Reverse engineering a scale

If you look at the Scala archive, you will find ten prefixed Keenan.
None of these could be counted as a tempering according to Dave's
definition, as there is no physical instrument and Scala scale files
are not like tun files--they are not given in terms of frequencies,
but (in one way or another) in terms of frequency ratios.

However, clearly tempering is often going on, and what sort of
tempering is sometimes explained in the header data. Even when it is
not, you can sometims figure it out. To do so you need to toss out
what Dave said about temperaments, since despite what he said he
clearly has in mind concepts of the kind he was deriding; one hardly
constructs a scale exhibiting a regular planar tempering by accident.

By example here is a 31-note scale called keenan5, which I decided to
reverse engineer. It is not a random collection of notes; it does not
even have a complex structure like a ciculating temperament often
will. It is straightforward, logical, and yes, mathematical in the way
it was evidently constructed.

By running it through Scala's equal temperament fitter, I found it
could be fitted, increasingly well, to 31, 41, 72, 125 and 166. This
strongly suggests it is 11-limit marvel, which you can discover by
putting together the corresponding standard vals.

By fitting it to 166-equal, and then replacing the steps of size 4, 5,
and 7 with steps of size 2048/2025, 16875/16384 and 128/125, I
produced an algebraically exact 5-limit version. This does exactly
what Dave objects to, namely, exhibits the scale as a planar
temperament. Approximations to 2,3 and 5 can be used to generate
marvel, and hence the 5-limit version tells you what the mapping from
11-limit JI is--the thing I was calling the temperament itself. At
this point choosing a particular tuning is a separate question which
does not involve what the temperament *is*, namely 11-limit marvel. I
say 11-limit because that is what the header information claims for
it, but also because of the precise tuning, which turns out to be the
11-limit minimax tuning.

Hence, Dave's scale can be *precisely* defined as the tempering by
11-limit minimax marvel of the following 5-limit scale:

16875/16384, 135/128, 16/15, 1125/1024, 9/8, 256/225, 75/64, 6/5,
10125/8192, 5/4, 32/25, 675/512, 4/3, 512/375, 45/32, 64/45, 375/256,
3/2, 1024/675, 25/16, 8/5, 3375/2048, 5/3, 128/75, 225/128, 16/9,
2048/1125, 15/8, 256/135, 2025/1024, 2

I think it is clear, first, that Dave cannot possibly have found this
scale without theoretical concepts in mind of the kind he is now
scoffing at. Moreover, I could not have reversed engineered it without
those same concepts.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> Hence, Dave's scale can be *precisely* defined as the tempering by
> 11-limit minimax marvel of the following 5-limit scale:
>
> 16875/16384, 135/128, 16/15, 1125/1024, 9/8, 256/225, 75/64, 6/5,
> 10125/8192, 5/4, 32/25, 675/512, 4/3, 512/375, 45/32, 64/45, 375/256,
> 3/2, 1024/675, 25/16, 8/5, 3375/2048, 5/3, 128/75, 225/128, 16/9,
> 2048/1125, 15/8, 256/135, 2025/1024, 2

Anyone want to draw a lattice diagram?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I think it is clear, first, that Dave cannot possibly have found
this
> scale without theoretical concepts in mind of the kind he is now
> scoffing at.

Gene,

You're beating a straw man. Maybe you should carefully reread my post
/tuning/topicId_55471.html#55576
Look particularly for the words "What if", and "Maybe we do".

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > Hence, Dave's scale can be *precisely* defined as the tempering by
> > 11-limit minimax marvel of the following 5-limit scale:
> >
> > 16875/16384, 135/128, 16/15, 1125/1024, 9/8, 256/225, 75/64, 6/5,
> > 10125/8192, 5/4, 32/25, 675/512, 4/3, 512/375, 45/32, 64/45,
375/256,
> > 3/2, 1024/675, 25/16, 8/5, 3375/2048, 5/3, 128/75, 225/128, 16/9,
> > 2048/1125, 15/8, 256/135, 2025/1024, 2
>
> Anyone want to draw a lattice diagram?

is it ok if i make it 8ve-equivalent and leave out 2?

please, Gene, fill out one of those temperament templates
for marvel.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> is it ok if i make it 8ve-equivalent and leave out 2?

Sure, but you should then put in 1. I'm following the Scala system,
which assumes 1 is always in the scale, and that the last note defines
the period.

> please, Gene, fill out one of those temperament templates
> for marvel.

Marvel and starling, among the planars, should go in, I guess.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > is it ok if i make it 8ve-equivalent and leave out 2?
>
> Sure, but you should then put in 1. I'm following the
> Scala system, which assumes 1 is always in the scale,
> and that the last note defines the period.

no, i didn't mean leave out the note 2, i meant
leave out the prime-factor 2 (and thus reduce is to 2-D).

anyway, i'll just make both 2-D and 3-D lattices,
what the hell.

it sure would have been easier for me to do this if
you had listed the pitches in monzos as well as ratios.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> it sure would have been easier for me to do this if
> you had listed the pitches in monzos as well as ratios.

Here they are, as octave-equivalent monzos:

[[0, 0], [3, 4], [3, 1], [-1, -1], [2, 3], [2, 0], [-2, -2], [1, 2],
[1, -1], [4, 3], [0, 1], [0, -2], [3, 2], [-1, 0], [-1, -3], [2, 1],
[-2, -1], [1, 3], [1, 0], [-3, -2], [0, 2], [0, -1], [3, 3], [-1, 1],
[-1, -2], [2, 2], [-2, 0], [-2, -3], [1, 1], [-3, -1], [4, 2]]

Temper it via marvel, and it has a large number of complete 11-limit
hexands.

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > it sure would have been easier for me to do this if
> > you had listed the pitches in monzos as well as ratios.
>
> Here they are, as octave-equivalent monzos:
>
> [[0, 0], [3, 4], [3, 1], [-1, -1], [2, 3], [2, 0], [-2, -2], [1, 2],
> [1, -1], [4, 3], [0, 1], [0, -2], [3, 2], [-1, 0], [-1, -3], [2, 1],
> [-2, -1], [1, 3], [1, 0], [-3, -2], [0, 2], [0, -1], [3, 3], [-1, 1]
,
> [-1, -2], [2, 2], [-2, 0], [-2, -3], [1, 1], [-3, -1], [4, 2]]
>
> Temper it via marvel, and it has a large number of complete
> 11-limit hexands.

thanks! i'll be doing the 3-D lattice too, so would
it be much trouble to give me the 2,3,5 monzos?

(if it is, i can do it myself ... no big deal.)

also, i'm still waiting on tha "marvel" family data.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> thanks! i'll be doing the 3-D lattice too, so would
> it be much trouble to give me the 2,3,5 monzos?

[[0, 0, 0], [-14, 3, 4], [-7, 3, 1], [4, -1, -1], [-10, 2, 3],
[-3, 2, 0], [8, -2, -2], [-6, 1, 2], [1, 1, -1], [-13, 4, 3],
[-2, 0, 1], [5, 0, -2], [-9, 3, 2], [2, -1, 0], [9, -1, -3],
[-5, 2, 1], [6, -2, -1], [-8, 1, 3], [-1, 1, 0], [10, -3, -2],
[-4, 0, 2], [3, 0, -1], [-11, 3, 3], [0, -1, 1], [7, -1, -2],
[-7, 2, 2], [4, -2, 0], [11, -2, -3], [-3, 1, 1], [8, -3, -1],
[-10, 4, 2], [1, 0, 0]]

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> Hence, Dave's scale can be *precisely* defined as the tempering
> by 11-limit minimax marvel of the following 5-limit scale:
>
> 16875/16384, 135/128, 16/15, 1125/1024, 9/8, 256/225, 75/64,
> 6/5, 10125/8192, 5/4, 32/25, 675/512, 4/3, 512/375, 45/32,
> 64/45, 375/256, 3/2, 1024/675, 25/16, 8/5, 3375/2048, 5/3,
> 128/75, 225/128, 16/9, 2048/1125, 15/8, 256/135, 2025/1024, 2

thanks for waiting so long, Gene. i finally made two
lattices of this -- one rectangular and one triangular, in here:

http://launch.groups.yahoo.
com/group/tuning_files/files/monz/gene-keenan/

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> http://launch.groups.yahoo.
> com/group/tuning_files/files/monz/gene-keenan/

Some tiny urls for that:

Triangular: http://tinyurl.com/6hg7a

Rectangular: http://tinyurl.com/5frqf

Note how the scale is oriented along the secor axis of 16/15. The 7/4
is represented by 225/128, and the 16/11, a 6/5 away, by 375/256; and
the scale streches itself out in this direction. The ratio (7/4)/(16/11) =
77/64 is equivalent under marvel to 6/5, so these end up next to each
other. It looks as if it is not too far off from being a Fokker block,
and it would be interesting to see if we can find 31-note 5-limit
Fokker blocks with similar properties.

Thanks, this was worth seeing.