6/2020

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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

In 72-edo with octave equivalence,

So stacking two downminor sixths (

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)

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

We can confirm by listening

THIRD

pi/phi^2 = [19] = 6:5 minor third (nearly exact)

3s = [21] = 11:9 neutral third

1/phi = pi^2 = [22] neutral third

3s+q = [23] = 5:4 major third

pi phi = [25] = 61:48 upmajor third (nearly exact)

ALSO, 1/pi = [25] = 14:11 upmajor third

FIFTH

6s = [42] = 3:2 perfect fifth

1/phi^2 = [44] = 61:40 (nearly exact)

pi = 7s-q = [47] 11:7 downminor sixth

ALSO, 1/(pi phi) = [47] = 96:61 (nearly exact)

phi = 1/pi^2 = [50] 13:8 downmid sixth

OCTAVE

pi/phi = [69] = 31:16 quartertone below octave (nearly exact)

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

Forming a

[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

[0 10 22 35 47 60 (72)]

If instead we pick a

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

We noted that traveling

What happens if we travel along the

After 12 moves by

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

Therefore, if we trace backward a move along

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

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

move | degree | interval |
---|---|---|

0 | 0 | 1/1 |

1 | 11 | ca. Φ |

2 | 22 | ca. sixthtone greater than perfect fifth |

10 | 2 | thirdtone greater than octave |

12 | 24 | ca. twelfthtone greater than π |

13 | 35 | sixthtone less than octave |

--------------------------------- | ||

23 | 1 | sixthtone greater than octave |

24 | 12 | ca. twelfthtone less than π^{-1} |

26 | 34 | thirdtone less than octave |

34 | 14 | ca. sixthtone less than perfect fourth |

35 | 25 | ca. (lowercase)φ |

36 | 36 (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 ⇐

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 ⇐

Maybe we can improve things by beginning with the rough approximation chain of 10 notes, and then attach another chain of 10 by using