Category breakers

While fooling around on my new 19edo guitar (thanks Igs!), I was wondering what I could play that would sound more "weird" or "xenharmonic", as a change from all the gorgeous ringing major and minor chords that put every 12-tone guitar I've ever played to shame.

One obvious thing is comma pumps, but a magic or hanson comma pump is actually pretty complicated and hard to play for a crappy guitarist like me. Even if I found some good ones and could play them perfectly at a moderate tempo, they're necessarily sort of long and listeners might not realize anything weird is going on unless they pay close attention.

But if you think about it, a comma pump is something designed to prove to you that you're not listening to JI, because it's something that can't happen in JI. But the thing everybody's used to already isn't JI, it's 12edo. Can we define an equivalent thing that proves to you that you're not listening to 12edo?

Given a 1-to-1 correspondence between some intervals of one scale and some intervals of a different scale, define a "category breaker" as a linear relationship between the intervals of one that doesn't hold in the other.

I wrote a (very short) program that will spit out the simplest possible category breakers if you supply the interval correspondence. This correspondence is the only arbitrary, hand-picked thing about it. For 19 I chose this:

19 ... 12

3 <-> 2
5 <-> 3
6 <-> 4
8 <-> 5
11 <-> 7
13 <-> 8
14 <-> 9
16 <-> 10

And the shortest results are:

(6, 6, 6, 6, 14)

(3, 3, 6, 6, 6, 14), (3, 6, 6, 6, 6, 11), (5, 5, 5, 5, 5, 13), (5, 5, 5, 13, 13, 16), (6, 6, 6, 6, 6, 8), (6, 6, 6, 11, 14, 14)

These are collections of intervals in 19edo that add up to some number of octaves, but whose corresponding intervals in 12edo do not.

So the simplest result is... a magic comma pump. You can reformulate it as many different equations, for example "M3 + M3 + M3 + M3 = m3 + octave" or "M3 + M3 + M3 = m6 + m3", but they all represent the same linear relationship between the intervals. In this case it's a familiar relationship because it's the same one between 6/5, 5/4, and 2/1 that's caused by 3125/3072 vanishing.

That's the only 5-interval relationship. The 6-interval relationships are:

(3, 3, 6, 6, 6, 14) - "magic comma pump with one M3 split"
(3, 6, 6, 6, 6, 11) - shows 2048/1875 is a M2 (either magic or negri)
(5, 5, 5, 5, 5, 13) - "hanson comma pump"
(5, 5, 5, 13, 13, 16) - shows 3456/3125 is a M2 (either magic or hanson)
(6, 6, 6, 6, 6, 8) - "magic comma pump"
(6, 6, 6, 11, 14, 14) - "magic comma pump"

Hmm, that's sort of disappointing. But we can add more correspondences and see what we get. If we add in 2 <-> 1 and 17 <-> 11, we get

(2, 2, 2, 13)

which is related to negri and indeed sounds quite unlike 12edo. If instead we add the alternative correspondences 1 <-> 1 and 18 <-> 11, we get

(1, 1, 1, 16), (1, 1, 3, 14), (1, 1, 6, 11), (1, 6, 6, 6)

which are also fruitful avenues to explore. The first three involve ultra-chromatically interpolating between familiar intervals (the second two very much reminiscent of the enharmonic genus), and the last expresses the fact that the 128/125 diesis does not vanish.

Here's the python program if anyone's interested:

from itertools import *
def category_breakers(d, m1, m2, n):
'''d - dict of (real EDO, category EDO) pairs
m1 - real modulus, i.e. EDO number
m2 - category modulus (often 12)
n - number of intervals to use at once'''
for x in combinations_with_replacement(d.keys(), n):
if sum(x) % m1 == 0 and sum(d[i] for i in x) % m2 != 0:
yield x

and you use it like this:

list(category_breakers({3:2,5:3,6:4,8:5,11:7,13:8,14:9,16:10}, 19, 12, 5))

Keenan

LOL, I was just gonna ask how you're liking the guitar! I take it you've got it intonating better than I ever did, since I had a hard time getting the chords to sound as nice as they did on my 19-tone electric.

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> One obvious thing is comma pumps, but a magic or hanson comma pump
> is actually pretty complicated and hard to play for a crappy
> guitarist like me. Even if I found some good ones and could play
> them perfectly at a moderate tempo, they're necessarily sort of long > and listeners might not realize anything weird is going on unless
> they pay close attention.

I'm surprised we agree so much about the short-comings of comma pumps!

My approach to the problem was to either a) use non-5-limit chords (subminors/supermajors sound xenharmonic but out-of-tune to everyone I played them for; oddly, 4:5:6:7:13 chords don't sound that xenharmonic, either...didn't have much luck with this approach in 19, actually, but try sensi[8] and see how that does ya), or b) use chord movements by non-5-limit intervals. This seems to be what your program works out. Negri's great for this. But again, I didn't have much luck. Which is why I gave you the guitar. I expect you'll fare better, as you have more patience than I!

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> LOL, I was just gonna ask how you're liking the guitar! I take it you've got it intonating better than I ever did, since I had a hard time getting the chords to sound as nice as they did on my 19-tone electric.

Yeah, I took it to The 5th String and the helpful guy there fixed it up real good. Tightened the truss rod (after warning me that the fretboard could possibly come unglued, lol), and sanded the nut way down. Cost me all of $10. Highly recommended.

Keenan

what would 3,2,3,3,3,2,3 be? a pseudo 12 diatonic?

as 19 edo degrees

0 3 5 8 11 14 16 19

On Mon, Jan 16, 2012 at 5:44 PM, Keenan Pepper <keenanpepper@...>wrote:

> **
>
>
> While fooling around on my new 19edo guitar (thanks Igs!), I was wondering
> what I could play that would sound more "weird" or "xenharmonic", as a
> change from all the gorgeous ringing major and minor chords that put every
> 12-tone guitar I've ever played to shame.
>
>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> what would 3,2,3,3,3,2,3 be? a pseudo 12 diatonic?
>
> as 19 edo degrees
>
> 0 3 5 8 11 14 16 19

Yep, that's meantone diatonic. (That's the dorian mode you're showing here.) When I play it on my 19edo guitar, no one bats an eye. Sounds just like 12.

(But I'd still call it "meantone diatonic" rather than "pseudo-12 diatonic".)

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Yeah, I took it to The 5th String and the helpful guy there fixed it up real good. Tightened
> the truss rod (after warning me that the fretboard could possibly come unglued, lol), and
> sanded the nut way down. Cost me all of $10. Highly recommended.

Whoa, I gotta check that place out! My 16-tone acoustic could use a similar treatment. How did the guy react to the "extra frets"?

-Igs

Some simple category breakers for selected EDOs (always assuming 12edo categories):

5edo

2+2+2+2+2 "blackwood comma pump"

7edo

1+1+1+4 "P4 divided into 3 equal steps"

9edo

2+2+5 "P4 divided into two m3's" / "bug"

10edo

1+1+1+1+6 "P4 divided into 4 equal steps" / "negri"
1+1+6+6+6
4+4+4+4+4 "blackwood comma pump"

11edo

2+3+3+3
3+4+4

13edo

1+3+3+3+3
2+2+3+3+3
1+4+4+4
2+3+4+4
3+3+3+4

15edo

1+1+1+1+11 "m3 divided into 4 equal steps"
1+1+1+6+6 "3 equal steps between P4 and P5"
1+1+6+11+11
1+6+6+6+11
4+4+4+4+14 "four m3's exceed an octave by a step"
4+4+4+9+9 "porcupine comma pump"
6+6+6+6+6 "blackwood comma pump"

16edo

5+9+9+9 "mavila comma pump"

17edo

4+4+4+4+4+14 ...not sure what to call this

19edo

6+6+6+6+14 "magic comma pump"

22edo

7+7+7+7+7+9 "magic comma pump"

2+4+4+4+4+4 "whole tones form machine scale"
2+4+4+4+15+15
4+4+4+4+13+15
7+7+7+7+18+20
7+7+7+9+18+18

24edo: NONE, because 12edo categories are unharmed

31edo

10+10+10+10+10+10+10+23 "wuerschmidt comma pump"

Keenan

Dang, 13 and 15 do some serious violence to 12edo categories! 13 doesn't surprise me, because all sorts of things add up "wrong" in that tuning (and it is also, on average, the furthest you can get from 12-TET by my totally unscientifc measurements). But 15...wow, I never thought before about how many ways you could violate 12edo-based categories with it. 5 whole-tones = an octave, or converseley 2 whole-tones = a perfect 4th, or two narrow whole-tones = a minor 3rd, or 3 narrow whole-tones = a perfect 4th, 4 minor 2nds = a minor 3rd, etc. etc. That's potentially important.

-Igs

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
>
>
> Some simple category breakers for selected EDOs (always assuming 12edo categories):
>
> 5edo
>
> 2+2+2+2+2 "blackwood comma pump"
>
> 7edo
>
> 1+1+1+4 "P4 divided into 3 equal steps"
>
> 9edo
>
> 2+2+5 "P4 divided into two m3's" / "bug"
>
> 10edo
>
> 1+1+1+1+6 "P4 divided into 4 equal steps" / "negri"
> 1+1+6+6+6
> 4+4+4+4+4 "blackwood comma pump"
>
> 11edo
>
> 2+3+3+3
> 3+4+4
>
> 13edo
>
> 1+3+3+3+3
> 2+2+3+3+3
> 1+4+4+4
> 2+3+4+4
> 3+3+3+4
>
> 15edo
>
> 1+1+1+1+11 "m3 divided into 4 equal steps"
> 1+1+1+6+6 "3 equal steps between P4 and P5"
> 1+1+6+11+11
> 1+6+6+6+11
> 4+4+4+4+14 "four m3's exceed an octave by a step"
> 4+4+4+9+9 "porcupine comma pump"
> 6+6+6+6+6 "blackwood comma pump"
>
> 16edo
>
> 5+9+9+9 "mavila comma pump"
>
> 17edo
>
> 4+4+4+4+4+14 ...not sure what to call this
>
> 19edo
>
> 6+6+6+6+14 "magic comma pump"
>
> 22edo
>
> 7+7+7+7+7+9 "magic comma pump"
>
> 2+4+4+4+4+4 "whole tones form machine scale"
> 2+4+4+4+15+15
> 4+4+4+4+13+15
> 7+7+7+7+18+20
> 7+7+7+9+18+18
>
> 24edo: NONE, because 12edo categories are unharmed
>
> 31edo
>
> 10+10+10+10+10+10+10+23 "wuerschmidt comma pump"
>
> Keenan
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Dang, 13 and 15 do some serious violence to 12edo categories! 13 doesn't surprise me, because all sorts of things add up "wrong" in that tuning (and it is also, on average, the furthest you can get from 12-TET by my totally unscientifc measurements). But 15...wow, I never thought before about how many ways you could violate 12edo-based categories with it. 5 whole-tones = an octave, or converseley 2 whole-tones = a perfect 4th, or two narrow whole-tones = a minor 3rd, or 3 narrow whole-tones = a perfect 4th, 4 minor 2nds = a minor 3rd, etc. etc. That's potentially important.

Of course, if a scale has some of these then it has an infinite number; they just get more and more complicated (and therefore less effective). So don't take the number of examples I present here as indicating a general tendency toward category-breaking.

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> While fooling around on my new 19edo guitar (thanks Igs!), I was wondering what I could play that would sound more "weird" or "xenharmonic", as a change from all the gorgeous ringing major and minor chords that put every 12-tone guitar I've ever played to shame.
> ...
>
> Keenan

One of my favorite progressions in 19 is the following, in which the weird stuff consists of melodic movement entirely by 1/3-tones:

F G B
E# G# B#
E G C

I'd call this an enharmonic progression. Note the expansion of a 5th into a minor 6th in the outer voices.

--George