Generalized?

I guess a lot of the posters here do not understand what a generalized
keyboard means. A generalized keyboard is one where intervals are
covered by the same key distance, no matter what the absolute pitch is.

It does not mean a keyboard that works for more than one different
tuning. It does not even imply that. That would be Multitonal,
Multimicrotonal or whatever you wish to call it. Feel free to discuss
that.

So please use "Generalized keyboard" only for what it really means!

/Ö

>It does not mean a keyboard that works for more than one
>different tuning. It does not even imply that.

Hate to disagree Mats, but on page 19 of Bosanquet's 1876
Treatise, he coins the term, saying,

"In the enharmonic harmonium exhibited by the writer at
the Loan Exhibition of Scientific Instruments, at South
Kensington, 1876, there is a keyboard which can be employed
with all systems of tuning reducible to successions of
uniform fifths; from this property it has been called the
generalized keyboard."

-Carl

Mats �ljare wrote:
> I guess a lot of the posters here do not understand what a generalized
> keyboard means. A generalized keyboard is one where intervals are
> covered by the same key distance, no matter what the absolute pitch is.

Does this mean that only Equal tempered scale work on Generalized Keyboards?

Is there a Generalized Keyboard for 19 tone, and one for 31 tone, etc...?

If you are refering to my posts on this, might you offer a recommendation of a keyboard layout for the 43 tone scale of Partch?

Thanks,

Mike

--- In tuning@y..., Michael J McGonagle <fndsnd@r...> wrote:
>
> Mats Öljare wrote:
> > I guess a lot of the posters here do not understand what a
generalized
> > keyboard means. A generalized keyboard is one where intervals are
> > covered by the same key distance, no matter what the absolute
pitch is.

As Carl Lumma stated, Bosanquet, coined the term "generalized
keyboard" for a keyboard that is able to accommodate tunings defined
by a "generating" interval (note the same word root), in this
particular case a fifth (either just or tempered).

> Does this mean that only Equal tempered scale work on Generalized
Keyboards?

No. The only requirement is that the tones of the scale or tonal
system all be members of a single series in which consecutive members
differ in pitch by the generating interval. So on Bosanquet's
keyboard you could use a temperament consisting of a series of fifths
of practically any size, even if two tones at opposite directions in
the series never matched (with octave reduction) to make a closed
circle.

However, this doesn't mean that a generalized keyboard won't work for
a system having mixed sizes of fifths (i.e., an unequal temperament)
or even just intonation. If you can find a logical way to map a set
of tones into some division of the octave (provided that the keyboard
will work for that division), then you could assign those tones to
the keyboard as if they were members of that division.

> Is there a Generalized Keyboard for 19 tone, and one for 31 tone,
etc...?

No. The whole point of a generalized keyboard is to be able to use
it for many different systems (as well as the uniform fingering
patterns). For example, Bosanquet's keyboard will handle 12, 17, 19,
22, 26, 27, 29, 31, 41, 46, and 53, just to name a bunch. It won't
handle 24, because its fifths are in two separate circles.

However, on my Scalatron I do have 24-ET as two sets of 12 -- the
standard 12-ET on the white, black, and red keys, and the
quartertones on the blue and green keys, so I can play quartertones
if I read them as if they were 31-ET. But this is at best a
makeshift arrangement. A different generalized setup (using ~9:11 as
a generating interval) would be able to handle 24 very nicely, as
well as some others such as 17, 31, and 41. But some very good
divisions such as 22 and 46 would be excluded. (Of course you would
have 12 as a subset of 24.)

> If you are refering to my posts on this, might you offer a
> recommendation of a keyboard layout for the 43 tone scale of Partch?

Graham Breed referred to a keyboard mapping that I proposed in
Xenharmonikon 3. I don't think the link was right; try this one:

http://www.anaphoria.com/secor.PDF

Here's the latest version of that (in a zipped file), complete with
key dimensions and colors. The order of the rows is now reversed and
then slanted to lateralize the octaves:

/tuning-math/files/secor/kbds/KbDec72.zip

Details are in my posting #37151. I was illustrating how 19-limit
ratios can be mapped onto 72, so some of these are different from
those in Partch's 43-tone set. You would need to look at Partch's
ratios in the Xenharmonikon article diagram to identify their
locations on this latest graphic.

This "decimal keyboard" will nicely accommodate a couple of the
larger divisions, 41 and 72, in addition to 31. It can also be used
for 53 (with a 5-degree generator; I don't think anybody has ever
suggested this before, so this might be news to a more than a few
people), but many of the intervals will occur in places different
from 31, 41, and 72. So fingerings in 53 will be different from the
others, just as 22 is different from 19 and 31 on the Bosanquet
keyboard.

I imagine that this is more information than you ever wanted or
needed, but once I get going it's sometimes hard to stop!

--George

> From: "gdsecor" <gdsecor@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, October 11, 2002 11:14 AM
> Subject: [tuning] Re: Generalized?
>
>
> --- In tuning@y..., Michael J McGonagle <fndsnd@r...> wrote:
>
> <snip>
>
> > Is there a Generalized Keyboard for 19 tone,
> > and one for 31 tone, etc...?
>
> No. The whole point of a generalized keyboard is to be able to use
> it for many different systems (as well as the uniform fingering
> patterns). For example, Bosanquet's keyboard will handle 12, 17, 19,
> 22, 26, 27, 29, 31, 41, 46, and 53, just to name a bunch. It won't
> handle 24, because its fifths are in two separate circles.
>
> However, on my Scalatron I do have 24-ET as two sets of 12 -- the
> standard 12-ET on the white, black, and red keys, and the
> quartertones on the blue and green keys, so I can play quartertones
> if I read them as if they were 31-ET. But this is at best a
> makeshift arrangement. A different generalized setup (using ~9:11 as
> a generating interval) would be able to handle 24 very nicely, as
> well as some others such as 17, 31, and 41. But some very good
> divisions such as 22 and 46 would be excluded. (Of course you would
> have 12 as a subset of 24.)

i just wanted to add that George's description of 24edo here
is referring to the "bike-chain" aspect of it. see:
/tuning/files/dict/bikechain.htm

any EDO that has a cardinality which is a multiple of a
lower-cardinality-EDO will have "bike-chains". i.e.,
22 is 2 bike-chains of 11,
36 is 3 bike-chains of 12,
72 is 6 bike-chains of 12, etc.

-monz
"all roads lead to n^0"

> Kensington, 1876, there is a keyboard which can be employed
> with all systems of tuning reducible to successions of
> uniform fifths;

Which means that it works for fifth/octave-based tunings.

/Ö

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

/tuning/topicId_39468.html#39494
>
> No. The whole point of a generalized keyboard is to be able to use
> it for many different systems (as well as the uniform fingering
> patterns). For example, Bosanquet's keyboard will handle 12, 17,
19,
> 22, 26, 27, 29, 31, 41, 46, and 53, just to name a bunch. It won't
> handle 24, because its fifths are in two separate circles.
>

***George, could you please explain to me, with reference to your
keyboard diagram, which I have, why the keyboard works for a series
of fifths in *one* circle, but won't for *two.* I'm not entirely
understanding that, and would like to!

Thanks!

Joseph Pehrson

--- In tuning@y..., "monz" <monz@a...> wrote:

/tuning/topicId_39468.html#39495

>>
> i just wanted to add that George's description of 24edo here
> is referring to the "bike-chain" aspect of it. see:
> /tuning/files/dict/bikechain.htm
>

***Monz... if I remember correctly, my own humble self was the
individual to come up with the "bike chain" metaphor.... not that it
really matters!

Joe Pehrson

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> /tuning/topicId_39468.html#39494
> >
> > No. The whole point of a generalized keyboard is to be able to
use
> > it for many different systems (as well as the uniform fingering
> > patterns). For example, Bosanquet's keyboard will handle 12, 17,
19,
> > 22, 26, 27, 29, 31, 41, 46, and 53, just to name a bunch. It
won't
> > handle 24, because its fifths are in two separate circles.
> >
>
> ***George, could you please explain to me, with reference to your
> keyboard diagram, which I have, why the keyboard works for a series
> of fifths in *one* circle, but won't for *two.* I'm not entirely
> understanding that, and would like to!
>
> Thanks!

Okay. First, here's the same diagram labeled for 31-ET, which will
work just fine for this explanation:

/tuning-
math/files/secor/kbds/KbScal31.zip

This differs from Bosanquet's original "generalised keyboard" of 1875
only in the size and shape of the keys (and in the Americanized, or
Americanised, spelling of the name, if you want to get picky). The
keyboard geometry is such that only tones in a single *chain* or
*series* of fifths can be put onto the keyboard. If are in 24-ET or
72-ET (for example) and you start on C and play tones along the
series of fifths in the sharp direction, i.e., C G D A E B F# and
then do the same thing in the flat direction, i.e., C F Bb Eb Ab Db
Gb, you will find that the F# and Gb are the same pitch -- a circle
of 12-ET, so the ends of the chain are now joined into a *bicycle
chain*. Observe that the vertical position of each key in the series
is slightly higher (in the sharp direction) or lower (in the flat
direction) as you go through the series of fifths.

Continuing the series in either direction will only repeat pitches
you already have. This is fine if you want duplicate keys for
alternate fingerings for 12-ET, but you'll never get any of the other
pitches of 24-ET or 72-ET by going through a series of fifths,
because you have no way of jumping out of the "bicycle chain" of
fifths into another one unless you introduce some other interval (one
not found in 12-ET) into the series. And if you do that, then you no
longer have a "generalized" arrangement of tones.

So there's no such thing as a "universal generalized keyboard" that
will handle everything, but I believe that the Bosanquet geometry
comes closest to that ideal.

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

/tuning/topicId_39468.html#39577

> Continuing the series in either direction will only repeat pitches
> you already have. This is fine if you want duplicate keys for
> alternate fingerings for 12-ET, but you'll never get any of the
other
> pitches of 24-ET or 72-ET by going through a series of fifths,
> because you have no way of jumping out of the "bicycle chain" of
> fifths into another one unless you introduce some other interval
(one
> not found in 12-ET) into the series. And if you do that, then you
no
> longer have a "generalized" arrangement of tones.
>
> So there's no such thing as a "universal generalized keyboard" that
> will handle everything, but I believe that the Bosanquet geometry
> comes closest to that ideal.
>
> --George

***Thanks so much, George. This now seems pretty well illustrated by
the distinctly different *colors* you have in your diagram.
Actually, they came out really nicely, at least on *my* computer
screen...

Joseph

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> /tuning/topicId_39468.html#39495
>
> >>
> > i just wanted to add that George's description of 24edo here
> > is referring to the "bike-chain" aspect of it. see:
> > /tuning/files/dict/bikechain.htm
> >
>
> ***Monz... if I remember correctly, my own humble self was the
> individual to come up with the "bike chain" metaphor.... not that
it
> really matters!
>
> Joe Pehrson

that's true -- and when i "hopped on", i used it in a much more
restrictive fashion than simply the number of notes being a composite
number!