twenty three x

The harmonic series can be thought, in one perspective, of as the sequence of real numbers.

1 2 3 4 5 6 7 8 9 10 11 …. ∞

Mathematically, the ratios of the frequencies between any two of these harmonics are exactly the ratio between the numbers.  For example, the ratio between the “first” harmonic (the unison) and the “second” — 2, the octave, is exactly 1:2.  The ratio between the 7th and the 11th harmonic is 7:11.  As just intonation ratios this is also notated as 11:7 or 11/7.

Powers of 2 are significant.  2 4 8 16 32 64 128… Powers of 2 are also each sequential octave (in the harmonic series, or otherwise).

This means we get these harmonic “realms”

1 2

2 3 4

4 5 6 7 8

8 9 10 11 12 13 14 15 16

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

I colour-coded harmonics as they show octave equivalence in the next realm.

Composite numbers such as 9 = 3*3 are both generated as a member of the harmonic series, being part of the sequence of whole numbers.  And are the combination of two or more harmonics.  In this case, the 9th harmonic to a frequency is the same as the 3rd harmonic applied twice to that frequency.

We can re-write segments of the harmonic series to be within an octave.  This is useful because it is a familiar and practical and perceptually relevant way to categorize a group of pitches.

Octave equivalence is created by multiplying or dividing any frequency by 2.  The 6th harmonic, 3*2, is an octave higher than the 3rd harmonic.

Since these harmonic realms are all in higher octaves, we can octave reduce all the frequencies by powers of 2 (some amount of octaves) to bring them all within the root octave or home octave.

1 2 just becomes 2 unisons

2 3 4 becomes 2/2, 3/2, 4/2 = 1/1, 3/2, 2/1 — this is the unison, perfect 5th, and octave

4 5 6 7 8 = 4/4, 5/4, 6/4, 7/4, 8/4 = 1/1, 5/4, 3/2, 7/4, 2/1 — this is a major triad with the harmonic 7th (a bit lower and more resonant / less beating than the minor 7th)

8 9 10 11 12 13 14 15 16 =

8/8, 9/8, 10/8, 11/8, 12/8, 13/8, 14/8, 15/8, 16/8 =

1/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8, 2/1

You can continue this pattern to the next harmonic series segment utilizing the denominator of 16, 32, 64, and so on.

But what about denominators that are not powers of two?  What do they mean?

Notice that the difference between the 11th harmonic and the 13th harmonic is 13/11.

So if we want to build a set related to the harmonic series, in particular the 11th harmonic, say starting from the 11th harmonic.  Lets define it at the unison 1/1.

Then the relationship between the unison (the 11th harmonic) and the 12th harmonic is 12/11.

So lets build a set

1/1, 12/11, 13/11, 14/11, 15/11, 16/11, 17/11, 18/11, 19/11, 20/11, 21/11, 22/11 = 2/1

So this set is exactly the set of harmonics from the 11th to the 22nd defined in relation to the 11th harmonic.  This is an overtone series.

We can also build a set of a constant numerator.  For example 23.  Say the 23rd harmonic.  Then 23/22 will be the distance from the 22nd harmonic to the 23rd harmonic.  And 23/11 will be the distance from the 21st harmonic to the 23rd harmonic.

Lets build a set.

1/1, 23/22, 23/21, 23/20, 23/19, 23/18, 23/17, 23/16, 23/15, 23/14, 23/13, 23/12, 2/1.

This is an undertone series of the distance between the 23rd harmonic and the harmonics from 12-22.  This is not a descending series however as 23/22, 23/21 etc are increasingly large intervals, but the description of the relationship is that of harmonic relations which are below 23.

At this set is described as 23/x (with x between 12 and 22), it can be spoken as twenty three x.

Take a listen to a recent piece composed in this 23 set 🙂

https://www.patreon.com/posts/among-dreams-23-16966657

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