Pi and Phi: Tone Network

Dan Nielsen
6/2020
Φ = inverse golden ratio
(φ = golden ratio)

72-edo is a temperament that approximates many common harmonic ratios very closely, and, as it so happens, it also works well for ratios involving π and Φ

In 72-edo with octave equivalence, π π equals Φ

nstep( x ) = round( 72 log2 x ) mod 72
nstep( π π ) = nstep( Φ ) = 22 (upmid third)
nstep( π ) = 47 (downminor sixth)

So stacking two downminor sixths (π) gives an upmid third (Φ)
A triad formed from these steps [0 22 47] sounds like this


It is interesting that several intervals are near to simple ratios
(octave-equivalent interval names in parentheses)

π / Φ ≈ 61/12 (upmajor third)
π Φ Φ ≈ 6/5 (minor third, like π π π Φ or π π π π π)
π π Φ ≈ 61/10 (double-up fifth, like Φ Φ or π π π π)
π π π ≈ 31 (quartertone less than octave, like π Φ)

Notice that our triad of steps [0 22 47] is approximately 31 : 6/5 : 61/10 ⇒ 155 : 6 : 61
We can confirm by listening


The picture at the top of the page depicts the tone network
You can see that after the triad [0 22 47], we get steps [69 44 19], the same triad shifted by a quartertone
It is generally desirable for notes in a scale to be somewhat separated, so let's skip every other set of three notes as we travel along
Then we wind up with four triads

[0 22 47]
[66 16 41]
[60 10 35]
[54 4 29]

This is like following the path π π Φ2 π π Φ2 π π Φ2 π π Φ2
Forming a 12-note scale from the collection gives something akin to the usual chromatic scale

[0 4 10 16 22 29 35 41 47 54 60 66 (72)]


If we again skip every other triad, we wind up with a 6-note scale similar to a wholetone scale

[0 10 22 35 47 60 (72)]


If instead we pick a 7-note diatonic subset, such as the nicely symmetrical Dorian, we get this

[0 10 16 29 41 54 60 (72)]


We noted that traveling π π π brings us back to a quartertone difference, which was considered to be undesirable
What happens if we travel along the Φ-direction instead of the π-direction?
After 12 moves by Φ, we come to step 48, very near to π's 47

0 ⇒ 22 ⇒ 44 ⇒ 66 ⇒ 16 ⇒ 38 ⇒ 60 ⇒ 10 ⇒ 32 ⇒ 54 ⇒ 4 ⇒ 26 ⇒ 48

If we travel another 12 moves, we would expect to wind up at π π, which we know is about equal to Φ
Therefore, if we trace backward a move along Φ, we would expect to arrive close to an octave, and in fact that's true - with 23 moves total, we arrive at step 2, close to the octave

48 ⇒ 70 ⇒ 20 ⇒ 42 ⇒ 64 ⇒ 14 ⇒ 36 ⇒ 58 ⇒ 8 ⇒ 30 ⇒ 52 ⇒ 2

But notice after only one step above, we had already arrived at 70, only 2 steps from the octave

After 3 more moves, we are again only 4 degrees from 0

2 ⇒ 24 ⇒ 46 ⇒ 68

Since all of the scale degrees are even, we can switch to 36-edo without any loss of precision and then write out the complete chain of 36 notes

0 ⇒ 11 ⇒ 22 ⇒ 33 ⇒ 8 ⇒ 19 ⇒ 30 ⇒ 5 ⇒ 16 ⇒ 27 ⇒ 2 ⇒ 13 ⇒
24 ⇒ 35 ⇒ 10 ⇒ 21 ⇒ 32 ⇒ 7 ⇒ 18 ⇒ 29 ⇒ 4 ⇒ 15 ⇒ 26 ⇒ 1 ⇒
12 ⇒ 23 ⇒ 34 ⇒ 9 ⇒ 20 ⇒ 31 ⇒ 6 ⇒ 17 ⇒ 28 ⇒ 3 ⇒ 14 ⇒ 25 ⇒ (0)

Let's make an account of special intervals in 36-edo, and how many moves along the chain of Φ it takes to reach them

movedegreeinterval
001/1
111ca. Φ
222ca. sixthtone greater than perfect fifth
102thirdtone greater than octave
1224ca. twelfthtone greater than π
1335sixthtone less than octave
---------------------------------
231sixthtone greater than octave
2412ca. twelfthtone less than π-1
2634thirdtone less than octave
3414ca. sixthtone less than perfect fourth
3525ca. φ (lowercase)
3636 (or 0)2/1

To construct the diatonic scale, we typically might choose 7 adjacent notes in the chain of fifths of 12-edo
For instance, to get Dorian, we might move down 3 fifths and up 3 fifths from a starting note

Ab ⇐ Eb ⇐ Bb ⇐ F ⇐ C ⇐ G ⇐ D ⇒ A ⇒ E ⇒ B ⇒ F# ⇒ C# ⇒ G#

36-edo requires the full chain of 36 notes, which seems a bit long
If we close a smaller chain through approximation, our choices for chain length are 10, 13, 23, or 26
Lengths 13 and 23 produce the closest approximations to ideal pitches, and length 13 is very close to 12, but they are odd lengths, so therefore we cannot form a harmonically symmetrical scale from them

1 ⇐ 12 ⇐ 23 ⇐ 34 ⇐ 9 ⇐ 20 ⇐ 31 ⇐ 6 ⇐ 17 ⇐ 28 ⇐ 3 ⇐ 14 ⇐ 25 ⇐ 0 ⇒ 11 ⇒ 22 ⇒ 33 ⇒ 8 ⇒ 19 ⇒ 30 ⇒ 5 ⇒ 16 ⇒ 27 ⇒ 2 ⇒ 13 ⇒ 24 ⇒ 35

Maybe we can improve things by beginning with the rough approximation chain of 10 notes, and then attach another chain of 10 by using π as glue. To be continued. . .