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What is an edo plus edo+1?

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Author Topic: What is an edo plus edo+1?  (Read 2225 times)

Easy Listening

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What is an edo plus edo+1?
« on: October 31, 2012, 03:38:39 PM »

I recently recorded an improv with a friend: 12 edo + 13 edo. It sounds fabulous.

Tonight I'm working on 11 edo + 12 edo. It sounds fabulous!

Have been very interested in combining edos altogether anyway (especially since it's taboo?) and previously had noticed [sadly not recorded yet] that 13 + 19 is super.

I don't have tremendous options for combination so far, but as * as the idea seems intuitively, the combination of adjacent edos really brings out what I want.

Any weird, or educational thoughts anyone?
« Last Edit: October 31, 2012, 07:37:21 PM by Easy Listening »
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Gedankenwelt

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Re: What is an edo plus edo+1?
« Reply #1 on: October 31, 2012, 06:31:14 PM »

When you combine 11- and 12-edo, you have a subset of LCM(11, 12) = 11 x 12 = 132-edo.
(LCM = least common multiple)

You can represent all intervals as steps in 132-edo; intervals from 11-edo become multiples of 12, and intervals from 12-edo become multiples of 11:

11-edo: 0  12 24 36 48  60 72  84 96 108  120  132
12-edo: 0 11 22 33 44 55 66 77 88 99 110 121 132

1 step in 132-edo is 1200/132 = 9.1 cent. As you can see, the difference between a note and the closest note from the other edo is smallest left and right (0), and increases towards the middle, where it is largest (6 steps ~54.6 cents, between 6\12, and 5\11 or 6\11).
(something that applies when you have edo and edo+1?)


You can also combine edos that are not coprime, like 12- and 15-edo. In this case, LCM(12, 15) = 60-edo:
(Benjamin Strange had a single guitar tuned to both edos)

12-edo: 0  5 10  15  20  25   30  35  40  45   50  55  60
15-edo: 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60

Here, 1 step is 1200/60 = 20 cents. The difference is lowest (0) where they overlap (0, 1\3, 2\3, 1), and highest (2 steps = 40 cents) in the middle between those spots.


It might be interesting to experiment with different string tunings.
Both guitar's strings could be tuned in their respective edo, or all strings from both guitars in the same edo (e.g. both in 11-edo). Or, if we use the example of 11- and 12-edo, the strings could be tuned to any scale degree of 132-edo.
« Last Edit: October 31, 2012, 06:34:04 PM by Gedankenwelt »
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Easy Listening

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Re: What is an edo plus edo+1?
« Reply #2 on: October 31, 2012, 07:36:45 PM »

Thank you for a perfect answer.
Yes, the divergence becomes large in the "middle" (4ths, 5ths).
You nailed my next probable question - how to they combine when nonconsecutive?
Brilliant!

I suppose my joy in these trials displays a desperate love of the semitone ...
« Last Edit: October 31, 2012, 07:40:11 PM by Easy Listening »
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Gedankenwelt

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Re: What is an edo plus edo+1?
« Reply #3 on: November 01, 2012, 08:31:29 AM »

You nailed my next probable question - how to they combine when nonconsecutive?

First of all, though 12- and 15-edo aren't consecutive edos, they're consecutive equal divisions of 1\3 octave (their gcd), since 12-edo divides 1\3 octave into 4 equal parts, and 15-edo divides 1\3 octave into 5 equal parts. So if we shift our focus a little, we can see them as consecutive.
I think it generally makes sense to view two edos as the equal division of their gcd when comparing them.

Some general thoughts:

The largest difference an edo's tone could have to the nearest adjacent tone from another edo is halft the step size from that other edo. If the first edo is even, and the second one is odd, this will always (and only) happen in the middle (between 1\2 octave and the adjacent notes from the odd edo).
However, the largest difference between a tone from the odd edo and the nearest adjacent tone from the even edo can still be larger (example: 2-edo and any large odd edo, like 53).
In addition, the difference doesn't have to increase continuously towards the middle, even if it is largest there.


I assume the reason for the described behaviour of consecutive edos is related to the fact that I compared them by stacking single step intervals, and that these used generators are farey pairs.
A single step in 11-edo can be written as 1\11 (= 1/11 octave), and a 12-edo step as 1\12. A farey pair is a pair of two reduced fractions a/b and c/d with a*d - c*b = +-1. So 1\11 and 1\12 are a farey pair, because 1*12 - 1*11 = 1, and the same holds true for any single step intervals from consecutive edos.

If we look at 13- and 19-edo, we realize that 1\13 and 1\19 is not a farey pair. However, 2\13 and 3\19 are, since 2*19 - 3*13 = -1. So, let's compare them by stacking these generators (2\13 = 38\247, and 3\19 = 39\247):

13-edo: 0 38 76 114 152 190 228                                                    266=19 57 95 133 171 209 247=0
19-edo: 0  39 78 117 156 195 234 [273=26 65 104 143 182 221] 260=13 52 91 130 169 208  247=0

Here again, the difference is largest in "the middle":
  • The greatest difference between a 13-edo note and the closest 19-edo note is between 12\13 = 228\247 and 18\19 = 234\247 (or between 1\13 and 1\19), i.e. 6 steps in 247-edo = 29.1 cents.
  • The greatest difference between a 19-edo note and the closest 13-edo note is between 8\19 = 104\247 and 5\13 = 95\247 (or between 11\19 and 8\13), i.e. 9 steps in 247-edo = 43.7 cents.

Note that both 26- and 19-edo support meantone/flattone, and 3\19 and 4\26 = 2\13 are the wholetones generated by stacking two meantone/flattone fifths. That's why stacking 3\19 and 2\13 leads to similar scales. Also note that the "bad representation" of 11\19 in 13-edo isn't a bad thing, because 11\19 is close to 15\26 (also a farey pair!), the meantone/flattone fifth that is "missing" in 13-edo, and makes a nice addition. ;)
If we generate a 13-tone MOS by stacking 3\19 wholetones (6 upwards, 6 downwards), it should contain the notes that are closest to 13-edo:

0 1 3 4 6 7 9 10 12 13 15 16 18 19

The remaining notes from 19-edo are probably* closer to notes that are in 26-edo, but "missing" in 13-edo:

2 5 8 11 14 17

* too lazy to check :P

Are you familiar with moments of symmetries, and the Stern-Brocot tree (aka Wilson's Scale Tree)? The latter is pretty useful for finding MOS's, farey pairs, or other generator pairs that are similar in size and lead to same or similar temperaments.
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Easy Listening

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Re: What is an edo plus edo+1?
« Reply #4 on: November 01, 2012, 10:09:54 AM »

Brilliant, brilliant, thank you!
Comes back around to one of my first questions - predicting tonal attributes of edos based on whether they're even, divisible by 3, etc.
What you wrote about 13 & 19 makes especial sense, and though I fail in math, the farey pairs make sense too.
I do know about MOS; however, although looking at possible MOS scales in a new system has been great start with a new edo, as it becomes less alien I quickly revert to intuitive playing, without intentional use of scales.
Thanks, this thread has blown my mind.
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Mat

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Re: What is an edo plus edo+1?
« Reply #5 on: November 01, 2012, 02:23:36 PM »

I'm impressed by your 11+12 track EasyL, it really is beautiful http://soundcloud.com/karmajinpa/room-for-identifying-with-a
It did unspeakable things to my mind last night.
« Last Edit: November 01, 2012, 02:25:57 PM by Mat »
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Easy Listening

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Re: What is an edo plus edo+1?
« Reply #6 on: November 01, 2012, 02:37:37 PM »

 ;D Thanks, that's so sweet of you!!!
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