Recently I visited a diary that had nothing to do with any of my interests or skills (probably a familiar experience for most of us). In this case, it was the diary title that caught my eye; this diary's title is an *hommage*.

*That* diary was about video aspect ratios, and it prompted a bemused comment from me, which was met with puzzlement from the diarist. *This* diary is about frequency ratios and their importance in music theory; perhaps it will prompt a few bemused comments from video nerds.

When I started learning music seriously I was a teenager. I'll turn 54 in a couple of weeks, and I'm still figuring out all this stuff, despite (or perhaps because of) being a professional musician and music teacher for three decades. Having a 7-year-old daughter is an enormous help.

In high school I took my first music theory class. The teacher's name was Mr. March, which should have been a clue. The first day, he said to the class, *"I'm going to test your musical ears."* He told us to take out a piece of paper. Then he said, *"I'm going to play two intervals on the piano. You write down which is bigger, the first or the second."*

Then he turned his back to us and pressed some keys on the piano.

*I did not have a freakin' clue what was going on.*

I did not recognize that he was hitting two keys simultaneously. What I heard was *a series of sounds*. What did he mean by "which one is bigger"? I'm pretty sure I just gave up on the exam.

Mr. March was operating under some default assumptions that were never stated. This is not uncommon in teaching, and it's practically a given in music teaching, where teachers are distressingly likely to start where *they are*, rather than *where their students are*.

Here's what I tell students who want to learn about music theory.

Musical sound concerns itself with vibration within the frequency range that our ears can perceive. Vibrations outside that range don't get picked up by our ears, so we won't talk about them.

Some vibrations have *periodicity*. Others do not. An example of the first kind is a tone played on a flute; an example of the second kind is crumpling a sheet of paper.

While musical performance uses both types of sounds, the study of harmonic relationships is only concerned with periodic sounds — the ones with identifiable frequencies, usually measured in cycles-per-second. Sounds with identifiable frequencies are called *tones*. If you take a series of rhythmic impulses and speed them up, they will turn into tones.

If you have two tones with the same frequency, they are in a very specific relationship. Their numbers match; they are in a 1-to-1 ratio. The musical term for this relationship is *unison*.

If you and I sing the exact same note, our vocal chords are vibrating at the exact same speed, and we are singing in unison. If we're *almost but not quite* at the exact same speed, the frequency ratio between our voices changes from 1:1 to something more complicated. 189.235147 : 193.772121 is almost the same as 190:190 (which reduces to 1:1) but it's a more complex relationship — and it's perceived by our ears as "out of tune." Obviously there are a lot more ways to be out of tune than to be in tune!

If you have two tones in the frequency ratio 2:1, their numbers no longer match, but their relationship is still simple. One vibration moves twice as fast as the other. The musical term for this relationship (in Western musical tradition) is *octave*.

Notice that the term "octave" means "eight," which has absolutely nothing to do with the actual mathematics involved.

To our ears, the frequency of any power of 2 seems to have the same "quality" as any other. Notes an octave apart are given the same name in nearly every world musical system that goes so far as to name the notes in the first place. This means that experientially, 2:1, 4:1, 8:1, 16:1... are all identical 1:1.

----------------------------------------------------------------------------------------------

Musical intervals can be quantified in various ways.

Keyboard or melodic distance simply measures how far you have to move your finger to get from one member of an interval pair to the other. From the lowest A on the piano to the highest is a finger distance of about a meter and a half. From "middle C" to the C-sharp immediately above it is a finger distance of about a centimeter. By this measure, the first interval is significantly "bigger."

*Ratio size* just addresses the distance between the two numbers, and it maps nicely onto the melodic distance measure. From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 (n.b., if you know this already, you also know that on the piano, thanks to the baffling miracle of equal temperament, this statement is untrue. Bear with me for the purposes of discussion, 'k?). 128 to 1 is a bigger jump than 16 to 15, so the first interval is significantly "bigger."

*Harmonic distance*, on the other hand, measures the complexity of the ratio involved. From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 — but 128:1 reduces to 1:1, and 16:15 doesn't reduce. An eight-octave jump has a harmonic distance of zero, while a "semitone" has a much greater harmonic distance. So when we use this measuring system, the second interval is "bigger."

------------------------------------------------------------------------------------

All harmonic intervals can be described as frequency ratios. Here are some of the ones we use most often:

3:2 is described in Western musical terms as a "fifth."------------------------------------------------------------------------------------

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol / 1-2-3-4-5), which has nothing to do with the actual mathematics involved.

4:3 is described in Western musical terms as a "fourth."------------------------------------------------------------------------------------

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa / 1-2-3-4), which has nothing to do with the actual mathematics involved.

5:4 is described in Western musical terms as a "Major Third."------------------------------------------------------------------------------------

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi / 1-2-3), which has nothing to do with the actual mathematics involved.

5:3 is described in Western musical terms as a "Major Sixth."------------------------------------------------------------------------------------

Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol-La / 1-2-3-4-5-6), which has nothing to do with the actual mathematics involved.

As my little videos demonstrate, rhythmic impulses turn into pitch when you accelerate them. If you record yourself tapping 2-against-3 for an hour, then accelerate the recording by multiple orders of magnitude, you'll wind up with two tones a fifth apart.

------------------------------------------------------------------------------------

You don't need to know about frequency ratios to use them effectively (just listen to the Beatles and you'll hear some dynamite frequency ratios rendered with exquisite fidelity by people who never gave the math a moment's thought). Most composers don't know. Most musicians don't know.

So why bother?

Speaking personally, I can say that learning all this has transformed my experience of music. I can spend a long time perfecting the tuning of a single interval — precisely because I have learned to perceive it as a source of deep experiential insight into simple mathematical relationships. Why bother? Because it's cool; because it's beautiful; because it's universal.

Okay, that's all for today.

Love,

WarrenS

## Comment Preferences