Has this been done?

I'm a mathematician who has done a lot of work with the mathematical
aspects of music, but I am short on knowledge of what has been done,
or what musicians would regard as practical. To introduce myself I'll
start with what may be a painfully obvious question, but it's one for
which I would be interested to hear an answer.

Suppose we had something like an ordinary keyboard, only with three
rows of white keys and two rows of black keys. The tuning would be
1/4 comma mean tone, so that for instance B# major is not the same as
C major.

The top row is to be tuned to D flat flat major, the middle row to C
major, and the bottom row to B# major. The top black keys are shared
between D flat flat and C, consisting of the usual black keys
regarded as flats--namely, D flat, E flat, G flat, A flat and B flat.
Similarly the bottom black keys consist of the black keys regarded as
sharps, and are shared between the B# white keys and the C white
keys, and consist of C#, D#, F#, G# and A#.

I said 1/4 comma temperament (that is, the fifth is flat by 1/4 of a
diatonic comma, meaning by the fourth root of 81/80, or 1.5*5^(-
1/4)), but we get almost the same system with 1/4.1511... comma
temperament, and also something perfectly acceptable for authentic
performance of older music. The advantage of this tuning is that now
we have 31 equal divisions of the octave, and this keyboard becomes a
way of playing music notated in the ordinary way in something like an
ordinary way, on a 31-tone division. (The 31 tones can be played in
succession by playing three white keys up, then two black keys up,
and so forth.)

My questions are:

(1) Has this already been done?

(2) Would such a keyboard be hard to make?

(3) Would such a keyboard be hard to play?

--- In tuning@y..., genewardsmith@j... wrote:

> My questions are:
>
> (1) Has this already been done?

Many times, mainly in the 16th and 20th centuries. Search for recent posts about Vicentino, and
Fokker's organ, and look under _Musical Instruments_ at John Starrett's website (which should
be everyone's "home" page in their browser).

31 is cool, isn't it? We might very well have been born in a 31-tone musical culture had not
economic considerations foisted the simpler solution of 12 upon our ancestors a couple of
centuries ago (big Vicentino fan here).

Hi Gene,

I've wanted to get in touch with you for a long time!

I'm very interested in the matters you discuss in

http://www.maths.ex.ac.uk/~mwatkins/zeta/zetatuning.htm

and other web pages I've found. May I entreat you to join 38 other math-music geeks over at

tuning-math@yahoogroups.com

We're discussing some topics that may interest you greatly, and I predict that your mathematical
understanding of groups and kernels can be applied directly to our concepts of periodicity
blocks and unison vectors, respectively, to prove some tentative theorems, etc.. You may very
well be a godsend to our hopeless floundering over there :) Meanwhile, you might entertain and
enlighten us with a discussion of what the Riemann zetafunction has to do with tuning!

-Paul

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> We're discussing some topics that may interest you greatly, and I
predict that your mathematical
> understanding of groups and kernels can be applied directly to our
concepts of periodicity
> blocks and unison vectors, respectively, to prove some tentative
theorems, etc..

I'll check it out. I didn't know this tuning group existed until
today, and now I learn there is a math-tuning group! Maybe I should
repost the article I just posted over there, and move any resulting
discussion.

As you can see from my question, there is lots I don't know about the
practical side of things, but I have certain aspects of theory up the
wazoo.

Hello, there, Gene Ward Smith and everyone -- and welcome to the
Tuning List!

Please let me join Paul in saying that circulating 31-note tunings
based on 1/4-comma meantone or 31-tET have indeed been done many
times, with the era of around 1550-1620 being one leading period
(Nicola Vicentino, 1555; Fabio Colonna, 1618).

One thing I'd emphasize is that precise mathematical closure isn't
required for musical circulation, and the most obvious effect of using
1/4-comma meantone rather than 31-tET for the system might be that the
"odd" fifth completing the circle would be notably close to pure; a
few remote thirds would also be a bit more impure than the others, but
to a far lesser extent than in 17th-19th century well-temperaments.

Thus the choice between 1/4-comma meantone or 31-tET is mostly a
matter of taste, or mathematical persuasion: pure major thirds, or
absolute symmetry?

As you have remarked, Paul, there's lots of discussion on these topics
in the archives, so I'll stick to a couple of points, while warmly
inviting any questions or comments, Gene -- or anyone else.

First, 1/4-comma meantone or 31-tET can fit either music in either a
Renaissance kind of modal system, or in major/minor tonality. In
either approach, different people might prefer different systems for
naming the notes -- Vicentino and Colonna, for example, take rather
divergent approaches.

Also, there are at least two approaches to arranging a keyboard for a
31-note cycle. One is to start with a usual keyboard design, and add
enough extra accidentals to include at least 31 notes -- possibly
using two manuals.

The other is a "generalized keyboard," for example Colonna's in 1618,
where there are five ranks of keys, each with a diatonic scale of
seven notes per octave, arranged so that each is a diesis or fifthtone
higher than the previous one. Interestingly, since this scheme
actually provides 35 keys for each octave although there are only 31
distinct notes, the effect is to make some notes available at more
than one place on the keyboard -- a kind of "redundancy" favored in
many later generalized keyboards also for easier playing.

Then, as now, a new type of keyboard has a certain unfamiliarity to
overcome: we are told by 16th-century sources that some players
excelled on these instruments, while many found them rather
intimidating with their many notes per octave. It may be largely a
question of musical acculturation, so to speak.

Anyway, welcome to our community, and thanks also for the JI
questions, which I'll address in another post.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <
MSCHULTER@V...> wrote:
>
> Also, there are at least two approaches to arranging a keyboard for a
> 31-note cycle. One is to start with a usual keyboard design, and add
> enough extra accidentals to include at least 31 notes -- possibly
> using two manuals.
>
> The other is a "generalized keyboard," for example Colonna's in 1618,
> where there are five ranks of keys, each with a diatonic scale of
> seven notes per octave, arranged so that each is a diesis or fifthtone
> higher than the previous one. Interestingly, since this scheme
> actually provides 35 keys for each octave although there are only 31
> distinct notes, the effect is to make some notes available at more
> than one place on the keyboard -- a kind of "redundancy" favored in
> many later generalized keyboards also for easier playing.

Today there are even better designs for a 31-tone
keyboard -- see Fokker's organ, for example, in
which all keys are fingered identically.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Today there are even better designs for a 31-tone
> keyboard -- see Fokker's organ, for example, in
> which all keys are fingered identically.

I've considered a system where the keys are circular buttons like
accordian buttons, in a hexagonal close packing array. One leg of the
triangle would be the small semitone (two steps) the other the large
semitone (three steps) and of course the third leg a tone (five
steps.) The whole thing to be slanted so that an octave is straight
across.

However, you both seem to be missing the point of my proposal, which
is to produce a keyboard as much like what keyboardists are used to
as possible. That way, they could perhaps migrate to using it with
relatively little difficulty.

If you look at the keyboard designs in Ellis' appendix, or the
description of the Colonna keyboard recently given, they seem to be
generally a little ad hoc and not well thought out in advance. I
think a little advance planning might make these things much more
useful.

Fokker sounds like he had a good approach--how does his keyboard work?

>> Today there are even better designs for a 31-tone
>> keyboard -- see Fokker's organ, for example, in
>> which all keys are fingered identically.
>
> I've considered a system where the keys are circular buttons like
> accordian buttons, in a hexagonal close packing array. One leg of
> the triangle would be the small semitone (two steps) the other the
> large semitone (three steps) and of course the third leg a tone
> (five steps.) The whole thing to be slanted so that an octave is
> straight across.

That sounds like Wilson's design. See:

http://catalog.com/starrlab/keyboards.html

> However, you both seem to be missing the point of my proposal,
> which is to produce a keyboard as much like what keyboardists are
> used to as possible. That way, they could perhaps migrate to using
> it with relatively little difficulty.
>
> If you look at the keyboard designs in Ellis' appendix, or the
> description of the Colonna keyboard recently given, they seem to be
> generally a little ad hoc and not well thought out in advance. I
> think a little advance planning might make these things much more
> useful.
>
> Fokker sounds like he had a good approach--how does his keyboard
> work?

His is a rectangular packing, and was implemented on a 31-tone
pipe organ which is currently located in The Netherlands. I believe
Fokker independently came up with this, but it is equivalent to
Bosanquet's circa 1870 design, implemented on a 53-tone harmonium
which is currently located in a museum in England.

Also with rectangular packing but without the slant is Janko's
well-known keyboard, which was primarily seen as an improved way
to access 12-tone equal temperament, being implemented on pianos
by Decker Bros. and others in the early part of the last century.

-Carl

--- In tuning@y..., genewardsmith@j... wrote:

> Fokker sounds like he had a good approach--how does his keyboard
work?

See http://www.xs4all.nl/~huygensf/english/instrum.html -- a
projection of the layout is clearly shown in white, black, and blue,
near the bottom of the page.

Hi,

> Suppose we had something like an ordinary keyboard, only with three
> rows of white keys and two rows of black keys...
>
> Has this already been done?

Lawrence Nalder includes a photo of Emanuel Moore's piano keyboard in
_The modern piano_ (1927; p.183-185), with illustrations reproduced from
on of the U.K. patents (UK161549, 180633)- this basically has the same
touch plate layout as you've described, however it's intended as a
12-tone double manual that permits coupling octaves. Unfortunately the
labelled side view, and Nalder's exerpt omit the second row of white
keys but which are obvious in the photo, between and slightly above the
sharps of the lower manual, apparently attached to the white keys of the
same. A few were produced in England by Aeolian.

> Would such a keyboard be hard to make?

There's a lot more to consider than just the arrangement of the key
tops, but this might be fairly straight forward for a reed organ - one
of the reasons they were popular for this purpose during the 19th
century (Paul - I read somewhere there's a 31-tone melodian at NEC).
Early 20th century quarter tone pianos took a bit more than twice as
much work than their 12 tone counterparts, and they used rather ordinary
parts.

Clark