Yahoo Tuning Groups Ultimate Backup tuning Re: Microtonal accordion challenge

At 00:24 20/06/02 -0000, you wrote:
>--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>Here's the challenge: If you're going to implement a tonal system on
>an accordion, you can't use more than one tuning on a given
>instrument, and it should probably not be any more than about 22
>tones per octave (considering space limitations). I would like to
>have an instrument that would:
>
>1) Permit playing most conventional diatonic music (i.e., without a
>comma problem);
>
>2) Offer a circle of more than 12 fifths around which I could freely
>modulate; and
>
>3) Allow me to play 15-limit otonalities in at least 3 different keys
>(temperament is okay, but not so much as to lose the identities of
>ratios of 11 and 13);
>
>4) Offer *all* of the above with uniform fingering patterns in *all*
>available keys (utilizing some duplicate keys, of course). (Left and
>right hand patterns are allowed to differ somewhat from each other.)
>
>Does this sound impossible? I have a solution for the above which
>will map onto a Bosanquet generalized keyboard. (One of the variable
>tuning banks of my Scalatron is currently tuned this way.) Would
>anyone venture a guess as to how it could be done (at least on the
>Scalatron)?

If you want the max error in your three 1:3:5:7:9:11:13:15 chords to be less than 7 cents, then it sounds impossible. But otherwise ...

I know nothing about accordion keyboards, e.g. what appears on the left and what on the right, but I do understand generalised keyboards (2D mapping of linear temperaments). However, all the other constraints seem like they couldn't leave much room to move, so I'll ignore the keyboard one for now.

A circle of more than 12 and no more than 22 fifths means either 17, 19 or 22. 22-ET is only 1,3,5,7,9,15-consistent (no 13) and its fifths are too far from meantone fifths to allow even one diatonic scale without the remaining fifths being too wide. I haven't yet entirely eliminated the possibility of a 17 note cycle of fifths with up to 5 additional notes providing the 5 and 15 odentities, since 17-ET is only 1,3,7,9,11,13-consistent (no 5 or 15). But it looks unlikely. Since you need diatonics and therefore meantone sized fifths, a cycle of 19 looks most promising, with an additional 3 notes providing the 11 odentities, since 19-ET is only 1,3,5,7,9,13,15-consistent.

Here is the cycle of 19 fifths I came up with:

694.7 1's region
694.7 3's region
694.7 9's region
694.7
694.7 5's region
694.7 15's region
698.7
698.7
698.7
694.7
694.7 7's region
694.1
694.1
694.1
694.7
694.7 13's region
690.0
690.0
694.7

Then we have another three notes in a chain of fifths
694.7 11's region
694.7
tuned to give just 11's relative to the 7's and 13's.

Everything is close to just except the fifths themselves which are 1/3-comma narrow. It doesn't seem possible to make them more accurate without making those two 690 cent fifths into even narrower wolves.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗gdsecor <gdsecor@yahoo.com>

--- In tuning@y..., David C Keenan <d.keenan@u...> wrote:
> At 00:24 20/06/02 -0000, you wrote:
> >--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> >Here's the challenge: ...
>
> If you want the max error in your three 1:3:5:7:9:11:13:15 chords
to be less than 7 cents, then it sounds impossible. But otherwise ...

I didn't say anything about maximum error, only this:

<< temperament is okay, but not so much as to lose the identities of
ratios of 11 and 13 >>

So it's a matter of how much temperament you think you can get away
with. Unfortunately, this gets a little tricky if you're only doing
it on paper -- you can't always be sure how it's actually going to
sound.

> I know nothing about accordion keyboards, e.g. what appears on the
left and what on the right, but I do understand generalised keyboards
(2D mapping of linear temperaments). However, all the other
constraints seem like they couldn't leave much room to move, so I'll
ignore the keyboard one for now.

Mapping it on the Bosanquet generalized keyboard is the only
requirement. (My accordion designs use the Bosanquet keyboard
geometry, with the only innovation being in the angles and dimensions
of the layout, which doesn't concern us here.) I have more comments
below.

> A circle of more than 12 and no more than 22 fifths means either
17, 19 or 22. 22-ET is only 1,3,5,7,9,15-consistent (no 13) and its
fifths are too far from meantone fifths to allow even one diatonic
scale without the remaining fifths being too wide. I haven't yet
entirely eliminated the possibility of a 17 note cycle of fifths with
up to 5 additional notes providing the 5 and 15 odentities, since 17-
ET is only 1,3,7,9,11,13-consistent (no 5 or 15). But it looks
unlikely. Since you need diatonics and therefore meantone sized
fifths, a cycle of 19 looks most promising, with an additional 3
notes providing the 11 odentities, since 19-ET is only
1,3,5,7,9,13,15-consistent.

Okay, you're on the right track!

> Here is the cycle of 19 fifths I came up with:
>
> 694.7 1's region
> 694.7 3's region
> 694.7 9's region
> 694.7
> 694.7 5's region
> 694.7 15's region
> 698.7
> 698.7
> 698.7
> 694.7
> 694.7 7's region
> 694.1
> 694.1
> 694.1
> 694.7
> 694.7 13's region
> 690.0
> 690.0
> 694.7

Yes, an unequally tempered circle of 19 is what I used. If you had a
chance to try this out in all 19 keys, you would probably make some
adjustments in the sizes of the fifths on the basis of how it sounds,
particularly to take care of the wolves that you mentioned below.
But you've got the general idea. You're halfway there!

Mapping 19 on the Bosanquet keyboard is fairly straightforward. You
would only need to specify how many keys (including duplicates) would
be used per octave: from which flat or double-flat to which sharp or
double-sharp).

> Then we have another three notes in a chain of fifths
> 694.7 11's region
> 694.7
> tuned to give just 11's relative to the 7's and 13's.

Okay! I also provided 3 auxiliary tones for the the 11's, tuned to
give just ratios relative to the 7's. (So you're 75 percent done.)
Now where will you put them on the keyboard? (I chose F, C, and G
as the three tones on which to build my 15-limit otonalities. On
that basis, I would assume that the first fifth in your list of 19
would be from C to G and the last from F to C.)

> Everything is close to just except the fifths themselves which are
1/3-comma narrow. It doesn't seem possible to make them more accurate
without making those two 690 cent fifths into even narrower wolves.

As to how I did it:

Hint 1: I paid as much attention to 9 as I did to 7.

Hint 2: I used only two different sizes of fifths.

Hint 3: I found that you can get away with tempering 13 more than you
would expect.

(And I hope you're having fun with this.)

--George

🔗David C Keenan <d.keenan@uq.net.au>

>--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>So it's a matter of how much temperament you think you can get away
>with. Unfortunately, this gets a little tricky if you're only doing
>it on paper -- you can't always be sure how it's actually going to
>sound.

Indeed.

>Mapping it on the Bosanquet generalized keyboard is the only
>requirement. (My accordion designs use the Bosanquet keyboard
>geometry, with the only innovation being in the angles and dimensions
>of the layout, which doesn't concern us here.) I have more comments
>below.

OK.

>Since you need diatonics and therefore meantone sized
>fifths, a cycle of 19 looks most promising, with an additional 3
>notes providing the 11 odentities, since 19-ET is only
>1,3,5,7,9,13,15-consistent.
>
>Okay, you're on the right track!

Funny you should say that, because when I take into account your hints below, and that it has to map to a linear temperament keyboard I get something that resembles a circular train track with a short siding.

>> Then we have another three notes in a chain of fifths
>> 694.7 11's region
>> 694.7
>> tuned to give just 11's relative to the 7's and 13's.
>
>Okay! I also provided 3 auxiliary tones for the the 11's, tuned to
>give just ratios relative to the 7's. (So you're 75 percent done.)
>Now where will you put them on the keyboard? (I chose F, C, and G
>as the three tones on which to build my 15-limit otonalities. On
>that basis, I would assume that the first fifth in your list of 19
>would be from C to G and the last from F to C.)

Yes. That's how I conceived of it. Now to attach the siding. We use the fact that 11:13 can be approximated by a stack of three perfect fourths (less octaves).

>> Everything is close to just except the fifths themselves which are
>1/3-comma narrow. It doesn't seem possible to make them more accurate
>without making those two 690 cent fifths into even narrower wolves.
>
>As to how I did it:
>
>Hint 1: I paid as much attention to 9 as I did to 7.
>
>Hint 2: I used only two different sizes of fifths.
>
>Hint 3: I found that you can get away with tempering 13 more than you
>would expect.

I think I've got it now. These hints and what you said about the wolves suggests that your 1:7 and 1:9 errors are the same, and you have Just 7:9's, in fact Just 7:9:11's.

So here's the open chain of 21 fifths (22 notes) that gets mapped to a straight line (gently sloping upward to the right) on the keyboard.

Na Fifth Odentity
me (cents) for middle octad
----------------------------
Bb 692.28
F 695.61
C 695.61 1
G 695.61 3
D 695.61 9
A 695.61
E 695.61 5
B 695.61
F# 695.61
C# 695.61
G# 695.61
D# 695.61
A# 695.61 7
E# 692.28
B# 692.28
G[ 692.28 (Fx)
D[ 695.61 (Cx)
A[ 695.61 13 (Gx)
E[ 723.20 to Bb] and 692.28 to Bb (Dx)
B[ 695.61 (Ax)
F] 695.61 11 (Ex)
C] (Bx)

[ and ] stand for half flat and half sharp.

Average fifth size is 696.29, so this can be used in my spreadsheet to generate the keyboard.
http://dkeenan.com/Music/KeyboardMapper.xls

Here's slightly more than one octave of the keyboard, showing cents.

Bx
Ax 34 Ex
1043 Dx 539 Ax
Cx 320 Gx 1043
B# 129 Fx 824
1144 E# 636 B#
D# 452 A# 1144
C# 261 G# 956
B 69 F# 765
1078 E 574 B
D 382 A 1078
C 191 G 887
Bb 0 F 696 C
1012 504 Bb 0
1012

You wrote:
>Mapping 19 on the Bosanquet keyboard is fairly straightforward. You
>would only need to specify how many keys (including duplicates) would
>be used per octave: from which flat or double-flat to which sharp or
>double-sharp).

I don't understand why we _need_ to use any duplicates. Since we can do it with 22 keys per octave (from Bb to Bx) as above. But I suppose you might _want_ some duplicates to make it easier when playing right around the cycle of fifths. One could put a copy of Dx in the Eb position, Gx in the Ab position and Cx in the Db position. It wouldn't make much sense to go back any further since we'd be into the near-wolves.

Here's the keyboard with these duplicates (for a total of 25 keys per octave) and showing the odentities for the middle octad (reduced to within a single octave).

Bx
Ax (11)
Dx Ax
Cx (13)
B# Fx
E# B#
D# (7)
C# G#
B F#
(5) B
(9) A
(1) (3)
Bb F 1
Dx Bb
Cx (Eb) Gx
(Db) (Ab)

Or, regarding duplicates, maybe I don't understand what additional constraints a Bosanquet keyboard places on the general idea of linear-temperament-based keyboards.

>(And I hope you're having fun with this.)

Oh yes! Thanks.

I would never have imagined such a thing was possible.

The errors in many of the rooted intervals are huge. 13 cents in the 1:7, 1:9 and 1:11 and 16 cents in the 1:13. I suppose it is the fact that they are all in the same direction that makes the tuning worthwhile.

Regards
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗gdsecor <gdsecor@yahoo.com>

--- In tuning@y..., David C Keenan <d.keenan@u...> wrote:
> >--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> >As to how I did it:
> >
> >Hint 1: I paid as much attention to 9 as I did to 7.
> >
> >Hint 2: I used only two different sizes of fifths.
> >
> >Hint 3: I found that you can get away with tempering 13 more than
you
> >would expect.
>
> I think I've got it now. These hints and what you said about the
wolves suggests that your 1:7 and 1:9 errors are the same, and you
have Just 7:9's, in fact Just 7:9:11's.

That's it! Not only does this improve the fifths and ninths in many
of the best keys, but you will also find that both the major and
minor thirds beat very slowly, something that you don't have in 19-
ET, 31-ET, or most of the variants of the meantone temperament. It
also gives rather good subminor (6:7:9) and supermajor (14:18:21)
triads, much better than in 19-ET; this is achieved with a relatively
small alteration in the size of the 19-ET fifth.

I wanted the circle of 19 fifths to be a well-temperament. To keep
the rest of the fifths from getting too large, I put all of my fifths
of that size in a single series -- just enough for the keys of F, C,
and G. So my ~695.61-cent fifths (twelve in number) are from F to E-
sharp, and the remaining seven fifths start on F-flat (same pitch as
E-sharp, but renamed) and end at F. These are ~693.23 cents, or
~8.72 cents narrow.

> So here's the open chain of 21 fifths (22 notes) that gets mapped
to a straight line (gently sloping upward to the right) on the
keyboard.
>
> Na Fifth Odentity
> me (cents) for middle octad
> ----------------------------
> Bb 692.28
> F 695.61
> C 695.61 1
> G 695.61 3
> D 695.61 9
> A 695.61
> E 695.61 5
> B 695.61
> F# 695.61
> C# 695.61
> G# 695.61
> D# 695.61
> A# 695.61 7
> E# 692.28
> B# 692.28
> G[ 692.28 (Fx)
> D[ 695.61 (Cx)
> A[ 695.61 13 (Gx)
> E[ 723.20 to Bb] and 692.28 to Bb (Dx)
> B[ 695.61 (Ax)
> F] 695.61 11 (Ex)
> C] (Bx)
>
> [ and ] stand for half flat and half sharp.
>
> Average fifth size is 696.29, so this can be used in my spreadsheet
to generate the keyboard.
> http://dkeenan.com/Music/KeyboardMapper.xls
>
> Here's slightly more than one octave of the keyboard, showing cents.
>
> Bx
> Ax 34 Ex
> 1043 Dx 539 Ax
> Cx 320 Gx 1043
> B# 129 Fx 824
> 1144 E#
636 B#
> D# 452
A# 1144
> C# 261 G# 956
> B 69 F# 765
> 1078 E 574 B
> D 382 A
1078
> C 191 G 887
> Bb 0 F
696 C
> 1012 504
Bb 0
> 1012

Your mapping (in the sharp direction) is exactly what I did. For
clarity, I also renamed the three 11 factors:
Ax as B-semiflat
Ex as F-semisharp
Bx as C-semisharp
We mapped these three tones in the same places that they would be in
31-ET, and the three tones that you renamed D[, A[, and E[ also occur
in the same places as for 31-ET. So you can play the 15-limit
otonalities on F, C, and G in this 19+3 temperament on the Bosanquet
keyboard as if you were in 31-ET -- exactly the same fingering
patterns in both systems!

> You wrote:
> >Mapping 19 on the Bosanquet keyboard is fairly straightforward.
You
> >would only need to specify how many keys (including duplicates)
would
> >be used per octave: from which flat or double-flat to which sharp
or
> >double-sharp).
>
> I don't understand why we _need_ to use any duplicates. Since we
can do it with 22 keys per octave (from Bb to Bx) as above. But I
suppose you might _want_ some duplicates to make it easier when
playing right around the cycle of fifths. One could put a copy of Dx
in the Eb position, Gx in the Ab position and Cx in the Db position.
It wouldn't make much sense to go back any further since we'd be into
the near-wolves.
>
> Here's the keyboard with these duplicates (for a total of 25 keys
per octave) and showing the odentities for the middle octad (reduced
to within a single octave).
>
> Bx
> Ax (11)
> Dx Ax
> Cx (13)
> B# Fx
>
E# B#
> D# (7)
> C# G#
> B F#
> (5) B
> (9) A
> (1) (3)
> Bb
F 1
> Dx Bb
> Cx (Eb) Gx
> (Db) (Ab)
>
> Or, regarding duplicates, maybe I don't understand what additional
constraints a Bosanquet keyboard places on the general idea of linear-
temperament-based keyboards.

The duplicates are necessary because we are mapping a *circle* of 19
onto a *linear* type of keyboard. One of the requirements of my
challenge was to have the same fingering patterns in every key.
Since you would want to be able to play a major scale and major triad
on E-flat, for example, the same way you would on C, you would need
at least 6 duplicate keys per octave to do this in every key.

An additional problem is posed by the thumb. My experience with the
generalized keyboard Scalatron is that it quickly gets tiring to have
to reach for the keys that are farthest from the edge of the keyboard
(i.e., the edge closest to the player). Providing duplicate keys in
the flat direction remedies this problem. My solution takes the
circle of 19 all the way to Abb, and the B[, F], and C] keys are also
duplicated in the Cbb, Gbb, and Dbb positions (which, again, is
exactly where they occur in 31-ET.) This is a grand total of 34 keys
per octave.

> > (And I hope you're having fun with this.)
>
> Oh yes! Thanks.
>
> I would never have imagined such a thing was possible.
>
> The errors in many of the rooted intervals are huge. 13 cents in
the 1:7, 1:9 and 1:11 and 16 cents in the 1:13. I suppose it is the
fact that they are all in the same direction that makes the tuning
worthwhile.

My E[ (as 13/8 of G) is about 18 cents too low, but it still sounds
convincing in a 7:9:11:13 chord in that key, inasmuch as the 13 is
false by only about 5.5 cents with respect to 7, 9, and 11. The
8:10:13 chord is not as successful, but it still works. The really
remarkable thing is that G to E[ must sound like 8:13, but in the key
of Eb this same interval must sound like 5:8 -- and heard in the
appropriate tonal context, it does! Before I discovered this, I,
too, would never have imagined such a thing was possible.

I'll be posting the specifics of the 19+3 temperament in a separate
message, along with a summary of its features and diagrams of the
left and right-hand accordion keyboards.

--George

🔗gdsecor <gdsecor@yahoo.com>

*BURIED TREASURE*
"Microtonal Accordion with 19+3 Temperament"
From: George Secor
June 28, 2002

Here's the microtonal accordion design I promised, complete with a
suitable multi-purpose tuning. (For prior discussion, including all
of the details of the microtonal accordion challenge, please see
messages #37869, 37887, 37930, 37954, 38010, 38076, 38109, 38132, and
38168.)

*The Challenge*

The tuning was required to be no more than more than about 22 tones
per octave, to permit playing most conventional diatonic music (i.e.,
without a comma problem), to have a circle of more than 12 fifths
around which one could freely modulate, and to have 15-limit
otonalities in at least 3 different keys (temperament is okay, but
not so much as to lose the identities of ratios of 11 and 13).

This was to be mapped onto a Bosanquet generalized keyboard so that
*all* of the above could be played with uniform fingering patterns in
*all* available keys.

*The Solution*

The tuning is a 19-tone well temperament (19-WT) with 3 auxiliary
tones to supply the 11 factors for the three 15-limit otonalities,
which are built on F, C, and G. The circle of 19 fifths has twelve
fifths from F to E-sharp of ~695.61 cents, or ~6.34 cents narrow
(such that 8 consective fifths give an exact 9:7). The remaining
seven fifths are of equal size and start on F-flat (same pitch as E-
sharp, but renamed) and end at F; these are ~693.23 cents, or ~8.72
cents narrow. The three auxiliary tones (B-semiflat, F-semisharp,
and C-semisharp) are tuned as just 11:9's above G, D, and A,
respectively.

This tuning has three purposes:

1) As a (~5/17 comma) variant of the meantone temperament, it has
slow-beating thirds, both major (~3.86 cents narrow) and minor (~2.48
cents narrow). The flat keys are not among the best, but old music
originally intended for the meantone temperament that is written in
these keys could be played a minor second lower, which would be
closer to the historical pitch.

2) It has a closed circle of 19 fifths, so it is an alternative to 19-
ET. (The pitches in 19-WT do not depart more than about +-7 cents
from 19-ET if A is tuned to the same frequency in both.)

3) By adding the three auxiliary tones, there are 15-limit
otonalities on F, C, and G, with the overall intonation being better
than with 31-ET (with 7:9:11 being exact). In just intonation, this
would require 18 tones, but in the 19+3 temperament 80:81 vanishes,
so 17 tones (out of the total of 22) are used to fulfill this
requirement.

The most amazing thing about this temperament is that F to D-flat, C
to A-flat, and G to E-flat must each function as 8:13 (neutral sixth)
in the 15-limit otonalities and as 5:8 (minor sixth) in the circle of
19 fifths -- and heard in the appropriate tonal context, it works!
Before completing the design of this temperament, I would never have
imagined such a thing was possible. (I devised it in the winter of
1978, and it was published in the first issue of _Interval_, which
appeared in the spring of that year.)

*The Treble Keyboard*

The treble keyboard proposed for the 19+3 temperament is a slight
modification of the design that was used on the generalized keyboard
Scalatron. It differs only in the size and shape of the keys. The
distance between the centers of adjacent Scalatron keys (~7/8 inch)
was made the same as the white keys of the piano; I subsequently
found that it would not have been necessary to make them that large.
Using round keys about 18 mm in diameter and 20 mm center-to-center
(as on the European chromatic button accordion) would have been
suitable, and with the space limitations posed by the accordion, that
is exactly what I have done for this microtonal instrument. A two
octave layout may be seen here:

/tuning-
math/files/secor/kbds/KbAc19p3.gif

An actual instrument would have about 3-2/3 octaves of keys for the
right hand. The keyboard is pictured as the right hand would "see"
it; when the instrument is played, it would be in a nearly vertical
plane, with the keys pictured at the left (i.e., the low notes) at
the top. The seven naturals are white, five sharps are black, and
five flats are red; the keys for the remaining two tones in the
remote part of the circle of 19 are a grayish-pink (or mauve).

There are a total of 28 keys per octave used for 19-WT tones
(including nine duplicates), which permit uniform fingering patterns
in all 19 keys. The three auxiliary tones are provided keys at both
the upper and lower edges of the keyboard (colored cyan), in the same
positions in which they would occur in 31-ET, making a grand total of
34 keys per octave. Since the fingering patterns are the same in all
keys, this keyboard would be much easier to learn than a piano
keyboard, since all scales, chords, etc. would need to be learned in
only a single key.

These duplicate keys also serve another function. My experience with
the generalized keyboard Scalatron has shown that it quickly gets
tiring to have to reach for the keys that are farthest from the edge
of the keyboard (i.e., the edge closest to the player) with the thumb
or little finger. Providing duplicate keys in the flat direction,
both for the sharps and for the three auxiliary tones remedies this
problem.

*The Bass System*

Implementation of the generalized Bosanquet geometry for the left
hand button-board requires some modification in light of the fact
that the left thumb cannot be used, which has at least two
consequences:

1) The cross-over-the-thumb scale technique used for the right hand
is not available for the left, so unless some modification were made,
unnatural twisting would occur in the fingering technique. (This
problem also occurs with the bassetti free bass system, but is
elegantly solved with the Moschino system.)

2) The physical distance the fingers can reach is less, so without
any modification the interval reach would be restricted.

The layout of buttons in my adaptation of the Bosanquet keyboard for
the left hand is shown here:

/tuning-
math/files/secor/kbds/AcLH19p3.gif

This shows an actual range of 3-2/3 octaves (with E1 as the lowest
note), which would be both useful and practical. As with the right-
hand keyboard, there are 34 buttons per octave, with nine duplicate
buttons per octave in the circle of 19 and the three auxiliary tones
placed at both edges. The bassboard is pictured as the left hand
would "see" it; when the instrument is played, it would be in a
nearly vertical plane, with the buttons pictured at the right (i.e.,
the high notes) at the top.

Twisting in scale passages is eliminated by the fact that if the
index finger or little finger is positioned on a given button, the
remaining fingers may easily access both higher and lower notes due
to the fact that the pitch changes in one direction along the nearly
lateral rows (tones separated by 3deg19) and in the opposite
direction along the nearly perpendicular rows (tones separated by
2deg19), a feature that was introduced in 1960 on the Moschino free
bass system.

The same modification that eliminates the above-mentioned twisting
also causes buttons an octave apart to be positioned closer to one
another, putting the interval of a tenth or twelfth within easy
reach. The octave distance is, in fact, almost exactly the same as
on the 12-ET bassetti free bass system; but since most
implementations of the bassetti system lack duplicate buttons and
require some finger-twisting to play scales, the 19+3 bass system
would definitely be easier to play.

The above diagram does not show the dimensions of the bass buttons,
inasmuch as they would not be large enough to label. The actual
dimensions of the layout are shown here:

/tuning-
math/files/secor/kbds/BassComp.gif

This diagram compares the 19+3 system with the Moschino system (the
latter ascending by major thirds in the eight lateral rows and
descending chromatically moving along a diagonal from the upper left
to the lower right); both are shown with a range of 2-2/3 octaves.
(Conventional accordion bass and chord buttons have the same angles
and spacing as shown for the Moschino system; the Moschino system
differs from the conventional accordion in that it has 8 rows of
buttons, half of these being duplicates.) Observe that the 19+3
generalized bass setup does not require much more area than a 12-ET
Moschino system with the same range (and most Moschino accordions
were built with a larger range: over 5 octaves for the left hand).

All of the bass buttons are black; color-coding is of no value since
the player cannot see the buttons. Instead some of the buttons would
be touch-coded, as indicated in the diagram.

*In Summary*

The accordion would be an excellent candidate for a microtonal
keyboard instrument. It's very portable, stable in pitch, produces
its sound acoustically (rather than electronically), and has a timbre
rich in harmonics that would easily define intervals above the 5
limit. It also has a naturally "straight" sound, lacking the sort of
devices intended to enhance or liven up electronic sounds (vibrato,
phase shifting, etc.) that tend to obscure the differences between 12-
ET and alternate tunings.

The 19+3 temperament which I am proposing for use on a microtonal
accordion is a multi-purpose tuning with a very reasonable number of
tones in the octave. In its best keys it comes close to
approximating the meantone temperament, while offering free
modulation within a circle of 19 fifths. It also offers 15-limit
otonalities with an accuracy that cannot be matched by any equal
division of less than 41 tones.

The proposed left and right-hand keyboards are generalized, with
uniform fingering patterns in all keys. Having had the experience of
learning two new keyboards (the Moschino accordion free bass system
in 1962 and the Bosanquet generalized keyboard Scalatron in 1975), I
have no doubt that an accordion with this design would be easy to
learn and a delight to play.

*In Closing*

I extend my congratulations to Dave Keenan, for successfully
participating in the microtonal accordion challenge. He figured out
just about everything that could have done on paper without actually
having a generalized keyboard instrument on which to try it out.
(Besides, he had only a few days, whereas it took me over two years
working with a Scalatron to come up with the 19+3 temperament.)

And thanks to Mats Öljare for expressing the desire that someday a
microtonal accordion might become a reality. I hope that my design
might help to make that happen.

--George

🔗manuel.op.de.coul@eon-benelux.com

🔗gdsecor <gdsecor@yahoo.com>

--- In tuning@y..., manuel.op.de.coul@e... wrote:
> My compliments too! Here's a scale file for it:
>
> ! secor-acc.scl
> !
> George Secor's Microtonal Accordion 19+3 temperament. TL 28-6-2002.

I didn't intend this to be a temperament especially for the
accordion -- it is intended to be used on any instrument of fixed
pitch as a more versatile alternative to 19-ET. (And I prefer using
a well-temperament to equal temperament in both 12 and 17.) Perhaps
a better filename would be "seco19p3.scl" (if you are restricting the
names to 8+3 characters).

> Aux=1,10,19
> 22
> !

I calculated some slightly different values for a few of these:

34.25140
69.30142
131.45629 instead of 131.44937
191.22898
260.53039
317.92090 instead of 317.90937
382.45795
451.75937
504.38551
538.63692
573.68693
638.22398 instead of 638.21937
695.61449
764.91590
824.68859 instead of 824.67937
886.84346
956.14488
1011.15321 instead of 1011.13937
1043.02243
1078.07244
1144.99168 instead of 1144.98937

Are your fifths on the far side of the circle ~691.23231 cents?

> I'm also fond of the sound of the accordion in jazz, and a fan
> of Dino Saluzzi (actually he plays bandoneon) and Richard
> Galliano. A rising star is Kimmo Pohjonen.
> Saluzzi's albums _Andina_ and _Mojotoro_ are his most lyrical.

I did quite a bit of commercial jobbing with my accordion in the
1960's and 1970's. Having the free bass in the left hand, I used the
instrument differently than most accordionists: since I tend to think
orchestrally, I would frequently use it as an acoustic synthesizer to
imitate a string or brass section when backing up a solo instrument.

For fun I played keyboard pieces from the classic and Baroque periods
(Bach's Italian Concerto is one of my favorites), which is an
altogether different (and truly "classical") sound from what most
people associate with the accordion. (But I don't have much time to
do that anymore.)

And thank you for making the file. I will be interested to see what
others think of this temperament.

--George

🔗manuel.op.de.coul@eon-benelux.com

🔗gdsecor <gdsecor@yahoo.com>

*BURIED TREASURE*
"Percussion Instruments in the 19+3 Temperament"
From: George Secor
July 1, 2002

After I said the following a few days ago regarding my 19+3
temperament (in message #38292), I realized that I set myself up for
a potentially embarrassing situation:

<< I didn't intend this to be a temperament especially for the
accordion -- it is intended to be used on any instrument of fixed
pitch as a more versatile alternative to 19-ET. >>

Other than keyboard instruments, to what instruments of fixed pitch
could I most likely have been referring? Why, percussion
instruments, of course, which brings up the question: How would the
19+3 pitches be arranged on a xylophone or metallophone?

While the slanting rows of the generalized keyboard arrangement could
be adapted to percussion instruments (in a series from Bb to Dx,
followed by the 3 auxiliary tones), I would prefer to keep the bars
in straight lateral rows to simplify the construction. This would
also allow the use of a single damper bar running the length of the
instrument for metallophones with a sustained tone (such as a
vibraphone or tubulongs). Since the 19+3 temperament (a 22-tone set)
is not a constant structure, there is no obvious way to do this, and
I didn't have a clue!

There are a couple of ways to do 19-ET in tubulongs. One is to use a
7+5+7 arrangement:

Cb Db Eb Fb Gb Ab Bb Cb
C# D# F# G# A#
C D E F G A B C

(If viewing this online, use the "expand messages" option to achieve
the proper spacing.)

Another (less compact) way would use a 7+7+5 arrangement:

C# D# F# G# A#
C D E F G A B C
Db Eb Fb Gb Ab Bb Cb

So where do you put the three auxiliary tones of the 19+3
temperament? I had struggled with that problem off and on for 24
years but didn't think about that when I made the statement that I
quoted above.

I'm happy to report that when I looked at the problem again this past
Saturday morning, it suddenly hit me to try something different, and
within twenty minutes I found a solution. It requires no more space
than the 7+5+7 arrangement for 19-ET:

C+ Db D# E# F+ Gb G# A# Bd B# C+
Eb Ab
C C# D E F F# G A Bb B C

Note: + (I really wanted a not-equals sign) indicates a semisharp,
and d indicates a semiflat.

For instruments not requiring both resonators and a damper bar, the
tones may be arranged in 3 rows of 10+2+10, as shown. Otherwise, E-
flat and A-flat may be moved into the nearest row for a vibraphone
(or it may be preferable to do *all* of the pitched percussion that
way -- does Jon Szanto have an opinion?). The tones of the nearest
row together with the middle row are those in a series of fifths from
A-flat to C-sharp, so a glissando played on these tones would
simulate a 12-tone chromatic scale.

If the five flats Db, Eb, Gb, Ab, and Bb are colored darker than the
rest, they will produce a two-and-three visual pattern that would be
helpful in finding one's way around.

Observe that all tones in the farthest row that are in the circle of
19 are 1 degree higher in pitch than those in the nearest row. Also,
all three auxiliary tones are in the farthest row and are placed
opposite tones in the nearest row that are 1/2-degree lower. If the
two tones in the middle row are considered to occupy both of the
other two rows simultaneously, then the sets of tones in the nearest
and farthest rows taken separately are constant structures. Thus
there is logic and consistency underlying the entire arrangement.

So now it's possible to build a set of tubulongs in the 19+3
temperament. Any takers?

--George