Fw: Roland SCB-55

>Someone recently gave me a Roland SCB-55 duaghter board, and I managed to
>get it working on my Soundblaster AWE-32 with very little hassle. I'm
>enjoying the sounds even though they are, of course, 12TET.
>
>The manual mentions a set of 12 messages for "Scale Tuning", one for each
of
>the 12 notes. The parameter is -64 to +63 cents.
>
>Can anyone here direct me to some software to control those parameters?
>
> -b0b- (Bob Lee) quasar@b0b.com http://www.b0b.com
>
>

Dave Keenan described his 12-note scale with 8 7-limit tetrads, etc.

>As offsets from 12-tET it is
>C +12.19
>C# +17.41
>D -17.41
>D# -12.19
>E -6.98
>F +6.98
>F# +12.19
>G +17.41
>G# -17.41
>A -12.19
>A# -6.98
>B +6.98

This tuning certainly seems to be a genuinely new and interesting
development. Whoever's keeping track of those out there, pay heed!

I'm assuming that no one else has paid any attention to my contest, but
in case anyone gave it a try, at this point I'm willing to allow for
some rule-bending. For example, a couple of 14 out of 26-tET scales look
good if you drop the requirement that three complete chords suffice to
cover the scale. If also you drop the requirement that otonal and utonal
chords be constructed from the same pattern of scale steps, a whole
array of many-noted scale open up. A couple of very well-tuned ones have
occured to us: 22 out of 41-tET to me, and 19 out of 31-tET to Dave. Let
me describe these scales a bit . . .

A maximally even 22 out of 41-tET has a very interesting melodic
structure. First, note that any octave span (C-c, D-d, etc.) of the
normal diatonic scale (maximally even 7 out of 12-tET, 19-tET, or
26-tET) has two tetrachords, each spanning a 4:3 frequency ratio and
consisting of several equally-sizes large steps plus one small step, and
that an additional large step (either at the bottom, at the top, or
between the two tetrachords) completes the octave. Well, any 2:1 span of
the 22-out-of-41 scale has three "octachords", each spanning a 5:4
frequency ratio and consisting of several equally-sized large steps
(2/41 oct) plus one small step (1/41 oct), and an additional large step
(either at the bottom, at the top, between the bottom two octachords, or
between the top two octachords). This is a remarkable similarity and may
have something to with the fact that 22 out of 41 is an MOS scale
generated by 41's 5:4 approximation, 13/41 oct.
Now the harmonic properties of this scale are at least as remarkable.
First, note that the normal diatonic scale in any meantone tuning
(including 12-tET, 19-tET, 26-tET, 31-tET, and many non-closed
temperaments) has close approximations to three Otonalities complete
through the 5-limit (major triads) and three Utonalities complete
through the 5-limit (minor triads). These have a maximum error of 15.6
cents in 12-tET and a maximum error of 5.4 cents in quarter-comma
meantone. Well, the 22-out-of-41 scale has ten Otonalities complete
through the 9-limit and ten Utonalities complete through the 9-limit
(two of each can be completed through the 11-limit). The maximum error
is 6.8 cents, which is the error of the 9:5. The largest error within
the 7-limit is that of the 5:3, namely, 6.3 cents.

A maximally even 19 out of 31-tET can be thought of as being constructed
from identical tetrachord-like blocks. But there are several instances
of both large (2/31 oct.) and small (1/31 oct.) step sizes in each
block. Therefore it may be difficult to know where one is in such a
scale just from the melodic succession. Richmond Browne has pointed out
the importance of rare intervals for position finding. The closest thing
to such a signpost is the three-step sequence small-large-small, which
only happens twice in the scale. But the scale is generated by the 4:3
approximation, something it has in common with the traditional diatonic
scale.
Harmonically, the scale is wonderful, having nine Otonalities complete
through the 9-limit and nine Utonalities complete through the 9-limit,
all with a maximum error of 11.1 cents (in the 9:5) and a maximum
7-limit error of 6 cents (in the 5:3).

A while back Andrzej Gawel described a maximally even 19 out of 36-tET
scale which was summarily ignored by the List. This scale is a 19-note
MOS generated by the 19/36 oct interval, much like the standard diatonic
scale is a 7-note MOS generated by the 7/12 oct interval, and including
the latter scale within it. Thus the tetrachord-like structure spans the
approximate 7:5, and only one small interval occurs within each of the
tetrachord-like blocks, which bodes well for position finding. The scale
has seven Otonalities complete through the 7-limit and seven Utonalities
complete through the 7-limit, with a maximum error of 15.8 cents (in the
7:5). This almost half the size of the smallest step. Attemting to
construct 9-limit chords forces one to have an error of more than half
the smallest step, since 36-tET is not consistent through the 9-limit.
Can anyone think of a way of improving Gawel's proposal?

Here's a version of my tuning that is easier to remember since it has whole cent offsets from 12-tET and reduces the two semi-wolves (12 cent error) to (barely) acceptable fifths with a 10 cent error. The price paid for this improvement is only an increase from 12 to 13.2 cents in the maximum 9-limit error (excluding the 7:5), so we see it is a tolerant tuning. (The 7:5 error is fixed at 17.5 cents for all versions since it is approximated by the half octave.)

This version has 8 fifths of 706 cents, 2 of 712, and 2 wolves of 664 cents.

Note Offset Step
from 12-tET size
(cents) (cents)
---------------------------
C + 6 106
C# +12 106
D +18 64
D# -18 106
E -12 106
F - 6 112
F# + 6 106
G +12 106
G# +18 64
A -18 106
A# -12 106
B - 6 112

This can be seen to be 12 from 200-tET, however it does not use the interesting approximations given by Robin Perry in this list on 17-Nov-1998.

Errata: The offsets in my earlier posting were as they would have been for a scale starting on G, not C as shown. My apologies.

For this version of the tuning the errors in the intervals of the 12 pentads and tetrads are:

2:3 4:5 5:6 4:7 5:7 6:7 4:9 5:9 6:9 7:9
4.0 -4.3 8.4 13.2 17.5 9.1 8.1 12.4 4.0 -5.1

Errors in the intervals of the other 4 triads are:

2:3 4:5 5:6
10.0 1.7 8.4

As a lattice of 7-limit tetrahedra it is
D#--------A#--------E#
/ \ / \ / \
/ \ / \ / \
/ A--\---/--E--\---/--B \
/ x/ \ / \ / \
A---------E---------B--/------F#--------C#--------G#
/ \ / \ / \/ x /
/ \ F#-/---\--C#-/--/\--G# /
/ Eb-\/--/--Bb-\---/--F \ /
/ x /\ / \ / \ /
Eb--------Bb--------F------/--C---------G---------D
\ / \ / \ /x /
\ C--/---\--G--/---\--D /
\ / \ / \ /
\ / \ / \ /
Gb Db Ab

Of course it wraps around due to D# = Eb, F# = Gb etc. The 'x's indicate that the 7:4 intervals in these positions don't work. Since lines representing 7:n intervals on these diagrams are not shown, but are understood, it was necessary to explicitly indicate where they don't exist.

I'm still waiting for someone to tell me they've seen it before.

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