Infraharmony

Eight octaves below sound.

Music has long known which tones go well together: those whose frequencies stand in simple ratios. A fifth is 3 : 2, a major chord 4 : 5 : 6. A rhythm is a frequency too, only a slow one, so the same ratios should tell which rhythms go well together. Take the C major scale and carry it eight octaves down, below anything an ear hears as pitch. Middle C, 262 vibrations a second, becomes one beat a second; G becomes a beat and a half.

In the grid below the rows are tones, the C major scale from middle C, and the columns are the same scale eight octaves lower, as rhythms. A cell joins the two: a tone that repeats in a rhythm. Put a note in the rhythm of its own note, and a chord sounds in pitch and in time at once: C, E and G beat four, five and six times in the same span.

Playing

• Left click on a cell starts a voice there. It sounds at once and comes back every period of its rhythm. Another click on the same cell adds another voice, in the phase of that click, so voices of one cell can be tapped into a pattern.
• Right click clears the cell. On a touch screen, hold the cell.
• On a keyboard the arrows move between cells, Enter adds a voice and Delete clears.

Each voice is a dot circling its cell once a period; the note sounds as the dot passes the mark at the top. The examples start from the intervals, from the octave to the tritone, and go on to chords, canons and a few things that are just fun to watch.

Two tunings

In just tuning every step of the scale is a ratio of whole numbers (9/8, 5/4, 4/3, 3/2…), so voices that start together come back together after a fixed time. The simpler the ratio, the sooner: an octave meets on every beat of its lower voice, a fifth every two, a major second every eight, and the tritone, 32 : 45, only after 23.5 seconds, when the whole scale meets as well. Consonance, heard in time, is voices that meet often.

Equal tuning, the one a piano uses, makes each step a little off those ratios. In a chord the error is heard as a slow beating; in a rhythm it shows as a drift. C and E in their own rhythms slide apart and meet again every 25 seconds, while the same pair in just tuning stands still.

Major and minor

A major chord is 4 : 5 : 6, a minor chord 10 : 12 : 15, and the second is the first turned over: in major the frequencies go in equal steps, in minor the periods do, 6 : 5 : 4. As rhythms, a major chord is one shared bar divided into 4, 5 and 6 parts at once, meeting fully at every bar. A minor chord is three loops of 6, 5 and 4 steps on a shared pulse: pairs meet often, all three rarely. Zarlino derived the two chords this way in the 16th century, from the harmonic and the arithmetic division of a fifth on a string. In the Borderland the same turn is heard in sound: a low major chord beats at one even rate, its shared fundamental; a minor one beats at two, and is held together instead by a shared overtone above it.

The Borderland

Between the slowest tone and the fastest rhythm there is a stretch, roughly 10 to 30 a second, that is neither: too fast to count, too slow to be heard as a pitch. It sounds as a flutter, a growl, a rattle. The Borderland grid looks at that stretch from both sides at once. Its tones run down the scale from C2, 65 vibrations a second, to C0, 16, close to the lowest pitch an ear can hear; its rhythms run up from C−2, 4 beats a second, to C0, 16. The two scales go in opposite directions and meet in the bottom right corner, where a tone and a rhythm are the same frequency.

Here a voice is a string that keeps vibrating and is struck again on every beat, so tones held together beat against each other, and that is where the two worlds cross. Two tones a whole step apart, C1 and D1, beat 4.09 times a second, which is C−2, the first column: a rhythm made of two sounds. A C major chord this low beats at C−1, the bass it lacks two octaves down. And the other way: four voices spread evenly in one cell strike 65 times a second, as often as their tone C2 vibrates, and the strikes vanish into the tone. A piano tuner listens in the same region, counting how fast an interval beats to hear how far it is from pure.

The tones here lie below what most small speakers can give out, so they carry many harmonics, and the ear finds the pitch from those, as it does on the lowest keys of a piano. Headphones or larger speakers help.

Sharing a piece

The address of the page keeps the piece: after # it holds every voice with its phase, the grid and the tuning, in a short code such as 1j8.AAAACAVVEAqr. Sending the link is enough; the piece opens with every voice in its place. Nothing goes to a server.

Where it comes from

The idea is not new. In 1931 Leon Theremin built the Rhythmicon for Henry Cowell, an instrument that played each tone of the overtone series together with a rhythm in the same ratio. This grid does the same on the ordinary major scale, and lets the voices be placed in time by hand. It started in 2012 as a handful of MIDI files, an interval or a chord each, and became something to play with only now. The code is on GitHub: bsulkowski/infraharmony.