Oddball ET divisions

What are ya'all trying to accomplish with oddball ET divisions? Let me show you some calculations to find consonant intervals. Is this what you are trying to accomplish? Jumping around in consonant intervals that exist starting on every note?

A 2 step ET octave has the square root of 2 close to
99/70. Not a very useful fraction to
work with there!

Go to this reference if you want to play along:
http://www.kylegann.com/Octave.html
I’m cutting up octaves into equal steps on the exponential
scale to see what kind of consonance I can come up with. I’ll stay within two cents of equal
temperament. That seems to be the gold
standard.

3 TET for three tone equal temperament; 34/27 63/50 121/96 29/23
27/17 46/29 100/63. Excuse me for not including the cents. They are obviously 400 and 800. Similar calculations are used for whatever number of steps/1200.

Let me skip ahead to 22, 24, and 44 TET.

22 TET
0 1/1
1 33/32 32/31
2 33/31 49/46
3 11/10
4 17/15
5 75/64
6 35/29 29/24
7 96/77
8 9/7
9 85/64
10 63/46
11 99/70
12 99/70
13 –
14 14/9
15 77/49
16 48/29 53/32 58/35
17 128/75
18 30/17
19 20/11 51/28
20 62/33
21 31/16
22 2/1

So it appears that in 22 TET three 11/10, eight 9/7,
fourteen 14/9 and nineteen 20/11, step intervals are useful intervals. The beauty of this is that you can jump
around the equal temperament and all the 3, 8, 14, and 19 step intervals are
going to be OK. Not my cup of tea, but
is that the idea?

For 24 TET the two 18/17, three 12/11, ten 4/3, eleven 11/8,
thirteen 16/11, fourteen 3/2, twenty one 11/6, and twenty two 17/9, step
intervals work.

For 44 TET you don’t gain anything by splitting the
intervals in 22 TET into two, and you get a whole bunch of useless intervals to
keep track of.

--- In MakeMicroMusic@yahoogroups.com, Daniel Bernard <danielbernard13@...> wrote:
>
> What are ya'all trying to accomplish with oddball ET divisions?

Politically, one thing I'm doing is trying to deconstruct what is considered "oddball" by
doing it over and over and over again, enough for patterns and tendencies and archetypes
of use to emerge from what was a mere open field for pioneering. (I do have in my head
some horrible metaphor of European expansion westward in North America, and it really
doesn't fit! An alternative tuning isn't an indigenous ecosystem waiting to be
colonized/homogenized, though many can be and have been treated insensitively...)

(freedom is in the choosing, the further constraining)

For example, now that so much 17-equal music has been written and performed, we can
listen and ask what some people thought was possible with 17, and we can write music
that takes that into consideration, whether by taking it for granted, challenging it, or
parodying it, or...? Eventually, there grows a tradition which, because it has a name and
adherents who identify with it, is at least legitimized in the sayscape of liberal correctness
(but only if it does not remain invisible from within the status quo, as microtonality
sometimes seems yet).

Another cue I'm taking is one from Ivor Darreg, who wrote about the characteristic sound
or "mood" particular to each different tuning, that "there are no bad scales", that each one
is able to do something that the others can't. And I like this way of thinking, this musical
practice, in the face of the monoculture pushed by the institutions of this society, with its
insistence on objectivity and dichotomizing and causal explanations. (I'm still trying to
find language that makes these distinctions effectively.)

Why ET's? I bask in the mood-variety of so many systems which sound so different but
were constructed, as lazily as possible, with the same type of organization.

> Let me show you some calculations to find consonant intervals.

Be not so quick to equate small-number-just with "consonance", or even usefulness. But
that high horse belongs to other people.

Draw an isosceles right triangle. If those were string lengths, you now have a 600 cent
interval. Simple, no?

> 3 TET for three tone equal temperament; 34/27 63/50 121/96 29/23
> 27/17 46/29 100/63.

I've cranked up the harmonic distortion and been able to make out the fundamental of
29/23. It's pushing the phenomenon. And I like difference tone progressions created in
this manner - <http://xenharmonic.wikispaces.com/DifferenceToneProgressions> - but
there often arises the problem (perhaps no problem at all) that the difference tone isn't
also present in the tuning. E.g. if an ET nails 11/10, there's no guarantee of it nailing
11/8 and 5/4 such that it can accurately do a chord like 1:10:11.

~./
Jacob

--- In MakeMicroMusic@yahoogroups.com, "Jacob"
<tricesimoprimalist@...> wrote:

Daniel wrote:
> > Let me show you some calculations to find consonant intervals.
>

Jacob wrote:
> Be not so quick to equate small-number-just with "consonance", or
even usefulness. But
> that high horse belongs to other people.

Jacob, I'm with you on this one; in fact, I just happened to have
finished changing the http://untwelve.org/why.html page to get away
from JI-centric thinking when I stumbled upon this response.

I also think I'm skeptical of my skepticism: there is _something_ to
the JI phenomenon, at least in the sense that non-tone-deaf singers
for instance naturally find themselves anchoring to those 'points of
stability'

The back and forth of this JI vs. ET vs. non-JI vs. non-JI/non-ET vs.
non-octave battle is both exasperating and fascinating. It sort of
gives away how rich this subject really is. Or how incredibly weird
and geeky we all are to be paying such close attention (or both?!)

> Draw an isosceles right triangle. If those were string lengths, you
now have a 600 cent
> interval. Simple, no?

Good one!

-A.

on this page you describe Indonesian music as non harmonic which i tend to disagree with in that difference tones seem to have some influence (at least). I do not think there is any culture though that is based on NOT doing something. What the something is that they go for is often unsolved, in which simple ratio explain only a few.
The investigation into Phi where the desire was to find the most dissonant merely lead to other type of phenomenon. So while i would agree with Ricks premise of waves, there are also other 'patterns' that humans and likewise the universe might serve as points of attraction.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Aaron Krister Johnson wrote:
>
> --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, "Jacob"
> <tricesimoprimalist@...> wrote:
>
> Daniel wrote:
> > > Let me show you some calculations to find consonant intervals.
> >
>
> Jacob wrote:
> > Be not so quick to equate small-number-just with "consonance", or
> even usefulness. But
> > that high horse belongs to other people.
>
> Jacob, I'm with you on this one; in fact, I just happened to have
> finished changing the http://untwelve.org/why.html > <http://untwelve.org/why.html> page to get away
> from JI-centric thinking when I stumbled upon this response.
>
> I also think I'm skeptical of my skepticism: there is _something_ to
> the JI phenomenon, at least in the sense that non-tone-deaf singers
> for instance naturally find themselves anchoring to those 'points of
> stability'
>
> The back and forth of this JI vs. ET vs. non-JI vs. non-JI/non-ET vs.
> non-octave battle is both exasperating and fascinating. It sort of
> gives away how rich this subject really is. Or how incredibly weird
> and geeky we all are to be paying such close attention (or both?!)
>
> > Draw an isosceles right triangle. If those were string lengths, you
> now have a 600 cent
> > interval. Simple, no?
>
> Good one!
>
> -A.
>
>

Daniel Bernard wrote:
> What are ya'all trying to accomplish with oddball ET divisions? Let me show you some calculations to find consonant intervals. Is this what you are trying to accomplish? Jumping around in consonant intervals that exist starting on every note?
> > A 2 step ET octave has the square root of 2 close to
> 99/70. Not a very useful fraction to
> work with there!
> > Go to this reference if you want to play along:
> http://www.kylegann.com/Octave.html
> I’m cutting up octaves into equal steps on the exponential
> scale to see what kind of consonance I can come up with. I’ll stay within two cents of equal
> temperament. That seems to be the gold
> standard.

Consonance is a little more complicated than simply finding a nearby just interval (I think harmonic entropy is a reasonably good model, but there are others that are roughly similar). For instance, while you've calculated that 7 steps of 22-ET is near 96/77, it's more likely to be heard as 4.5 cents flat of a 5/4. You might as well call it a "major third" since that's what it will sound like.

What I see as the big attraction of ET is that all the steps are the same size, allowing any melody to be transposed to any degree of the tuning system. Each ET has its own musical character which is related to how closely it approximates JI to some extent, but goes beyond that. One interval in an ET can represent two or more different JI intervals, such as 9/8 and 10/9 in the meantone ET's like 12, 19, 31, 43, etc. This opens up the possibility for chord progressions that would drift if played in just intonation ("comma pumps" as they're called), and unique chords that only exist in ET's (C-E-G#-C in 12-ET, with three major thirds adding up to an octave).

There's also the challenge of working with a "difficult" ET like 11-ET or 13-ET and trying to get something that sounds musically interesting out of it.

the thing that is nice about any prime ET is that any interval can be described and treated as the superposition of any other interval.
This allows structural possibilities.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Herman Miller wrote:
>
> Daniel Bernard wrote:
> > What are ya'all trying to accomplish with oddball ET divisions? Let > me show you some calculations to find consonant intervals. Is this > what you are trying to accomplish? Jumping around in consonant > intervals that exist starting on every note?
> >
> > A 2 step ET octave has the square root of 2 close to
> > 99/70. Not a very useful fraction to
> > work with there!
> >
> > Go to this reference if you want to play along:
> > http://www.kylegann.com/Octave.html > <http://www.kylegann.com/Octave.html>
> > I’m cutting up octaves into equal steps on the exponential
> > scale to see what kind of consonance I can come up with. I’ll stay > within two cents of equal
> > temperament. That seems to be the gold
> > standard.
>
> Consonance is a little more complicated than simply finding a nearby
> just interval (I think harmonic entropy is a reasonably good model, but
> there are others that are roughly similar). For instance, while you've
> calculated that 7 steps of 22-ET is near 96/77, it's more likely to be
> heard as 4.5 cents flat of a 5/4. You might as well call it a "major
> third" since that's what it will sound like.
>
> What I see as the big attraction of ET is that all the steps are the
> same size, allowing any melody to be transposed to any degree of the
> tuning system. Each ET has its own musical character which is related to
> how closely it approximates JI to some extent, but goes beyond that. One
> interval in an ET can represent two or more different JI intervals, such
> as 9/8 and 10/9 in the meantone ET's like 12, 19, 31, 43, etc. This
> opens up the possibility for chord progressions that would drift if
> played in just intonation ("comma pumps" as they're called), and unique
> chords that only exist in ET's (C-E-G#-C in 12-ET, with three major
> thirds adding up to an octave).
>
> There's also the challenge of working with a "difficult" ET like 11-ET
> or 13-ET and trying to get something that sounds musically interesting
> out of it.
>
>