
Pegasus12: An offshoot of Kraig Grady's Centaur
Using prime 59 as aurally equivalent to prime 7

Kraig Grady's 12note Centaur tuning for primes 2357 has led me to
an offshoot or variation based on primes 2359, with prime 59
producing approximations of 7based ratios found in the original
Centaur varying from these by a factor of 14337:14336 (0.121 cents).
At the same time, as suggested by the great philosopher, physician,
and music theorist Ibn Sina (9801037), 59based ratios of 64/59 and
72/59 reasonably approximate 13/12 and 39/32, for example, the
difference being a factor of 768:767 (2.256 cents).
The Pegasus12 tuning, inspired by Centaur and also by Ibn Sina's
mention of 72/59, proposed as one possible neutral third fretting for
the `oud by Safi alDin alUrmawi (1216?1294), draws on these just
intonation "puns" involving prime 59 so as to provide some aurally
equivalent versions of septimal (primes 237) tetrachords and modes
like those of Archytas and Ptolemy in the Greek tradition, as well as
tetrachords and modes with middle or neutral intervals in the Near
Eastern tradition of the Islamic Renaissance (8th15th centuries).
For much fascinating information on Centaur and some of its possible
extensions and close relatives, please visit Kraig Grady's web page on
this system at .
In the Scala scale archive, Centaur is included as grady7.scl.
Here is a Scala file for Pegasus 12:
! bamm24bpegasus12_D.scl
!
Offshoot of Kraig Grady's Centaur: Rast/Penchgah plus Archytaslike modes on 1/1
12
!
531/512
9/8
4779/4096
59/48
4/3
177/128
3/2
1593/1024
27/16
14337/8192
59/32
2/1

1. A lattice diagram: one possible perspective

As shown in the lattice diagram below, naming the notes according to
their keyboard locations in the Bamm24b tuning with two 12note
Pythagorean chains (EbG#) at 531/512 apart (63.082 cents), and with
the note located at D on the lower keyboard taken as the 1/1,
Pegasus12 actually consists of only two chains of fifths. There is a
fivenote or pentatonic including the 1/1 (GB, or 4/327/16), plus a
sevennote or heptatonic chain identical to a Pythagorean diatonic
(F*B*, or 59/4814437/8192).
From one practical viewpoint, however, we may find it convenient to
regard the lower four notes of this second heptatonic chain as forming
septimal ratios with the 1/1 of the kind so prominently featured in
Kraig Grady's Centaur (here ~28/27~7/4). The upper three notes of
this second chain form neutral ratios with the 1/1 of 59/48, 59/32,
and 177/128, which have their own fine nuances of shading but could be
compared, for example, to 16/13, 24/13, and 18/13 at 768:768 larger.
The lattice reflects this perspective on Pegasus12:
(this chain continues downward below) > o

765.0 267.0 470.9 
A* E* B* 
1593/1024 4779/4096 14337/8192 
~14/9  7/6  7/4 
/ / / 
/ / / v
/__________/__________/__________________ 
4/3 1/1 3/2 9/8 27/16 
G D A E B 
498.0 0 702.0 203.9 905.9 
    
    
    
++++ < o
59/48 59/32 177/128 531/512
~896/729 ~448/243 ~112/81 ~28/27
F* C* G* D*
357.2 1059.2 561.1 63.1
What Centaur and Pegasus notably share in common is a 1379 hexany
(or in Pegasus an aurally equivalent structure differing by the factor
of 14337:14336), shown in the upper portion of the lattice diagram by
slanted lines showing minor thirds at ~7:6 (4779/4096) which form two
adjacent slanted quadrangles representing sonorities of ~12:14:18:21.
In the lower portion of the diagram, there are three dashed vertical
lines showing 59:48 neutral thirds.
From the standpoint of our chosen 1/1, the system thus features minor
septimal intervals and larger neutral steps and intervals. If we shift
our focus to a note such as D* (531/512 or ~28/27) on the upper chain,
however, then we encounter many major septimal intervals or virtual
equivalents (e.g. D*G, ~9/7; D*D, ~27/14) and smaller neutral steps
or intervals (e.g. D*E, 64/59 or ~13/12; D*B, 96/59 or ~13/8).
Often a given mode may focus on either septimal or neutral intervals.
Thus this mode from the 1/1 of D could be described as one variation
on the Diatonic of Archytas, or Tonic Diatonic of Ptolemy, with steps
of 28:27, 9:8, and 8:7, using disjunct tetrachords.
Tonic Diatonic 9:8 Tonic Diatonic
......
D D* E* G A A* B* D
1/1 ~28/27 ~7/6 4/3 3/2 ~14/9 ~7/4 2/1
28:27 9:8 8:7 9:8 28:27 9:8 8:7
This mode, with its larger neutral steps and intervals from the same
1/1 of D, is a basic version of a possible shading of a modern
disjunct Arab Rast:
Rast 9:8 Rast
......
D E F* G A B C* D
1/1 9/8 59/48 4/3 3/2 27/16 59/32 2/1
9:8 59:54 64:59 9:8 9:8 59:54 64:59

2. Pegasus as a Centaur offshoot: interval categories

A quick and general overview might note that Centaur and the Pegasus
offshoot share in common a set of Pythagorean and septimal intervals
generated from primes 237  or, in Pegasus, "virtual septimal
equivalents" from ratios of 59.
Additionally, Centaur has a variety of ratios of 235 or 2357,
where in Pegasus a corresponding number of scale steps often produce
neutral ratios of 59. While both 12note systems, according to Scala,
are strictly proper as well as constant structures, this contrast
between 5based and neutral flavors of intervals correlates some
differences between the two tunings in the organization of their
interval types and sizes.
In a very effective graphical overview of Centaur's 50 interval sizes
, Kraig Grady
presents two concepts helpful for such a comparison: "span" and "gap."
If we group the intervals of a tuning into categories based on the
number of scale steps subtended, for example 3step minor thirds and
4step major thirds in Centaur, then the "span" of a given category is
equal to the range of sizes within it. The "gap" between two
categories is equal to the difference or "jump" between the largest
3step and smallest 4step interval, for example.
In Centaur, while there is considerable variation as we move through
the groups of intervals formed by from 1 to 11 scale steps found
between the 1:1 unison and 2:1 octave, we find that these spans and
gaps are often comparable, with spans for categories other than 6step
tritones at 48.856.5 cents, and gaps at 35.763.0 cents. (Tritones or
6step intervals have a somewhat wider overall span of 78.0 cents.)
A general impression might be one of balance between spans and gaps.
In Pegasus, this equation is considerably altered by the presence of
smaller and larger neutral categories which often result in much
smaller and more subtle gaps between adjacent categories. The
following table, modelled after Kraig's also less elegant, shows some
of these patterns:

Pegasus12 interval categories and sizes

1step intervals 11step intervals
(minor and small neutral 2nds) (large neutral and major 7ths)
531:512 63.1 1024:531 1136.1
256:243 90.2 span: 77.7 243:128 1109.8
64:59 140.8 59:32 1059.2
____________________________________________________________________
gap: 12.5

2step intervals 10step intervals
(large neutral and major 2nds) (minor and small neutral 7ths)
59:54 153.3 108:59 1046.7
9:8 203.9 span: 77.7 16:9 996.1
16384:14337 231.1 14437:8192 968.9
___________________________________________________________________
gap: 35.9

3step intervals 9step intervals
(minor and small neutral 3rds) (large neutral and major 6ths)
4779:4096 267.0 8192:4772 933.0
32:27 294.1 span: 77.7 27:16 905.9
72:59 344.7 59:36 855.3
____________________________________________________________________
gap: 12.5

4step intervals 8step intervals
(large neutral and major 3rds) (minor and small neutral 6ths)
59:48 357.2 96:59 842.8
81:64 407.8 span: 77.7 128:81 792.2
2048:1593 435.0 1593:1024 765.0
___________________________________________________________________
gap: 35.9

5step intervals 7step intervals
(fourths and superfourths) (subfifths and fifths)
43011:32768 470.9 65536:32768 729.1
4:3 498.0 span: 77.7 3:2 702.0
81:59 548.6 118:81 651.4
____________________________________________________________________
gap: 12.5

6step intervals
(tritonic)
177:128 561.1 256:177 638.9
1024:729 588.3 span: 77.7 729:512 611.7
____________________________________________________________________

A curious characteristic of Pegasus12 is that every category has a
span of 77.7 cents. This range of 32768:31329 (77.747 cents) is equal
to the difference, for example, between 531:512 (63.1 cents) and 64:49
(140.8 cents) as complements adding up to a 9:8 tone.
The smaller gaps at 12.5 cents are equal to the difference, for
example, between smaller and larger neutral second steps at 64:59
(140.8 cents) and 59:54 (153.3 cents), complements adding up to a
32:27 minor third. This difference of 3481:3456 (12.478 cents) makes
for a gentle and subtle transition.
A clearer transition occurs between categories such as 2step and
3step intervals, for example, with 16384:14337 (231.1 cents) as the
largest interval of the first category, and 4779:4096 (267.0 cents) as
the smallest of the second, these sizes being almost identical to the
more familiar 8:7 (231.2 cents) and 7:6 (266.7 cents) of Centaur.
With these simpler ratios, the "gap" would be 49:48 or 35.697 cents;
since the Pegasus ratios involve a very slightly smaller ~8:7 tone and
larger ~7:6 minor third, the difference is 68516523:67108864 (35.938
cents), a minutely larger gap.
To note these ~49:48 gaps in either Centaur12 or Pegasus12 is
another way of saying that in these areas of transition neither tuning
set happens to include what might be termed "interseptimal" intervals,
for example hemifourths significantly larger than 8:7 but smaller than
7:6. However, extensions of either system might well add such
intervals.
One rather arbitrary although sometimes possibly useful distinction
made in the above table is between "superfourths" or "subfifths"
formed by adding to a fourth around 4:3, or subtracting from a fifth
around 3:2 an intervals somewhat larger than a septimal comma (64:63,
27.264 cents) but smaller than a usual "semitone" of around 28:27
(62.961 cents) or larger; and "tritonic" intervals involving the
addition or subtraction of such semitones. By this distinction, for
example, the 5step interval of 81:59 (548.6 cents), equal to a 4:3
fourth plus a diesis at 243:236 (50.6 cents) could be considered a
"superfourth" (a term for which I thank Dave Keenan, who takes 11:8 as
an example), while the 6step interval of 177:128 (561.1 cents), a 4:3
fourth plus a semitone or thirdtone at 531:512 (63.1 cents) could be
considered a more usual "tritonic" interval.
More generally, we should note that Centaur offers a greater variety
of interval sizes, 50 as opposed to 34 in Pegasus; and that Centaur
has four sizes of adjacent steps, all superparticular (28:27, 21:20,
16;15, and 15:14), while Pegasus as three sizes, none of them
superparticular (531:512, 256:243, 64:59).
While it would generally be informative to say that ratios of 235 or
2357 in Centaur often correspond to neutral ratios of 59 in
Pegasus, this does not fully convey the greater range of interval
sizes in the former tuning. For example, Centaur has a 4step interval
of 80:63 (413.6 cents) which would nicely add to the shadings of major
thirds in Pegasus in the Pythagoreantoseptimal range (81:649:7 or
so).
Another way of expressing this last point is to say that while Centaur
sometimes has large "gaps" between categories (e.g. from a 15:14
semitone as the largest 1step interval to a 10:9 tone as the smallest
2step interval, a difference of 28:27 or 63.0 cents), Pegasus
consistently has its largest "jumps" between interval sizes occurring
_within_ categories. These jumps of 243:236 or 50.6 cents occur, for
example, between the usual Pythagorean limma at 256:243 or 90.2 cents
and the small neutral second at 64:59 or 140.6 cents  both 1step
intervals.
Further, Pegasus consistently has its smallest "jumps" or differences
of 12.5 cents between interval sizes located as gaps _between_
categories. Within any given category, the smallest jump is found only
among the 6step or tritonic intervals, between the Pythagorean
diminished fifth at 1024:729 (588.3 cents) and augmented fourth at
729:512 (611.7 cents), a difference equal to the Pythagorean comma at
531441:524288 at 23.460 cents. Otherwise, all differences within a
category are equal either to 131072:129033 (27.143 cents), virtually
equivalent to the familiar 64:63 septimal comma (27.264 cents); or to
243:236 at 50.6 cents as mentioned in the previous paragraph.
In contrast, intracategory differences in Centaur range from 225:224
(7.712 cents, e.g. 16:15 at 111.7 cents and 15:14 at 119.4 cents) to
the septimal or Archytan comma at 64:63. These often subtle gradations
within a given category show one very sophisticated side of just
intonation, especially in so compact a system.
A lesson of these contrasts is that simply referring to a given tuning
as "strictly proper" or a "constant structure" leaves opens many
details and nuances of intervallic patterns and spacing along the
continuum.

3. Pegasus and the 729:728 connection between ratios of 7 and 13

While from one perspective, as reflected in the lattice diagram at the
beginning of this article, Pegasus12 can be seen as featuring a
mixture of Pythagorean, quasiseptimal, and neutral intervals, the
latter two categories are in fact realized alike by steps forming a
single heptatonic chain of fifths.
This structure raises an interesting question: if Pegasus includes
59based ratios virtually equivalent to more familiar ratios of 7,
couldn't an aurally equivalent version of Pegasus be realized simply
by using pure ratios of 7?
The quick answer is yes! For purposes of tuning by ear, in fact,
Pegasus could be defined in precisely such terms. Thus, in theory 
considerations of the safety of strings could modify the order of
tuning in practice, as explained below!  one might first tune the
pentatonic chain including the 1/1:
4/3 1/1 3/2 9/8 27/16
G D A E B
498.0 0 702.0 203.9 905.9
The next step might be to tune either a 7/6 or a 7/4 step, for
example, thus setting the first note of the second chain of fifths,
which from there would be completed in usual Pythagorean fashion:
896/729 448/243 112/81 28/27 14/9 7/6 7/4
F* C* G* D* A* E* B*
357,1 1059.1 561.0 63.0 764.9 266.9 968.8


4/3 1/1 3/2 9/8 27/16
G D A E B
498.0 0 702.0 203.9 905.9
In practice, a safer procedure for stringed instruments such as the
harpsichord might be to tune the _upper_ heptatonic chain first at
pitch level known to involve safe levels of tension, and then, for
example, taking E* as a reference, to tune D at a 7:6 minor third
_below_ this note, or G at a 7:4 minor seventh below it!
Here the complex septimal ratios of 896/729, 448/243, and 112/81 are
virtually identical to 59/48, 59/32, and 177/128 in Pegasus, which
differ by only 14337:14336. These 59based ratios are founded in the
convenient arithmetic division on a monochord or `oud of a 32:27 minor
third into neutral seconds of 64:59:54 (140.8153.3 cents).
However, as Ibn Sina's original suggestion in the 11th century that
such a division might reasonably approximate his favored division of
this minor third into steps of 13:12 and 128:117 (138.6155.6 cents)
may move us to consider, in either a 59based or pure septimal version
of Pegasus, these intervals can be taken as "justly tempered"
variations on ratios of 13. Our lattice in the 7based version could
thus be read:
(~16/13) (~24/13) (~18/13) (~27/26)
896/729 448/243 112/81 28/27 14/9 7/6 7/4
F* C* G* D* A* E* B*
357,1 1059.1 561.0 63.0 764.9 266.9 968.8


4/3 1/1 3/2 9/8 27/16
G D A E B
498.0 0 702.0 203.9 905.9
With 7based ratios, the difference between these complex septimal
forms and their pure 13based counterparts is equal to that between
thirdtone steps of 28/27 and 27/26: 729:728 (2.376 cents). Between
59based and pure 13based counterparts such as 72/59 and 39/32, it is
a very slightly smaller 768:767 (2.256 cents). And the yet smaller
difference between these differences is 14337:14336 (0.121 cents).
A scenario to illustrate the 729:728 has us start with a 9/7 septimal
major third at 435.1 cents. Now we form a characteristic type of
septimal augmented fourth by adding a 9:8 tone, arriving at 81/56 or
639.0 cents (an interval present in Centaur and virtually identical in
the 59based version of Pegasus at 256:177 or 638.9 cents). Either the
7based or 59based version of this interval nicely approximates a
pure 13/9 (636.618 cents).
Now if we move up by a 3:2 fifth and subtract a 2:1 octave, we have
arrived at 243:224 or 140.9 cents, a step famous for its inclusion in
the Chromatic of Archytas, where together with a 28/27 step it forms a
9/8 tone (1/128/279/84/3 or 063.0203.9498.0 cents). This
interval is larger by only 729:728 than a 13:12 neutral second, to
which it may serve as a good approximation, as may 64:59.
Now if we move up by another 3:2 fifth, we arrive at 729:448 or 842.9
cents, wider by only a 729:728 than a just 13.8 at 840.5 cents.
In a system like Centaur, while the chain of septimal intervals goes
far enough to produce ratios of 112:81 (~18:13) and 81:56 (~13:9), it
does not reach quite far enough to bring about neutral seconds,
thirds, sixths, or sevenths. Rather, this 12note system complements
its septimal (i.e. 237) intervals with others based on factors of
235 or 2357.
In Pegasus, we might say that the septimal or quasiseptimal chain is
sufficiently lengthened to bring about these neutral intervals, so
that all intervals are based on ratios of 2359 in the standard
version, or 237 in the modified version based on tuning a pure
interval of 7/6 or 7/4 to locate the first note of the second or
heptatonic chain of fifths.
While Ibn Sina recognized the kinship between ratios of 13 and 59
(differing by 768:767), and the 729:728 relationship between ratios of
7 and 13 has also been recognized, I am not sure how widely the almost
perfect coincidence of 7 and 59 and their 14337:14336 relationship has
been realized, a recognition celebrated by the Pegasus tuning.
Most appreciatively,
Margo Schulter
mschulter@calweb.com