What does the Well-Tempered Clavier sound like in actual well temperament?

First, some niche Twitter comedy:

The Well-Tempered Clavier is a book of JS Bach compositions for keyboard instruments in each of the twelve major and twelve minor keys. The name refers to Bach’s preferred tuning system, which made it possible to play (sort of) in tune in every key. This was a big deal, because in the usual tuning systems of Bach’s era, only some of the keys sounded good, while others sounded horrible. The history of tuning in Western music is complicated and abstruse, and I won’t go into detail about it in this post, but you can learn some of how it works here. The key facts:

  1. Western tuning systems, keys and scales are based on the natural harmonic series.
  2. Harmonics are based on prime numbers.
  3. Prime numbers don’t divide into each other evenly.

The practical consequence is that your music can either be in perfect tune, or it can use more than one key, but it can not do both. In Hindustani classical tradition, they opted for being in tune, so everything is in a single “key” defined by the omnipresent drone. Western Europeans wanted to be able to change keys, however, and that required some tuning compromises.

Continue reading

Teaching note values

Western music notation is a graph of pitch (on the vertical axis) and time (on the horizontal axis.) It’s mostly self-explanatory on the pitch axis, but it’s harder to understand on the time axis. It helps if you visualize your rhythms on a circle, like the Groove Pizza does. Everything I talk about in this post will assume that we’re in 4/4 time, because it’s the default rhythmic setting for Western music, and all note values are determined in reference to it. Also, 4/4 is the time signature of most of the music that you probably like.

Continue reading

Scales, keys and modes on the circle of fifths

If you want to understand Western music theory, the circle of fifths is an invaluable tool. For one thing, it can help you understand how key signatures work. But it also helps explain how the major scale and diatonic modes relate to each other, and gives a possible explanation for why they sound good.

Here’s the C major scale on the circle of fifths:

The purple notes are the ones that form “perfect” intervals above the root C: unison, octave, fourth and fifth. The green notes form major or “natural” intervals above the root. The numbers refer to the scale degrees.

Continue reading

RIP Godfried Toussaint

I was sad to learn about the recent death of Godfried Toussaint, whose work on the geometry of musical rhythm has been a major inspiration for me.

Godfried Toussaint

I never met Godfried, but I have read and re-read his work. His rhythm necklace diagrams were the direct inspiration for the Groove Pizza – I saw them and thought, I would love to have a drum programming interface like that.

Rhythm necklaces Continue reading

The Groove Pizzeria

For his NYU music technology masters thesis, Tyler Bisson created a web app called Groove Pizzeria, a polyrhythmic/polymetric extension of the Groove Pizza. Click the image to try it for yourself.

Note that the Groove Pizzeria is still a prototype, and it doesn’t yet have the full feature set that the Groove Pizza does. As of this writing, there are no presets, saving, or exporting of audio or MIDI. However, you can send MIDI via the IAC bus to the DAW of your choice (Mac OS Chrome only). You can also record the Groove Pizzeria’s output using Audio Hijack.

Like the Groove Pizza, the Groove Pizzeria is based on the idea of the rhythm necklace, a circular representation of musical rhythm. The Groove Pizza is a set of three concentric rhythm necklaces, each of which controls one drum sound, e.g. kick, snare and hi-hat. The Groove Pizzeria gives you two sets of concentric rhythm necklaces, each of which can have its own time duration and subdivisions. This means that you can use the Groove Pizzeria to make polyrhythm and polymeter.

Continue reading

Why can’t you tune your guitar?

Short answer: because math. Longer answer: because prime numbers don’t divide into each other evenly.

To understand what follows, you need to know some facts about the physics of vibrating strings:

  • When you pluck a guitar string, it vibrates to and fro. You can tell how fast the string is vibrating by listening to the pitch it produces.
  • Shorter strings vibrate faster and make higher pitches. Longer strings vibrate slower and make lower pitches.
  • The scientific term for the rate of the string’s vibration is its frequency. You measure frequency in hertz (Hz), a unit that just means “vibrations per second.” The standard tuning pitch, 440 Hz, is the pitch you hear when an object (like a tuning fork or guitar string) vibrates to and fro 440 times per second.
  • Strings can vibrate in many different ways at once. In addition to the entire length of the string bending back and forth, the string can also vibrate in halves, in thirds, in quarters, and so on. These vibrations of string subsections are called harmonics (or overtones, or partials, they all mean the same thing.) Continue reading

Kumbaya

When you look up “Kumbaya” on Urban Dictionary, you get an adjective meaning “blandly pious and naively optimistic.” This is the sense in which Fox News often uses the word to make fun of bleeding heart liberals like me. I learned the song from numerous earnest white folk singers, many of whom learned it from Joan Baez:

But then I read on Anne C Bailey’s blog that “Kumbaya” is a Gullah song, named for the dialect version of the phrase “come by here.” Bailey’s post links to the earliest known recording, a 1926 wax cylinder whose performer is listed only as “H. Wylie.” This version is surprisingly funky for those of us raised on the white folkie version.

Continue reading

Chord pizzas

The Groove Pizza uses geometry to help visualize rhythms. The MusEDLab is planning to create a similar tool for visualizing music theory by merging the aQWERTYon with the Scale Wheel. When you put the twelve pitch classes in a circle, you can connect the dots between different notes in a chord or scale to form shapes. My hypothesis is that seeing these shapes along with hearing the notes will help people learn music theory more easily. In this post, I’ll talk through some concept images.

First, let’s look at two different ways to represent the pitch classes on a circle. On the left is the chromatic circle, showing the notes in the order of pitch height (the way they are on a piano keyboard.) On the right is the circle of fifths. These two circles have an interesting relationship: the circle of fifths is the involute of the chromatic circle. Notice that C, D, E, G-flat, A-flat and B-flat are in the same places on both circles, while the other six notes trade places across the circle. Pretty cool!

The chromatic circle and the circle of fifths

Continue reading

Four bars of Mozart explains everything humans like in music

I’m not arguing here that everyone loves Mozart, or that I’m about to explain what all humans enjoy all the time. But I can say with confidence that this little bit of Mozart goes a long way toward explaining what most humans enjoy most of the time. The four bars I’m talking about are these, from “Eine Kleine Nachtmusik.”

These four bars of music demonstrate that humans like:

  1. Repetition
  2. Breaks in the repetition
  3. Repetition of the breaks in the repetition
  4. Breaks in the repetition of the breaks in the repetition
  5. Recursive layers of patterns of breaks and repetitions

In order to prove this to you, I’m going to talk you through these eighteen notes one at a time.

Continue reading

The vocoder and Auto-Tune

The vocoder is one of those mysterious technologies that’s far more widely used than understood. Here I explain what it is, how it works, and why you should care.

Casual music listeners know the vocoder best as a way to make the robot voice effect that Daft Punk uses all the time.

Here’s Huston Singletary demonstrating the vocoder in Ableton Live.

You may be surprised to learn that you use a vocoder every time you talk on your cell phone. Also, the vocoder gave rise to Auto-Tune, which, love it or hate it, is the defining sound of contemporary popular music. Let’s dive in!

Continue reading