of a note you get the same note an octave higher.  But 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 us E1.  And clearly, we can then halve the frequency of E1 and get E0.  So, multiplying any given frequency by 1.5x gives us 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.  In other words, if A0=440Hz, then E0=660Hz.

Next, we’ll look at the fourth harmonic of A0.  This gives us A2 at 17600Hz, so we’ll move right on to the fifth harmonic, 2200Hz.  Subjectively, this gives us a note which is a 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 at 550Hz.  The notes A, C#, and E (at 440Hz, 550Hz, and 660Hz) together make the triad chord of A-Major, which is very harmonious on the ear.

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.  Using this procedure we can map out every last note between A0 and A1.  Starting with A0, the first fifth gives us the note E0.  The next one B0.  Then F#0.  Then C#0, and so on, eventually ending up at A1.  If you are sufficiently interested (and understand the geography of a piano keyboard), it is instructive at this point to get out a piece of paper and a pencil and do the math.  Because, somehow, this calculation ends up defining A1 as 892 Hz – when it should be 880 Hz!  Why is there a discrepancy?  Something is obviously not right.

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 systematic 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 mathematical approach defines an “Equal-Tempered” scale and gives us a structure where we can play any tune, in any key, but our harmonic-based approach is based on specific intervals that “sound” qualitatively better but don’t add up if we try to extend the principle to other notes.  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.  All this took place in the early part of the 18th Century, and 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?  It turned out that there wasn’t – and still isn’t – a simple answer for that.  Every tuning that isn’t equally-tempered will, by pure definition, have the property that a piece played in one key will sound subtly different if played in another key.  So you have a choice.  You can have a tuning that sounds identically bad in every key, or one where each different key has a sound that may vary slightly in character, but none of which sounds “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 loudspeakers, even of digital vs analog sound.  However, by Johann Sebastian Bach’s time (~1750’s), a consensus emerged in favor of what is termed “Well-Tempered” tuning.  Bach famously wrote a collection of keyboard pieces entitled “The Well-Tempered Klavier” whose purpose is to illustrate musically the different tonal characters of the different tonal keys resulting from this tuning.  I won’t go into the specifics regarding how Well-Tempered tuning is obtained, but in practice it has been adopted as the basis of all modern Western music.  Even so, you may (or, being audiophiles, may not) be surprised to learn that there is no modern consensus on what, precisely (since in 2017, unlike in 1717, we can be as precise as we want), a “well-tempered” tuning actually is, or why.

One thing which emerges as a result of all this is that the tonal palette of a composition is affected, to a certain degree, by the key in which it is written.  Below is a web site that explores the “character” of each of the different tonal keys.  You may (or, being audiophiles, may not) be surprised by the level of inconsistency, not to mention ambiguity!

This is what is behind the preference for classical composers to name and identify their major works by the key in which they are written.  Or even to decide what that key was going to be in the first place.  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.  Well, now you know!

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