Alternative Mapping Paradigm (AMP)

Why an Alternative Mapping Paradigm (AMP)?

The purpose of the Alternative Mapping Paradigm (AMP) is to
permit a better understanding "at a glance" of some of the
melodic as well as vertical or harmonic resources of tunings and
temperaments. For example, a tuning system designed for Near
Eastern Maqam or Dastgah music might offer a variety of neutral
or Zalzalian steps and intervals; and might also happen, not
necessarily by design, to have some interesting subsets evoking
gamelan with its Pelog and Slendro modes.

Designing and exploring a tuning system often involves a
delightfully quirky mix of intention and serendipity: making the
most of what was meant to be there, and happening upon "scales
within the scale" (or better "subsets within the gamut") which
take us in new directions.

A musician-friendly "map" of a tuning to be used for these
purposes should be reasonably accessible and at the same time
give lots of information.

In its prototype form, the AMP focuses on these types of tunings:

(1) Simple Pythagorean JI, with a single chain of 3/2 fifths;

(2) Simple regular temperaments, with a single chain of
fifths or other generators;

(3) JI systems with two chains of 3/2 fifths at the distance
of some rational interval, or "mixed" systems where the
distance is measured by an irrational interval;

(4) Tempered systems with two chains of fifths or other
generators at some distance apart; and

(5) "Shaggy" systems where there is a slight irregularity in a
chain of fifths or other generators, as with synthesizer
temperaments that alternate two fifth sizes at one tuning
unit apart in order to obtain an intermediate "average
fifth."

The AMP should be extensible to more irregular temperaments,
which are an ideal way to maximize parsimony and variety -- a
description which can also apply to JI systems. Including
"shaggy" systems is a first step in that direction.

--------------------------------------------------------
AMP and Melodic Variety: Are we getting superparticular?
--------------------------------------------------------

While Erv Wilson's Moment of Symmetry (MOS) with only two
adjacent step sizes is a theoretical concept with rich
applications, tetrachords in the Ancient Greek or medieval and
later Near Eastern tradition often feature three step sizes. Possibly we could call this a Threefold Order of Symmetry, or
TOS. And when diverse types of tetrachords are combined, as in
many of the maqamat, we can get a lot more than three step sizes
in even a textbook "octave species" version of a given maqam!

To survey some of the possibilities of a given tuning system, the
AMP uses as a default table of superparticular steps from 3:2
to 14:13, plus any smaller steps more or less closely
approximated. Especially when a tuning system is designed to
include or closely approximate superparticular ratios, these
ratios can serve as useful guideposts to the continuum.

However, other relevant ratios can also be included: in Maqam
music, for example, al-Farabi's step of 88:81 (143 cents), or Ibn
Sina's of 128:117 (156 cents). At times, we may specifically
desire values somewhere between the nearest epimore or
superparticular ratios, as with the frequent hemifourths of
gamelan between 8:7 and 7:6. Here 15:13 or 22:19 might be a
better guide, and the table of ratios and melodic steps could be
modified to include these other ratios.

---------------------------------------------------------------
A Shaggy Temperament Story: Starting with the Prime-Odd Listing
---------------------------------------------------------------

To show how AMP can operate in practice, let's consider a
temperament I often use: the Milder Extended Temperament in 24
notes, or MET-24 for short. Here's a Scala file:

<http://www.bestII.com/~mschulter/met24.scl>

! met24.scl
!
Milder Extended Temperament, 5ths avg. 703.658c, spaced 57.422c
24
!
57.42188
126.56250
183.98438
207.42187
264.84375
289.45313
346.87500
414.84375
472.26563
496.87500
554.29688
622.26563
679.68750
704.29688
761.71875
829.68750
887.10938
911.71875
969.14063
992.57813
1050.00000
1119.14063
1176.56250
2/1

We have three generators -- or is it four?

Generator 1 (G1): 2/1 (period)
Generator 2 (G2): 703.125-704.297 cents
Generator 3 (G3): 57.422 cents

The notation "703.125-704.297 cents" for the second generator, or
G2, means that we have a "shaggy" or slightly irregular system
where the values of 703.125 and 704.297 cents alternate along a
chain.

Note that this definition leaves open the size of the system,
which in MET-24 has two 12-note chains of fifths. From another
viewpoint, this version of the system has 11 G2 generators in
each chain (six at 703.125 cents and five at 704.297 cents), and
one instance of G3 to define the distance between the two chains.

We could also have, for example, a MET-34 with two 17-note
chains, and the three generators (or four) would remain the
same. Here, however, we'll focus on MET-24.

A first step is the Prime-Odd Listing (POL), which shows how
relevant prime factors or nonprime odd factors of interest (often
specifically 9) are represented in a given system. "Relevant"
factors are typically those a system is meant to support.

For example, MET-24 is designed to support prime-odd factors of
2-3-7-9-11-13, and also the superparticular step 11:10. Since
11:10 is part of the optimization, it's not surprising that we
get a few locations with approximations of prime 5 (5/4). We can
place prime 5 in parentheses to show its "serendipitous" if
rather predictable nature:

===========================================================
Prime-Odd Listing: MET-24
-----------------------------------------------------------
2/1 3/2 (5/4) 7/4 9/8 11/8 13/8 59/32
1200.0 702.0 386.3 968.8 203.9 551.3 840.5 1059.2
-----------------------------------------------------------
G1 +1 +9u +3u +2 -1u -4u +5d
J 703.1 390.2 968.0 207.4 553.1 842.6 1060.5
704.3 391.4 969.1 554.3 1061.7
...........................................................
J +1.2 +3.9 -0.9 +3.5 +1.8 +2.1 +1.4
+2.3 +5.1 +0.3 +3.0 +2.5
===========================================================

In addition to showing some of the intervals of the tuning
system, the POL shows the mapping for these prime-odd
approximations (and/or divergences). For example, 9/8 has a
mapping of "+2" -- that is, two G2 generators or fifths up, at
207.4 cents.

For 13/8, the mapping is a bit more complex: "-4u". This means
four G2 generators or fifths down, and then a G3 up ("u"). Another way of putting this is that a regular minor sixth at
785.2 cents (-4) plus a G3 at 57.4 cents (u) gives us an
approximate 13/8 neutral sixth at 842.6 cents.

With a "shaggy" temperament like MET-24, lots of interval
categories are going to have two values, and some appear on the
POL. An interesting example is 59/32, included because of the
important role of the division 72:64:59:54 in medieval Near
Eastern theory. The mapping "+5d" means we go up five G2
generators, and then down a G3 -- in other words, a regular major
seventh (1118.0 or 1119.1 cents) less a G3 at 57.4 cents. We get
either 1060.5 or 1061.7 cents. Note that in this type of system
the values may vary a bit, but the mapping for a ratio remains
constant, here +5d.

In addition to showing where to find an interval, the mapping
tells us -- at least once we've defined the size of the system --
how many locations we'll find it at. Thus 7/4 is "+3u", so we'll
find it from any note on the lower chain of fifths where we have
a regular major sixth (+3) -- nine locations in all. We'll get an
approximate 13/8 or "-4u" wherever on the lower chain we have a
regular minor sixth -- eight locations. In contrast, the 5/4
approximation or "+9u" is rarer: we'll get it from wherever on
the lower chain we have an augmented second (a small neutral or
supraminor third at 332.8 or 334.0 cents) -- three locations.

As George Secor and others have noted, prime approximations can
be handy in figuring out the accuracy of other ratios. For
example, let's consider 9/7. We know that 9/8 is +3.5, while 7/4
may be +0.3 or -0.9 -- so we take the difference, +3.2 or +4.4.

How about 13/11. Here 13/8 is +2.1, while 11/8 is +1.8 or +3.0. Thus 13/11 will be either +0.3 or -0.9.

A handy rule for MET-24 -- your mileage or kilometrage may vary
with other "shaggy" systems -- is that if two prime factors
listed in the POL are both shaggy or variable, their difference
will be a constant. Let's consider 7/6 -- with both 7/4 and 3/2
variable at +0.3/-0.9 and +2.3/+1.2. Applying the rule, we can
simply take the difference of +0.3 and +2.3, or of -0.9 and +1.2
for example, so that 7/6 is -2.0 or -2.1 (rounding inaccuracies
produce these slightly different results where a more precise
figure would be -2.027 cents).

At any rate, this discussion has hopefully illustrated some
aspects and uses of the POL; and possibly also moved you to ask,
"Why are we spending so much time on this cheap imitation of
George Secor's 29-HTT when I could be tuning up the real thing?"
(see Scala archive, secor29htt.scl).

----------------------------------------
Listing Some Steps, Melodic or Otherwise
----------------------------------------

The List of Steps (LOS) shows superparticular or other
approximations of interest, using the same mapping notation as
the POL:

===========================================================
List of Steps: MET-24
-----------------------------------------------------------
2/1 3/2 4/3 (5/4) (6/5) 7/6 8/7 9/8
1200.0 702.0 498.0 386.3 315.6 266.9 231.2 203.9
-----------------------------------------------------------
G1 +1 -1 +9u -8d +2u -3d +2
J 703.1 495.7 390.2 312.9 264.9 230.9 207.4
704.3 496.9 391.4 232.0
...........................................................
J +1.2 -2.3 +3.9 -2.8 -2.0 -0.3 +3.5
+2.3 -1.2 +5.1 +0.9
===========================================================
(10/9) 11/10 12/11 13/12 14/13 (17/16) 22/21 27/26
182.4 165.0 150.6 138.6 128.3 105.0 80.5 65.3
-----------------------------------------------------------
+7u -10 +2d -5u +7 -10d -5 +7d
182.8 162.9 150.0 138.3 125.4 105.5 80.9 68.0
184.0 139.5 126.6 82.0 69.1
...........................................................
+0.4 -2.1 -0.6 -0.3 -2.9 +0.5 +0.3 +2.6
+1.6 +0.9 -1.7 +1.5 +3.8
===========================================================
28/27 91/88 33/32 64/63 78/77
63.0 58.0 53.3 27.3 22.3
-----------------------------------------------------------
G3 G3 G3 -5d -5d
57.4 57.4 57.4 23.4 23.4
24.6 24.6
...........................................................
-5.5 -0.6 +4.1 -3.8 +1.1
-2.7 +2.3
===========================================================

What ratios to list can be a matter of judgment, with the same
tempered interval (here G3, 57.4 cents) listed as the relevant
approximation of 28/27, 91/88, and 33/32. Often, as here, the LOS
includes all steps of some kind of "tone" or smaller, say from
8/7 or so down.

-----------------------------
Listing Families of Intervals
-----------------------------

While the accuracy or divergence of a system with respect to
prime, superparticular, or other ratios may often be of interest,
there's also the pleasure of simply looking at a system in its
own terms, and seeing what resources are available. The List of
Interval Familes (LIF) is designed for this purpose, and is here
presented in a format for "chains of fifths" tunings as David
Keenan has called them. With other types of generators, the
format might change a bit.

We look first at intervals found within either chain of fifths,
and then at G3 (here 57.4 cents) and the additional intervals it
makes possible.

==========================================================
List of Interval Familes: MET-24
----------------------------------------------------------
Intervals within a chain of G2's
----------------------------------------------------------
+1 +2 +3 +4 +5 +6
703.125 207.422 910.547 414.844 1117.969 622.266
704.297 911.719 1119.141
...........................................................
-1 -2 -3 -4 -5 -6
495.703 992.578 288.281 785.156 80.859 577.734
496.875 289.453 82.031
===========================================================
+7 +8 +9 +10 +11
125.391 829.688 332.812 1037.109 540.234
126.563 333.984
...........................................................
-7 -8 -9 -10 -11
1073.437 370.312 866.016 162.891 659.766
1074.609 867.188
===========================================================
Intervals up from lower chain (G3 = 57.422)
-----------------------------------------------------------
+1u +2u +3u +4u +5u +6u
760.547 264.844 967.969 472.266 1175.391 679.688
761.719 969.141 1176.563
...........................................................
-1u -2u -3u -4u -5u -6u
553.125 1050.000 345.703 842.578 138.281 635.156
554.297 346.875 139.453
==========================================================-
+7u +8u +9u +10u +11u
182.812 887.109 390.234 1094.531 597.656
183.984 391.406
...........................................................
-7u -8u -9u -10u -11u
1130.859 427.734 923.438 220.312 717.186
1132.031 924.609
===========================================================
Intervals down from upper chain (G3 = 57.422)
-----------------------------------------------------------
+1d +2d +3d +4d +5d +6d
645.703 150.000 853.125 357.422 1060.547 564.844
646.875 854.297 1061.719
...........................................................
-1d -2d -3d -4d -5d -6d
438.281 935.156 230.859 727.734 23.437 520.312
439.453 232.031 24.609
===========================================================
+7d +8d +9d +10d +11d
67.969 772.266 275.391 979.688 482.813
69.141 276.563
...........................................................
-7d -8d -9d -10d -11d
1016.016 312.891 808.594 105.469 602.344
1017.188 809.766
-----------------------------------------------------------
===========================================================

The LIF may take a while to compile or to read, but is handy in
giving an overview of the system that shows the range of
intervals, where to find them, and how often they occur.

----------------------------------------------
A Maqam Extra: The Zalzalian Action List (ZAL)
----------------------------------------------

With tuning systems intended for Maqam or the closely related
Persian Dastgah tradition, the AMP supports an extra kind of
mapping or listing: the Zalzalian Action List (ZAL) giving an
overview of neutral steps and intervals.

The ratio comparisons chosen may vary from system to system. For
example, MET-24 doesn't have anything very close to Ibn Sina's
step of 128:117 (155.6 cents); but many other systems do. The
idea is to focus mainly on the interval sizes actually present,
so that one can see at a glance some of the options and
possibilities.

==================================================
Zalzalian Action List: MET-24
--------------------------------------------------
Neutral or middle seconds
--------------------------------------------------
14:13 13:12 12:11 11:10
128.3 138.6 150.6 165.0
..................................................
+7 -5u +2d -10
125.4 138.3 150.0 162.9
126.6 139.5
--------------------------------------------------
Neutral or middle thirds
--------------------------------------------------
40:33 11:9 16:13 26:21
333.0 347.4 359.5 369.7
..................................................
+9 -3u +4d -8
332.8 345.7 357.4 370.3
334.0 346.9
--------------------------------------------------
Superfourths or small tritones
--------------------------------------------------
15:11 11:8 18:13 88:63
537.0 551.3 563.4 578.6
..................................................
+11 -1u +6d -6
540.2 553.1 564.8 577.7
554.3
--------------------------------------------------
Large tritones or subfifths
--------------------------------------------------
63:44 13:9 16:11 22:15
621.5 636.6 648.7 663.0
..................................................
+6 -6u +1d -11
622.3 635.2 645.7 659.8
646.9
--------------------------------------------------
Neutral or middle sixths
--------------------------------------------------
21:13 13:8 18:11 33:20
830.3 840.5 852.6 867.0
..................................................
+8 -4u +3d -9
829.7 842.6 853.1 866.0
854.3 867.2
--------------------------------------------------
Neutral or middle sevenths
--------------------------------------------------
20:11 11:6 24:13 13:7
1035.0 1049.4 1061.4 1071.7
..................................................
+10 -2u +5d -7
1037.1 1050.0 1060.5 1073.4
1061.7 1075.6
==================================================

----------
Conclusion
----------

The AMP is no substitute for Manuel Op de Coul's Scala or other
invaluable tools such as X. J. Scott's Li'l Miss' Scale Oven
(LMSO); nor does it exclude other tuning and temperament
approaches. And for sheer compactness, Scala's famous
SHOW /LINE INTERVALS is hard to outdo!

However, if one wants to get a quick and concrete overview of a
tuning system, and maybe to delve into the interval patterns,
then the AMP provides a perspective which may supplement other
approaches.

Most respectfully,

Margo Schulter
mschulter@...