Temperament, Cents, and Hertz

Hello again, all. Thanks to those of you who decided to read the first installment in this musical series of mine. Last time we discussed the Circle/Spiral of 5ths, some things about Pythagoras, and the basics of the Harmonic Series. Today the focus of our discussion is the concept of Temperament, but we will also cover the idea of Cents and the measurement known as Hertz.

I'll pick up where we left off in the Circle of 5ths lecture. People were stumped by the Pythagorean Comma and wished to find a tuning system that circumvented this problem. Attempts to do so resulted in what are called Temperaments, different tuning systems devised to capture certain natural elements of music; no matter what elements of nature's harmony a temperament replicates, it always falls short in other areas.

I will be discussing three types of temperament: Just, Meantone, and Equal Temperament. I mentioned the Pythagorean musical scale last time, but this is not a tempered scale; instead, it derives its pitches directly from the Spiral of 5ths, and contains the very issues that Temperaments were designed to snuff out.

Just Intonation is designed to tune chords with absolute accuracy. In J.I., the Major chord is 4-5-6 exactly, the natural harmonics in series. The octaves, fifths, major thirds, harmonic sevenths, ninths -- indeed every interval of the diatonic scale (Do Re Mi Fa Sol La Ti Do) is pure. There is no harmonic compromise; this is exact harmony as nature intended. Sounds spiffy, doesn't it?

It certainly does, but there's a significant problem involved. You can tune an instrument--say, a keyboard--to play in Just intonation with a diatonic (=major) scale. The scale of C-D-E-F-G-A-B-C, for example. So you have this beautiful, pure C Major scale on your instrument--BUT. Because you've tuned every interval in relationship to the note C, only the diatonic scale on C (or the relative A Minor scale) can be played in tune. Every other interval against C will sound out of tune and strikingly dissonant, ruining the wonderful harmony.

The basic point here is that when you're using J.I., your instruments cannot transpose. You can't play a piece and C and immediately afterward play a piece in F#. For this reason, if you listen to music of the Baroque era you'll find that almost every song remains in a single key, and uses either a Major or Minor scale (probably both) exclusively. They are beautiful compositions but restricted to a certain number of notes, namely the diatonic seven as opposed to all 12 Chromatic pitches. That is the downfall of Just Intonation: crisp, pure intervals, but no (or very limited) transposition to other keys.

Meantone Temperament was an attempt to reconcile accurate harmonicity with the ability to transpose. There were all kinds of meantone temperaments, each with a different focus. Some aimed to preserve pure Thirds, others pure Fifths, others to approximate all the intervals while allowing for freedom of transposition to other keys. There are enough types of meantone tunings that I don't want to get into them right now, but if you're curious there's a fine Wiki article that can tell you things I wouldn't even know about.

An honorable mention goes to J.S. Bach, a Baroque composer who invented a mythical tuning called Well Temperament. There is no consensus on what exactly Well Temperament sounded like, but it was said to sound in tune in every chromatic key signature while allowing each key signature to have its own unique sound. So the exact intervals used in C Major and F# Major were somewhat different, but the harmony in each of them would be essentially smooth and pleasing to the ear. Some people point to a certain Baroque tuning as likely being Well Temperament, but after listening to example of the tuning I'm not sure I believe them. Others say that Well Temperament was only lost in history for so long because another clever temperament popped up at roughly the same time.

Equal Temperament. 12-tone, equal, temperament. All rise for the pledge of allegiance! *hand over heart*

So 12-tone Equal Temperament, it's the guy we in the Western World look to for our musical tuning system. It caught on like wildfire when the concept appeared toward the end of the Baroque era. So what's so fab about it? First off, it approximates Perfect 5ths almost perfectly. In this case, "almost" is a VERY good thing. You recall last time I said people were looking for a way to close the Spiral of 5ths? Yeah, 12-TET did that. Like a boss. To close the Spiral into a Circle of 5ths, each 5th used in the circle was made ever-so-slightly flat. That way, when you start on A and go up 12 Equal Tempered 5ths, you land on the same A as if you'd gone up 7 Octaves. This notion of a perfect acoustic circle was too good to be true, and it quickly began to dominate the world of instrument tuning /even though/ some people thought it sounded weird at the time.

Alright, we make the Perfect 5ths a little flat. But how much flatter than a real one? The answer: almost exactly 2 Cents flatter.

Wait WAT? What the devil are "Cents"? Well Cents are a notion that only came into being once 12-TET took over the world. There are 12 chromatic pitches in an octave, right? Well, each one of those pitches in 12-TET is equidistant from its neighbors. That is to say, the distance between C and C# or Gb and G is 100 cents. In this way, 12 notes having 100 cents distance between each other make an octave comprising 1200 cents, and 12 notes. Not difficult to process, is it? This also gave us a new way of quantifying intervals from a Root, other than talking about pure harmonic relationships.

Here are the different Chromatic intervals, showing their Cent values:
0 c. = Root
100 c. = half step / semitone
200 c. = whole step / whole tone
300 c. = minor third
400 c. = Major third
500 c. = Perfect Fourth
600 c. = Tritone (I'll talk about this guy)
700 c. = Perfect Fifth
800 c. = minor 6th
900 c. = Major 6th
1000 c. = minor 7th
1100 c. = Major 7th
1200 c. = Perfect Octave (fun fact: in some Meantone temperaments the octaves were NOT perfect!)

These cent values are convenient and easy to remember, perhaps easier to remember than the specific numbered harmonic that is ascribed to each note. Is it easier to remember a Major 6th as 900 cents or the 27th harmonic? Maybe that depends on the person; I know not. In any case, those intervals are the ones we (read: the entire Western world and all "modernized" countries) use on our instruments, our synthesizers, our pitch pipes and all the rest. 12-EQT is actually a bold compromise, as the only TRULY perfect interval is the octave while the 4th and 5th are almost exact but not quite. The Major 3rd we use is 14 cents sharp; our Major 7th is 12 cents sharp; our "harmonic seventh" is 31 cents sharp of the true one, and that's a lot.

Then we have the lovely little lady called the Tritone.

Throughout history the interval of the Tritone was called by many unpleasant names. The most famous of these is "diabolus in musica" -- which means "THE DEVIL IN MUSIC." Why did it get such a bad rap? Quite simply, it was almost impossible to sing in tune. This is partly because there are many different "tritones" in the Harmonic series, rather than a single example to emulate.

The Equal Tempered Tritone, whose ratio is the square root of 2 (1.414...), became the quintessential version of the interval. Because it's an equal division of the octave, it has a strange kind of "stability" while being nonetheless dissonant to our ears (dissonant =/= ugly!). Our tritones have become on a wide scale much easier to sing after years and years of acoustic conditioning. I am personally so comfortable with tritones now that they are no longer dissonant to my ears. Not everyone has this luxury; some still find them hideous or unpleasant.

I suppose now's as good a time as any to begin on Hertz. Hertz are especially easy to understand if we're talking about Just Intonation. The word Hertz refers to a unit of measurement, and what the unit measures is Cycles-Per-Second. This basically means, in acoustic terms, the frequency of a wave. A sound wave.

Every note has a value in Hz, and the best example to begin with is A. In our tuning, we tune the note A to 440 Hz. Thus, every other note A is an octave up or down from 440, which means dividing or multiplying by 2. There's A 440, A 880, A 1760...and on the other hand, A 220, A 110, A 55. This direct, perfect relationship of doubling or halving a frequency is what Octaves are all about.

In J.I. you can take the hertz values even further in their relationship. Let's consider A 110. Recall the harmonic series: 1, 2, 3, 4, 5 etc. If we treat A 110 as the Root, or first harmonic, this is what follows:

110, 220, 330, 440, 550
Which from the Root spells an Octave, Perfect 5th, Double Octave, and Major Third.
Isn't Math fun? :)

Hz and Cents can both be used to compare the similarities and differences of different types of tuning. They can show numerically how the 12-EQT Major Third is considerably sharper than the acoustic/numeric reality, and the slight, nigh-imperceptible difference between an acoustic P5 and a tempered one. I often check the cent values of my tempered notes against the true harmonic series' pitches when I am crafting new chords for my compositions. I also use a waveform analyzer to prove to me visually whether a chord is consonant or dissonant, and in what way.

I took a course on the Physics of Music, and I took from it a strong understanding about Cents and Hertz, on a level that I can't fully communicate here. If any of you want to ask me about it in person, I can show you materials I still have from that class which would back up everything I've said here and reinforce it in new ways. Otherwise, if you're curious enough you can take the mindspark I've shared here and ride it across the web searching for answers I haven't provided.

That wraps up my rant on Temperament, Cents, and Hertz. I hope you found it on some level informative--and once again, if I'm missing any crucial information or didn't cover something you've wondered about just let me know and I'd gladly make an addendum to this post.

Next time: ??? - I dunno, maybe Combination Tones and their relevance to harmony. We'll see. :)