Yahoo Tuning Groups Ultimate Backup makemicromusic microabc -- sagittal Peppermint mapping

Dear Hudson and all,

Thank you for the suggestions I read in MMM Digest 1639. Here I'm
going to focus on some ideas for a sagittal notation file for
Peppermint 24, which I guess could be named -ipepper24.txt or the
like.

[Please note that in the next couple of days I hope that I might be
closer to understanding the implementation problem, since your helpful
posts and e-mails have filled me in on some helpful procedures and the
new release may provide opportunities for experimentation. Here I'm
mainly explaining the musical context of Peppermint and some of the
equivalent sagittal spellings that I describe. I give a basic
keyboard diagram with a few basic symbols, and then some examples
where equivalents might be helpful. I'm trying to catch on the
material you've so helpfully posted and e-mailed in the last couple of
days.]

Here's a table of symbols with some information about Peppermint
interval sizes with reference to C.0 or midi note 60. It's not code,
but a kind of "pseudocode" that might help in approaching some actual
code for abc utilities including the microabc release and related
programs.

One of my problems is that at this point I'm not sure which notation
or abc-related programs would be best to use -- for example, whether
cat is still required (one thing I saw suggested otherwise now) or
abcpp. I've included both ASCII sagittal notation and decimal codes in
the following table of symbols for Peppermint 24, under the
understanding that either style of coding could be helpful depending
on the chosen approach.

Note that Peppermint mainly approximates integer ratios based on odd
factors of 2-3-7-9-11-13, but here and there has approximations of
other factors such as 5 and 17.

index ASCII Sag name(s) Decimal codes cents JI approximations
0 C . ^/144C . 0.000 1/1
1 C/|\ D!!!) ^/168C _/80D 58.680 28/27 33/32 1053/1024
2 C/||\ . ^/196C . 128.669 14/13
3 C/|||\ D!) ^/220C _/132D 187.349 10/9 392/351 19/17
4 D . ^/144D . 208.191 9/8 44/39 273/242
5 D/|\ E!!!) ^/168D _/80E 266.871 7/6 (just)
6 E\||/ . _/92E . 287.713 13/11 33/28
7 E(!) D)||| _/116E ^/197D 346.393 11/9 39/32
8 E . ^/144E . 416.382 14/11 33/26
9 E/|\ F!) ^/168E _/132F 475.062 21/16
10 F . ^/144F . 495.904 4/3 117/88 121/91
11 F/|\ G!!!) ^/168F _/80G 554.584 11/8
12 F/||\ . ^/196F . 624.574 56/39
13 F/|||\ G!) ^/220F _/132G 683.253 49/33
14 G . ^/144G . 704.096 3/2 176/117 182/121
15 G/|\ A!!!) ^/168G _/80A 762.775 14/9
16 G/||\ . ^/196G . 832.765 34/21 21/13
17 G/|||\ A!) ^/220G _/132A 891.445 5/3 847/507
18 A . ^/144A . 912.287 22/13 56/33
19 A/|\ B!!!) ^/168A _/80B 970.967 7/4
20 B\||/ . _/92B . 991.809 16/9 39/22 484/273
21 B(!) A)||| _/116B ^/197A 1050.488 11/6
22 B . ^/144B . 1120.478 21/11
23 B/|\ C!) ^/168B _/132C 1179.158 63/32 77/39 196/99

Here I have tried to be generous with equivalent spellings. As I'll
explain below, these can be helpful in certain directed progressions
in signalling the anticipated direction of the resolution.

First the sagittal basics for Peppermint 24, which George Secor, David
Keenan, and I developed together around the end of 2002 on tuning-math.
As they pointed out, it is often desirable to map a JI or nonlinear
tuning using an ET (or as we now often say EDO) sagittal set. We
decided on 121-EDO as a basic model, with a bit of modification.

For Peppermint 24, the relevant signs (which might occur in either
direction depending on the chosen midi note as "1/1") are as follows:

....................................................................
Symbol 121-ET steps Pep24 cents Sagittal name ratio / cents
--------------------------------------------------------------------
!) 2 20.842 7-comma 64:63 27.264
---------------------------------------------------------------------
/|\ 6 58.680 11-diesis 33:32 53.273
(13-diesis) 1053:1024 48.348
---------------------------------------------------------------------
(!) 7 69.990 11'-diesis 729:704 60.412
(13'-diesis) 27:26 65.337
---------------------------------------------------------------------
/||\ 13 128.669 apotome 2187:2048 113.685
---------------------------------------------------------------------
)||| 14 138.202 ? [~13:12] 13:12 138.573
---------------------------------------------------------------------
!!!) 15 149.512 apotome+7-comma 243:224 140.949
---------------------------------------------------------------------
/|||\ 19 187.349 apotome+11-diesis 72171:65536 166.958
---------------------------------------------------------------------

In Peppermint, a whole tone of around 208 cents is roughly 21 steps of
121-ET. Such a tone like C-D is often divided into a regular chromatic
semitone or apotome (C-C# or C-C/||\) of 128 cents or ~13 steps plus a
regular diatonic semitone of 80 cents (C#-D or C/||\-D) or ~8 steps.

The following keyboard diagram shows how the symbols for 6, 7, and 13
steps suffice to define the tuning; each keyboard has a regular chain
of 12 notes in fifths (Eb-G#), with 59 cents or about 6 steps between
corresponding notes on the two keyboards (e.g. C-C/|\):

19 35 69 90 106
C/|||\ E(!) F/|||\ G/|||\ B(!)
C/|\ D/|\ E/|\ F/|\ G/|\ A/|\ B/|\ C/|\
6 27 48 56 77 98 119 127
--------------------------------------------------------------------
13 29 63 84 100
C/||\ E\!!/ F/||\ G/||\ B\!!/
C D E F G A B C
0 21 42 50 71 92 113 121

These symbols should in many circumstances also suffice for abc
microtonal notation, as for example in this variation on the tuning
of Zalzal with its many neutral intervals of a kind characteristic of
medieval and more recent Near Eastern musics alike:

G A B(!) C D E(!) F G
0 208 346 496 702 842 992 1200

Here the spellings G-B(!) and C-E(!), for example, nicely suggest the
idea of a neutral third, here around 11:9, while G-E(!) is around 13:8.

However, there are situations where a step like B(!), or about 7 steps
below C, might be respelled as A)!!!, or about 14 steps above A, a
variation on the usual A/||\ or A#. The respelling has appeal when we
have a 13th-14th century European variety of progression where this
step acts in effect as a substitute for a regular A/||\ (not present
in Peppermint 24), resolving to B by a compact semitone of 7 steps, or
one step smaller than usual diatonic semitone of 8 steps.

E(!) E D)||| E
B(!) B A)||| B
F/||\ E or F/||\ E

The first notation might be more "keyboardistic," since it tells us
that we are for the first sonority to play a sharp on the lower manual
and two flats on the upper manual (the latter at 7 steps below E and B
on the lower keyboard). The second spelling, however, with its
accentuated sharps (14 steps above D and A), suggests a "striving"
toward the resolution of lower 426-cent major third to fifth and outer
922-cent major sixth to octave, the sagittal signs psychologically
reinforcing the expected upward motion of the higher two voices by
70-cent semitones or thirdtones A)|||-B and D)|||-E. This spelling
also suggests that B(!) and E(!) are in this context serving as
substitutes for the regular steps A/||\ and D/||\, which are not
present in Peppermint 24.

In this example the complementary pair of signs are (!) at -7 steps
and )||| at +14 steps, with absolute values adding up to 21 steps or a
usual whole tone. The !) and /|||\ pairs at -2 and +19 steps form
another such pair.

For example, let us consider these two spellings of a common
resolution in Peppermint with a septimal flavor:

E F/||\ E F/||\
B C/||\ B C/||\
F/|||\ F/||\ or G!) F/||\

The first spelling expresses a familiar keyboard pattern: septimal
progressions often combine voices moving by regular whole tones on a
single keyboard (B-C/||\ and E-F/||\) with others moving by the
59-cent interval between the two keyboards (F/|||\-F/||\). The opening
sonority of 0-437-933 cents, close to 7:9:12, thus resolves to a
near-2:3:4.

The second spelling emphasizes that the step F/|||\ -- or indeed G!)
-- is acting as a form of G lowered by 21 cents, an adjustment serving
as a septimal comma so that G!)-B is a near-9:7 and G!)-E a pure 12:7.
This notation more generally tells us both that G!) is available as a
substitute for a regular G\!!/ (not present in Peppermint 24), and
that its use may involve a septimal comma shift.

A similar situation arises with this common resolution:

G G/|\ G A!!!)
D/|\ C/|\ E!!!) D!!!)
C C/|\ or C D!!!)

The first, more conventional spelling reflects the usual keyboard
pattern that a major second plus the distance between the two
keyboards (about 21 + 6 or 27 steps of 121-ET) yields a pure 7:6 minor
third, here in a near-6:7:9 sonority. The outer voices ascend by
very efficient 6-step intervals, C-C/|\ and G-G/|\, while the middle
voice descends by a regular tone, D/|\-C/|\.

The second spelling may emphasize that C/|\ is acting in effect as a
lowered (and thus accentuated) D\!!/ a septimal comma closer to C than
the regular step (again not present in Peppermint 24) would be, and
likewise G/|\ as a septimal A\!!/.

Alternative spellings using septimal comma symbols might sometimes be
especially attractive when two versions of a progression are
available, one with regular steps available on a single manual, and
the other with septimally-flavored steps and intervals.

regular septimal -- two spellings

B C B B/|\ B C!)
G F F/|||\ E/|\ G!) F!)
E F E E/|\ or E F!)

The regular version has a minor third E-G at 288 cents, near 13:11,
and a major third G-B at 416 cents, near 14:11, with usual diatonic
semitones E-F and B-C at 80 cents, near 22:21 (about 8 steps in
121-EDO). The altered version has smaller 6-step semitones or
thirdtones near 28:27, and thirds at 7:6 and a near-9:7, as in the
previous example. While the first spelling shows the characteristic
keyboard motions E-E/|\ and B-B/|\ for the "accentuated" 6-step
semitones, the second more expressly treats these as septimal
variations on the usual E-F and B-C motions of 8 steps, made smaller
by the 2-step comma !), thus E-F!) and B-C!). This version implies
that the notes F!) and F, or C!) and C, might represent two "versions"
of the same basic note. Both versions might be used in a piece,
possibly involving some interesting comma shifts.

My purpose here is to explain some musical reasons for preferring a
symbol set with certain sagittal equivalents for the same pitches, and
also to give some idea of the "neo-medieval" qualities of the tuning
as applied to European or Near Eastern music of that era.

Of course the object is to seek out some methods for realizing the
notation in microabc or related utilities.

Peace and love,

Margo Schulter
mschulter@...

microabc -- sagittal Peppermint mapping

🔗Margo Schulter <mschulter@...>

🔗Hudson Lacerda <hfmlacerda@...>