I made some music using modes of the harmonic series

Posted byEthan

It’s a cliche to say that the harmonic series is the basis of all of music. It is true that the first five harmonics are the basis of Western tuning. The first seven harmonics are a possible basis for the blues. You don’t tend to hear much music based on the higher harmonics, but they can be a great way to build exotic scales that still have a feeling of coherence and order.

 

Here are some tracks I made using the pitch ratios derived from the first forty-three natural harmonics.

I made this music using Ableton Live and a tuning plugin called MTS-ESP. I got it so I could try out historical tuning systems, first for my own understanding, and then for teaching music theory. While I was experimenting with the software, I started plugging in weird scales of my own invention. To understand what I was doing, you need to know a little math.

You can express tuning systems as multiples of some arbitrary starting frequency. In standard Western tuning, you start with A at 440 Hz, and you find the frequency of all the other notes by multiplying or dividing by the twelfth root of two (about 1.059463). Multiplying 440 Hz by the twelfth root of two gives you B-flat at 466.16 Hz. Multiplying that by the twelfth root of two gives you B at 493.88 Hz. And multiplying that by the twelfth root of two gives you C at 523.25 Hz. Simple!

MTS-ESP lets you create tuning systems by multiplying your base frequency by any ratio, not just the twelfth root of two. For example, if you multiply and divide by 3/2 instead, you get Pythagorean tuning. I decided to try inventing my own tunings by picking some fractional denominator and systematically creating every interval I could with it. For example, I made a scale using the ratios 5/5, 6/5, 7/5, 8/5, 9/5, and 10/5. It turns out that this is a known approach to just intonation called primodality that uses sections (modes) of the natural overtone series. My “over five” scale gives the same ratios as harmonics five through ten. The scales you get from modes of the harmonic series are weird, but they are weird in a clearly patterned way, and they have their own attractive internal logic. 

To make my tracks, I used a formulaic approach. I put a scale into MTS-ESP, put down some beats, and recorded myself improvising via MIDI. I edited my takes down to the good parts, and then repeated those sections in groups of two, four, or eight. My goal was to make the scales sound good, and to not overwhelm the listener with too much musical information. All of my tracks are in C, and the basslines are mostly just C pedals. 

First, here’s Mode Four, also known as quartal primodality. The samples are from “The Dry Cleaner from Des Moines” by Joni Mitchell, “Bemsha Swing” by Thelonious Monk and “PYT” by Michael Jackson.

The image in the Bandcamp player shows a diagram of the tuning system. The inner circle shows the twelve tones in 12-tone equal temperament, as measured in cents. So if C is at twelve o’clock, then 100 cents is C-sharp/D-flat, 200 cents is D, 300 cents is D-sharp/E-flat, 400 cents is E, and so on. The outer circle shows the intervals in the current scale. In mode four, those are:

  • 4/4 (1/1) – unison
  • 5/4 – just major third
  • 6/4 (3/2) – just perfect fifth
  • 7/4 – harmonic seventh
  • 8/4 (2/1) – octave

This is not so much a “scale” as it is an arpeggiated just intonation dominant seventh chord. The fifth is almost identical to its 12-TET tuning, but the third is noticeably flatter, and the seventh is much flatter. 

You can take primodal scales into their “second octave,” which means using the (n*2)th mode of the harmonic series. You can also think of it as doubling the denominator and building all the possible fractions on it that way. Or you can think of it as arithmetically splitting each interval in half. Here’s Mode Eight, the second octave of quartal primodality. The samples are from “Heavy Soul Slinger” by Bernard Purdie, “Chameleon” by Herbie Hancock and “Who Is It” by Björk.

  • 8/8 (1/1) – unison
  • 9/8 – just major second
  • 10/8 (5/4) – just major third
  • 11/8 – undecimal superfourth
  • 12/8 (3/2) – just perfect fifth
  • 13/8 – (lesser) tridecimal neutral sixth
  • 14/8 (7/4) – harmonic seventh
  • 15/8 – just major seventh
  • 16/8 (2/1) – octave

The scale still has lots of familiar intervals, but it also includes a very flat 11/8 tritone and a neutral sixth that’s right in between major and minor six. 

Here’s Mode Five, also known as quintal primodality. The samples are from “Vital Transformation” by the Mahavishnu Orchestra.

  • 5/5 (1/1) – unison
  • 6/5 – just minor third
  • 7/5 – narrow tritone
  • 8/5 – just minor sixth
  • 9/5 – just minor seventh
  • 10/5 (2/1) – octave

There’s no fifth in here, but otherwise everything sounds pretty familiar. The narrow tritone is especially bluesy.

Mode Ten, the second octave of quintal primodality, opens up lots of intriguing new possibilities. The samples are from “A Funky Kind of Thing” by Billy Cobham.

The odd-numbered intervals are seriously weird. There’s a very flat major second, a note right between the major third and fourth, and a very sharp major sixth and major seventh. Spicy! 

Here’s Mode Six, also known as seximal primodality. The samples are from “I Walk On Gilded Splinters” by Johnny Jenkins.

  • 6/6 (1/1) – unison
  • 7/6 – subminor third
  • 8/6 (4/3) – just perfect fourth
  • 9/6 (3/2) – just perfect fifth
  • 10/6 (5/3) – just major sixth
  • 11/6 – undecimal neutral seventh
  • 12/6 (2/1) – octave

Most of this scale is warmly familiar. The only oddball note is the neutral seventh right in between the minor and major sevenths. The subminor third is a great blue note. Taken together, this all feels like an asymmetrical whole tone scale.

Mode Twelve, the second octave of seximal primodality, has twelve pitches like the familiar chromatic scale, but they are distributed differently. The samples are from “Don’t Let Me Down” by Charlotte Dada.

The scale includes the familiar major third and the relatively familiar harmonic seventh, but also a few novelties. The neutral second between the minor and major seconds is very exotic. There is also have a super sharp major seventh. 

Mode Seven is also known as septimal primodality.

All of these intervals are exotic, but their clear harmonic relation to each other still gives them a folkloric integrity. There’s a wide major second, a very wide major third, a wide tritone, a slightly narrow minor sixth, a wide major sixth, and a narrow major seventh. The scale sounds like an mbira tuning to me. The samples are from “Poinciana” by Ahmad Jamal and “Donna Lee” by Tito Puente.

Mode Fourteen, the second octave of septimal primodality, is even more exotic. The samples are from “Hunk O’ Funk” by Brother Jack McDuff.

There’s a perfect fifth in here, but otherwise, everything is unusual-sounding. I love the 19/14 “Hendrix fourth.” 

Mode Nine is also known as nonary primodality. The samples are from “God Make Me Funky” by the Headhunters and “Half The Fun” by the Duke Ellington Orchestra.

This scale approximates several friendly and familiar intervals: perfect fourth, major sixth, and flat seventh. That makes the weird intervals sound that much weirder by comparison. The 11/9 neutral third is especially wonderful.

Mode Eighteen, the second octave of nonary primodality, adds lots of extremely strange intervals. The samples are from “Keharwa” by Maede Shafie and “Ko-Ko” and “Trains” by the Duke Ellington Orchestra.

  • 18/18 (1/1) – unison
  • 19/18 – large undevicesimal semitone
  • 20/18 (10/9) – small whole tone
  • 21/18 (7/6) – subminor third
  • 22/18 (11/9) – neutral third
  • 23/18 – unnamed
  • 24/18 (4/3) – just perfect fourth
  • 25/18 – augmented fourth
  • 26/18 (13/9) – tridecimal diminished fifth
  • 27/18 (3/2) – just perfect fifth
  • 28/18 (14/9) – subminor sixth
  • 29/18 – unnamed
  • 30/18 (5/3) – just major sixth
  • 31/18 – unnamed
  • 32/18 (16/9) – Pythagorean minor seventh
  • 33/18 (11/6) – undecimal neutral seventh
  • 34/18 (17/9) – large septendecimal major seventh
  • 35/18 – septimal semidiminished octave
  • 36/18 (2/1) – octave

I don’t even know what to say about these notes, there are so many of them, it’s hard to keep them all in your mind’s ear. 

Finally, here’s Mode Eleven, also known as undecimal primodality. The samples are from “The Funky Drummer Parts One and Two” by James Brown, “Diamonds on the Soles of Her Shoes” by Paul Simon, “Symphony for Dot Matrix Printers” by The User and “Caravan” by Thelonious Monk.

This scale is uniformly weird. There are notes that come pretty close to the minor and major thirds but otherwise we are completely off the 12-TET grid. 

Mode Twenty-Two, the second octave of undecimal primodality, uses such finely spaced intervals that they start feeling like out-of-tune versions of each other rather than distinct notes. The samples are from “Quimbara” by Celia Cruz.

  • 22/22 (1/1) – unison
  • 23/22 – greater vicesimotertial semitone
  • 24/22 (12/11) – neutral second
  • 25/22 – undecimal super-second
  • 26/22 (13/11) – neo-Gothic minor third
  • 27/22 – rastmic neutral third
  • 28/22 (14/11) – undecimal major third
  • 29/22 – unnamed
  • 30/22 (15/11) – undecimal augmented fourth
  • 31/22 – unnamed
  • 32/22 (16/11) – minor fifth
  • 33/22 (3/2) – just perfect fifth
  • 34/22 (17/11) – septendecimal subminor sixth
  • 35/22 – unnamed
  • 36/22 (18/11) – undecimal neutral sixth
  • 37/22 – unnamed
  • 38/22 (19/11) – undevicesimal semitwelfth
  • 39/22 – unnamed
  • 40/22 (20/11) – small undecimal neutral seventh
  • 41/22 – unnamed
  • 42/22 (21/11) –  large undecimal diminished octave
  • 43/22 – unnamed
  • 44/22 (2/1) – octave

We are truly in another harmonic dimension here. At least there’s a perfect fifth to anchor yourself with. 

If you want to try these scales out for yourself, here are a couple of considerations. If the scale has fewer than twelve notes in it, then it’s easy to map it to the MIDI keyboard. Just pick a subset of the chromatic pitches that make sense to you and jam out. If the microtonal notes happen to fall on or near “normal” intervals like the perfect fifth, that makes for more obvious mapping choices. If the scale has more than twelve notes in it, you will need to spread it across multiple octaves of the MIDI keyboard. I didn’t do this in any systematic way, just piled them up on successive chromatic keys. This means that patterns won’t repeat at octaves. I found that if I was dealing with a very complex scale, one octave worth is plenty of musical material anyway.

I find that timbre makes a huge difference in how “out of tune” a given scale sounds. One synth preset will make the scale sound harsh and alienating, while another makes it intriguing and inviting. (This is true of “normal” scales too!) I also find that repetition quickly changes how weird the scales sound. After enough repetition, the lower modes all start sounding perfectly normal. The higher modes retain their strangeness just because there are so many fine intervals to keep track of, and it’s more than my short-term memory can accommodate. I bet that if I spent enough time listening to them, though, even the highest modes would become comfortable and familiar.