Yahoo Tuning Groups Ultimate Backup tuning Re: [MMM] 11-EDO and Godzilla

Igs,

I'm cc'ing this to tuning, because I think we should continue anything from
this thread over there......but I appreciate your taking the time to lay
this out here.

Much of this I already knew, but I think this is a nice executive summary
for newbies and seasoned experts nonetheless. I didn't know the last part
about a "Breed", but the dimesionality part (an N-dimensional space
corresponding to given primes) is of course old hat to me. You also managed
to make the val concept much less ambiguous in such a way that it will
really stick this time.

But more importantly, this shows that you are engaged in making people
understand things in as intuitive a way as possible, which is a big plus in
our field of work. I think using what the brain tends to do best---visual
and geometric intuition, is crucial to get these concepts across.

Anyway, Igs--you or Carl I think are excellent candidates for bridging the
"comprehension gap" in the prose at the xenharmonic wiki---i.e., bringing it
down to the mere mortal level.

Best,
AKJ

On Wed, Mar 9, 2011 at 4:46 PM, cityoftheasleep <igliashon@...>wrote:

> --- In MakeMicroMusic@yahoogroups.com, Aaron Krister Johnson <aaron@...>
> wrote:
> >
> > This kind of answer basically makes me want to reiterate that we need a
> > 'tuning math for dummies' page in the xenwiki.
>
> It's pretty simple when you break it down to brass tacks, actually. Start
> with JI. Imagine a scale where the only note is a 1/1--no octaves, even.
> This is a 0-dimensional scale. We can't get any other notes because 1 to
> any power is always 1. Now add a 2nd interval, any interval will do but
> let's say 2/1 for convention's sake. We now have a 1-dimensional scale,
> because we can go up or down from 1/1 by powers of 2/1. Adding another
> interval---3/1, for instance--gets us to a 2-dimensional tuning-space where
> we can move along either or both of two axes (powers of 2 and/or powers of
> 3). Adding another interval--5/1, say--gets us to a 3-dimensional space.
> Every time we add an interval that is not expressible as some combination
> of the intervals we already have, we are adding a dimension to tuning space.
>
> In JI, all the dimensions of tuning space are open and "flat" and
> infinite--you can move as far as you want along any combination of them and
> you'll never return to the origin. When we temper, we are essentially
> "folding" or "curling" tuning space in such a way that it becomes possible
> to return to the origin of 1/1 by moving a finite number of steps along some
> dimensions/axes. This happens because we equate an interval (usually a
> comma) with 1/1 so that that comma ceases to exist--it becomes "one" with
> the unison. I think of this in terms of "fudging" the rules of math so that
> two distinct numbers become equated--it's like we're defining a universe in
> which 2+2=5. What this also does is equate movement along one dimension
> with movement along other dimensions. IOW, with the example of meantone
> temperament, we can now get to places on the 5 axis by moving only along the
> 2 and 3 axes.
>
> This where mappings come in: depending on the comma we temper out, we will
> create different rules for movement in our folded space. A mapping is
> nothing more than a set of directions that tells you how far you far you
> have to go along the available axes of your tempered tuning space to get to
> the places that existed in the untempered tuning space.
>
> In a rank-2 temperament, we have only two axes: the period and the
> generator. When we give a mapping for the temperament, as determined by the
> comma, one way to give it is by giving a val for each axis such that the two
> vals together are coordinates that correspond (usually) to the successive
> primes. A rank-2 temperament can be derived from any level of JI
> dimensionality, but we will have to temper out more commas in order to
> reduce higher dimensions down to two. For instance, we can give 5-limit
> meantone with the two vals <1 1 0] and <0 1 4], where the first val is the
> period, the second is the generator. Taking the first number from each val,
> we have the coordinate (1, 0) or "one period up, zero generators up", and
> that is how we get to 2/1. The second number from each val gives (1, 1), or
> "one period up, one generator up" and that gets us to 3/1. The third
> numbers give (0, 4) or "zero periods up, four generators up" and that gets
> us to 5/1.
>
> We can reduce rank-2 temperaments to rank-1 temperaments, which are equal
> tunings (though not necessarily EDOs), by further tempering out another
> comma. Any time we temper out a comma, we reduce the dimensionality of
> tuning-space. Doing this means we can specify the mapping with a single
> val, and our generator is always going to be one step of the equal scale.
> So the 5-limit val for 12-TET would be <12 19 28]; a rank-1 val is also
> known as a "Breed". Of course, we can give mappings for tunings that are
> not the most accurate mappings; for 7-EDO, we could map 5/1 to its perfect
> 4th rather than its neutral 3rd, and give a val of <7 11 17]. A "patent
> val" is the val that corresponds to the most accurate mapping of JI to steps
> of an ET.
>
> Does that help?
>
> Also, the whole random naming thing--this used to bother the crap out of
> me, but I've since come to accept that with the plethora of things that need
> naming, there's no good clear logical way to go about it.
>
> -Igs
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

Re: [MMM] 11-EDO and Godzilla

🔗Aaron Krister Johnson <aaron@...>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>