Bosanquet keyboards and linear temperaments

Here is a url for a picture of a Bosanquet keyboard:

http://monxmood.free.fr/bosanquet/bosanquet.html

I will assume this is laid out in a flat array, and that the hexagons
are actually regular hexagons. I will also assume it is a meantone
keyboard, rather than specifically a 31-et keyboard. It then has the
following properties:

(1) There is a "sharp" axis, c, c sharp, etc., where if q is a 5-
limit rational number, then h5(q) (where h5=[5, 8, 12] is the 5-limit
standard val for the 5-et) counts the number of steps along this axis
of the "sharp" coordinate. This axis is inclined at an angle of
76.102 degrees upward.

(2) 120 degrees away there is a "flat" axis, c, d flat, e double flat
and so forth, inclined at an angle of -43.898 degrees downward. Given
q, the number of steps along the flat axis for the flat coordinate is
h7(q), where h7 = [7, 11, 16] is the 5-limit standard val for the 7-
et.

(3) Because of the inclination of the h5 and h7 axis, octaves are
exactly along a horizontal line.

(4) The keys are divided into white and black colors. If [a, b] are
the coordinates for a key, with "a" being the h5 coordinate and "b"
being the h7 coordinates, then if a+b mod 12 is 0, 2, 4, 5, 7, 9, or
11 the key is colored white; if it is 1, 3, 6, 8, or 10 mod 12, it is
colored black. This is a 7-note MOS (the diatonic scale) in 7+5=12
equal terms.

The values of for the inclination of the h5 and h7 axes follow from
the assumption that the keys are regular hexagons. This means the
axes are 120 degrees apart, and so we can use the inner product H
where H([a1, a2], [b1, b2]) = a1b1 + a2b2 - (a1b2+b1a2)/2 to
determine angles and distances. Afterwards, however, we are free to
apply an affine transformation if we prefer another shape of hexagon.

We can now generalize this for any linear temperament. Let us instead
take miracle, and h10 and h11 in the place of h5 and h7. The h10 axis
is now inclined 64.715 degrees upward, and h11 is inclined -55.285
degrees downward. To get the key coloring, we take a version of the
11-note miracle MOS in 21-equal; and reduce it to a set of eleven
numbers mod 21. If a+b mod 21 is a member of this set, we color
white; if not, we color black.

Gene Ward Smith wrote:

> I will assume this is laid out in a flat array, and that the hexagons > are actually regular hexagons. I will also assume it is a meantone > keyboard, rather than specifically a 31-et keyboard. It then has the > following properties:

Bosanquet didn't use hexagons. Erv Wilson did, and you can see his original papers at

http://www.anaphoria.com/wilson.html

It's generally a keyboard for scales generated by an octave and fifths. Bosanquet used 53 notes with a schismic mapping (the tuning may have been JI, I forget).

> We can now generalize this for any linear temperament...

Erv did that a long time ago.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:

> Bosanquet didn't use hexagons. Erv Wilson did, and you can see his
> original papers at
>
> http://www.anaphoria.com/wilson.html

Do you think Wilson lattices would be a better name than Bosanquet
lattices for these?

> > We can now generalize this for any linear temperament...
>
> Erv did that a long time ago.

I'll see if I can figure out what Erv is saying (not always easy for
me, since he says it in pictures) and if I agree. Do you have a more
exact reference?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
> > Gene Ward Smith wrote:
>
> > Bosanquet didn't use hexagons. Erv Wilson did, and you can see
his
> > original papers at
> >
> > http://www.anaphoria.com/wilson.html
>
> Do you think Wilson lattices would be a better name than Bosanquet
> lattices for these?
>
> > > We can now generalize this for any linear temperament...
> >
> > Erv did that a long time ago.
>
> I'll see if I can figure out what Erv is saying (not always easy
for
> me, since he says it in pictures) and if I agree. Do you have a
more
> exact reference?

I found Xenharmonikon 3. This does generalize Bosanquet somewhat, but
it is hardly the same as what I was saying. Is this what you meant?

Gene Ward Smith wrote:

> Do you think Wilson lattices would be a better name than Bosanquet > lattices for these?

I don't know. I usually say "Bosanquet/Wilson" when there's any doubt. The layout is Bosanquet's but the shape of the keys is Wilson's. Bosanquet only considered 12-based temperaments as well, so some of the generalisation is Wilsonian.

> I'll see if I can figure out what Erv is saying (not always easy for > me, since he says it in pictures) and if I agree. Do you have a more > exact reference?

I think that's what the multi-keyboard grid iron is about. It's only octave-based scales, but we can always define temperaments by the period.

He doesn't line them up so that the octaves are all in a horizontal line, so you may have some new geometry.

I also wrote this some time ago:

http://x31eq.com/notakey.htm#gengenkey

Graham

>I don't know. I usually say "Bosanquet/Wilson" when there's any
>doubt. The layout is Bosanquet's but the shape of the keys is
>Wilson's. Bosanquet only considered 12-based temperaments as well,
>so some of the generalisation is Wilsonian.

"12-based"? Bosanquet issued layouts for negative, positive, and
doubly-positive temperaments, IIRC.

-Carl

Carl Lumma wrote:

> "12-based"? Bosanquet issued layouts for negative, positive, and
> doubly-positive temperaments, IIRC.

"Negative" means the fifth is smaller then that of 12-equal. "Positive" means the fifth is greater than that of 12-equal. "Doubly-positive" means the fifth is larger than that of 12-equal, and the Pythagorean comma divides in 2. This is all based on 2.

I had a brief look at Bosanquet's book, and didn't see a doubly-positive mapping. But he did class all equal temperaments using this scheme. To get an ET's positivity, take the number of notes modulo n. 17 is singly positive, and 17 = 5 mod 12. An n note scale is mthly positive if n = 5*m mod 12. For singly negative scales, m=-1 and -1*f = 7 modulo n. So the singly negative scales are 7, 19, 31, 43, ...

Wilson in one of the Xenharmonikon 3 papers extended this scheme to singly positive and negative scales with respect to 5 and 7, which allows more ETs to be played on a Bosanquet keyboard. Other files in the Wilson Archive show he has considered more general cases as well.

Graham

>> "12-based"? Bosanquet issued layouts for negative, positive, and
>> doubly-positive temperaments, IIRC.
>
>"Negative" means the fifth is smaller then that of 12-equal.
>"Positive" means the fifth is greater than that of 12-equal.
>"Doubly-positive" means the fifth is larger than that of 12-equal,
>and the Pythagorean comma divides in 2. This is all based on 2.

Based on 12, you mean. It's simpler to say it's based on the
pythagorean comma. What did Bosanquet leave out?

>I had a brief look at Bosanquet's book, and didn't see a
>doubly-positive mapping.

That's because it isn't in the book. But you almost certainly
were looking at Rasch's edition, in which he includes the doubly-
positive mapping (from a later paper by B.) in the introduction.

Yep, Wilson definitely showed a lot of mappings that Bosanquet
didn't.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I don't know. I usually say "Bosanquet/Wilson" when there's any
> >doubt. The layout is Bosanquet's but the shape of the keys is
> >Wilson's. Bosanquet only considered 12-based temperaments as well,
> >so some of the generalisation is Wilsonian.
>
> "12-based"? Bosanquet issued layouts for negative, positive, and
> doubly-positive temperaments, IIRC.
>
> -Carl

all of which are defined with respect to 12.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> "12-based"? Bosanquet issued layouts for negative, positive, and
> >> doubly-positive temperaments, IIRC.
> >
> >"Negative" means the fifth is smaller then that of 12-equal.
> >"Positive" means the fifth is greater than that of 12-equal.
> >"Doubly-positive" means the fifth is larger than that of 12-equal,
> >and the Pythagorean comma divides in 2. This is all based on 2.
>
> Based on 12, you mean. It's simpler to say it's based on the
> pythagorean comma. What did Bosanquet leave out?

tunings based on more than one chain of fifths, like 72, for example.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
> > We can now generalize this for any linear temperament...
>
> Erv did that a long time ago.

I'm sure he would have, however he apparently hasn't explained it very
well. I didn't try to explain it either, but instead just
_implemented_ it -- in a spreadsheet that draws the layout, given the
period, the generator, the number of notes and an aspect-ratio
parameter. See
/tuning/topicId_25575.html#25742
The URL for the spreadsheet given in that message will soon be
obsolete. The current location is
http://dkeenan.com/Music/KeyboardMapper.xls

The spreadsheet doesn't do the colouring, but that's still somewhat a
matter of taste. To generalise that, I'd colour white the most central
mode of that proper MOS whose notes per octave is between 5 and 9 and
is closest to the magic number 7 (I'd use an improper MOS if there's
no proper MOS with 5 to 9 notes and failing that I'd use the smallest
MOS with 10 or more notes per octave) and then expand that
symmetrically to the next proper MOS with black, then the next with
red, then blue, yellow, green, orange, purple as required.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...>
wrote:

> http://dkeenan.com/Music/KeyboardMapper.xls

I'm planning to put up a page on Bosanquet lattices on xenharmony,
and would like to link to this. I presume that is OK?

We really need to set up a system of links, and I still like the idea
of a web ring.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...>
> wrote:
>
> > http://dkeenan.com/Music/KeyboardMapper.xls
>
> I'm planning to put up a page on Bosanquet lattices on xenharmony,
> and would like to link to this. I presume that is OK?

Sure, no problem.

> We really need to set up a system of links, and I still like the idea
> of a web ring.

I'm afraid I prefer trees to rings (you don't end up where you've
already been without knowing it). I think of John Starret's site as
the root.

http://www.nmt.edu/~jstarret/microtonalists.html

Ask John to list you there and link to your site.

I've not described the algorithm in words anywhere before now,
although I hope I made it easy for anyone to reverse engineer the
spreadsheet. Here it is in a nutshell:

I calculate the pitches in cents (not octave reduced) for sufficiently
many chains of generators (36 in the spreadsheet), with the centers of
the chains spaced out by periods, and each chain having N/2 notes on
either side of its center. So these pitches are calculated in an array
indexed by period-number (+-big_number) in one dimension and
generator-number (+-N/2) in the other dimension.

To plot the pitches onto the keyboard, the left/right coordinate (from
the players point of view) is simply the pitch in cents, and the front
back coordinate is simply the pitch's generator-number multiplied by
the aspect ratio parameter. The lines joining the dots on the plot
show the chains of generators.

Given that one usually has to tweak not only the aspect ratio
parameter but also the generator size in order to obtain an
equilateral hexagonal layout, it strikes me that there are only a
finite number of distinct hexagonal layouts corresponding to linear
temperaments having a generator between say 1/12 and 1/2 of the
period. I'd be interested to know how many, and how far apart their
generators typically are.