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Eagle 53: John O'Sullivan's Marveldene Tuning and Some Alternatives
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John O'Sullivan has asked me to comment on his article _How I arrived at
Eagle 53 (2017). Here I will focus on a few points.
First, the logic of his construction, which effectively reinvents a scheme
which has become known as Marveldene, seems to me to be a general preference
for 5-limit just intonation (JI), or some near-just tempering of this. Many
of the choices John makes at locations of his tuning where a variety of
ratios he considers "high-strength" or "medium-strength" appear most simply
to follow this 5-limit logic.
! john_o-sullivan_eagle53.scl
!
John O'Sullivan's Eagle 53 (Marveldene in 53-ed2)
12
!
113.20755
203.77358
316.98113
384.90566
498.11321
588.67925
701.88679
815.09434
883.01887
1018.86792
1086.79245
2/1
.5/3---------5/4--------15/8........7/5
884.4.......386.3......1088,3......582.5
883.0.......384.9......1086.8......588.7
..|...........|...........|..........|
..|...........|...........|..........|
.4/3---------1/1---------3/2--------9/8
498.0.........0.........702.0......203.9
498.1.........0.........701.9......203.8
..|...........|...........|..........|
..|...........|...........|..........|
16/15 -------8/5---------6/5--------9/5
111.7.......813.7.......315.6......1017.6
113.2.......815.1.......317.0......1018.9
This lattice shows just or tempered intervals of 3:2 fifths as dashed
horizontal lines, and likewise 5:4 major thirds, just or tempered, as
vertical lines. Note that, at the upper right portion of the diagram, the JI
version has the interval 15/8-7/5 as a dotted line, since this narrow fifth
has a size of 112/75 (694.2 cents) which is smaller than 3:2 by 225:224 or
7.7 cents. However, in the tempered Eagle 53 version, this becomes a regular
fifth at 1086.8-588.7 cents (701.9 cents, less than 0.1 cent narrow), or
virtually pure.
Either 53-ed2 as an overall system, or this Eagle 53 subset, has the
distinction of providing these excellent fifths, in addition to near-just
versions of 5/4 (1.4 cents narrow); 6/5 (1.3 cents wide); and also 9/5 (1.3
cents wide) and 16/15 (1.5 cents wide). These values are all rather more
accurate than in Pythagorean intonation with pure 3:2 fifths, where these
5-prime intervals are narrow or wide by a schisma of 32805:32768 or not
quite 2.0 cents.
If we take Pythagorean tuning as a starting place, then the step here
representing 7/5 is actually closer to just (582.5 cents) with pure fifths,
at 1024/729 or 588.3 cents, a difference of 5120/5103 (5.8 cents), than in
Eagle 53 (588.7 cents, or 6.2 cents wide). This is an excellent
representation of the just 5-based tritone at 45:32 (590.2 cents, 1,5 cents
wide), but can, like a Pythagorean 1024/729, be treated as a not-so-accurate
representation of the simpler 7:5.
Why incorporate 7/5, at least conceptually, into this mainly 2-3-5-based
scheme? One reason might be the elegance of the simpler ratio, which in
John's theory is not only a conceptual nicety, but an imperative in order to
legitimize this note as forming at least a "medium-strength" interval with
the 1/1. If I were labelling this diagram, I might call the step 45/32,
since I have no problem with complex intervals -- while recognizing that in
terms of the simpler 7/5, Eagle 53 is closer than 5-based JI (at 225/224
wide) but a tad further than Pythagorean JI (at 5120/5103 wide).
John points out a ramification of the system, clarified by the 7/5 notation,
which to me offers a highly practical and satisfying musical progression,
although not one likely in a usual 5-limit style. If we take the notes
7/5-9/5-6/5, we get a sonority of 0-430.2-928.3 cents, which approximates a
just 7:9:12 (0-435.1-933.1 cents), with differences of 4.9 and 4.8 cents on
12:7, and offers a fine medieval or neomedieval European resolution:
Here the lowest voice descends by a virtually just 9:8 tone (203.8 cents),
while each upper voice ascends by 67.9 cents, not too far from a just 28:27
thirdtone or small semitone much favored by Archytas in a purely melodic
context (63.0 cents). Thus we have available a fine medieval cadence in an
overall system mostly lending itself to 5-limit styles. While Pythagorean
would be slighly less accurate with 5-limit intervals, here it would be
slightly more accurate with ratios of 2-3-7, with its small semitone or
thirdtone at 66.8 cents, and approximations of 9:7 (431.3 cents) and 12:7
(929.3 cents), with differences here of 3.8 cents.
Thus Eagle 53 is comparable to the Marveldene structure as implemented in
Pythagorean intonation, with some small nuances: Eagle 53 minutely
compromises 3:2 in the narrow direction, slightly improving ratios of 5, and
making ratios of 7 (including 7/5) slightly less accurate.
Now let us consider one possible alternative solution that I arrived at
intuitively, and which is not based on any ed2 system:
! eagle-700.5_384.0.scl
!
Eagle duodene with fifth 700.5 cents, ~5/4 major third 384.0 cents
12
!
115.50000
201.00000
316.50000
384.00000
499.50000
585.00000
700.50000
816.00000
883.50000
1017.00000
1084.50000
2/1
.5/3---------5/4--------15/8........7/5
884.4.......386.3......1088,3......582.5
883.5.......384.0......1084.5......585.0
..|...........|...........|..........|
..|...........|...........|..........|
.4/3---------1/1---------3/2--------9/8
498.0.........0.........702.0......203.9
499.5.........0.........700.5......201.0
..|...........|...........|..........|
..|...........|...........|..........|
16/15 -------8/5---------6/5--------9/5
111.7.......813.7.......315.6......1017.6
115.5.......816.0.......316.5......1017.0
In this system, the defining parameters are 3:2 at 700.5 cents (1.5 cents
narrow) and 5:4 at 384.0 cents (2.3 cents narrow). Here the emphasis is on
getting a near-just 7/5 step, here 585.0 cents as compared to a just 582.5
cents (2.5 cents wide).
If we used a just 3:2 and 5:4, then 7/5 would be represented by a just
45/32, at 7.7 cents wide. Here a relevant relationship is 7/5 = 9/8 x 5/4.
Thus, since a tempered 9:8 is formed from two fifths each about 1.5 cents
narrow, 9:8 at 201.0 cents is not quite 3 cents narrow of a just 203.9
cents; and 5:4 at 384.0 cents is narrow by 2.3 cents. More precisely 9:8 is
narrow by 2.910 cents, and 5:4 by 2.314 cents, so that the 45/32 of 5-limit
JI is narrowed by 5.224 cents in all -- leaving 7/5 wide by the 225/224 less
this amount, or 2.488 cents.
An obvious compromise, as compared with Eagle 53, is that fifths at 1.5
cents narrow are tangibly although subtly impure, or "near-just," as
compared to the virtually just 53-ed2. Likewise 5:4 at 384.0 cents is less
accurate than in Eagle 53 (384.9 cents) or Pythagorean (384.4 cents). As it
happens, 6/5 and 5/3 are within a cent of just; but 16/15 and 15/8 are off
by 3.8 cents, as compared to 1.5 cents in Eagle 53 or 2.0 cents in
Pythagorean.
Although this is hardly the centerpiece of the Marveldene system, the
neomedieval progression we noted in Eagle 53 is interesting to compare in
this temperament:
12/5---- 5/2
9/5 ----15/8
7/5 ---- 5/4
Here the 7:9:12 sonority is at 0-432.0-931.5 cents, with 9:7 at 3.1 cents
narrow and 12:7 at 1.6 cents narrow, somewhat more accurate than in either
Pythagorean or Eagle 53. The descending 9:8 step at 7/5-5/4 is tempered at
201.0 cents, less accurate than in the other systems where it is just or
virtually just; and the ascending 28:27 thirdtone steps at 67.5 cents, very
slightly closer to just than in Eagle 53 (67.9 cents), but less accurate in
Pythagorean (66.8 cents).
I might add that the POTE based on Graham Breed's methods appears to have a
fifth at 700.408 cents and a 5:4 major third at 383.638 cents -- with 7/5 at
584.453 cents. Here is my understanding of this POTE version, subject to
correction by Graham or others:
! eagle-pote.scl
!
Eagle, POTE result from Graham Breed's algoriths
12
!
115.95490
200.81500
316.76990
383.63760
499.59250
584.45260
700.40750
816.36240
883.23010
1017.17740
1084.04510
2/1
.5/3---------5/4--------15/8........7/5
883.4.......386.3......1088,3......582.5
883.2.......383.6......1084.0......584.5
..|...........|...........|..........|
..|...........|...........|..........|
.4/3---------1/1---------3/2--------9/8
498.0.........0.........702.0......203.9
499.6.........0.........700.4......200.8
..|...........|...........|..........|
..|...........|...........|..........|
16/15 -------8/5---------6/5--------9/5
111.7.......813.7.......315.6......1017.6
116.0.......816.4.......316.8......1017.2
While Marveldene, as I have frequently noted, is mainly a 2-3-5 oriented
system, there are two locations, at a just or tempered 16/15 and 8/5, with
minor sevenths approximating 7:4. At 16/15, we have what could be described
as Erv Wilson's 1-3-7-9 hexany (1/1-9/8-7/6-21/16-3/2-7/4), with either 14/9
or 27/16 as a possible seventh step to fill out a variation on one
permutation of the Archytan Diatonic. Here it is interesting to compare four
implementations:
JI:.........1/1.....9/8...7/6.....21/16.....3/2...(14/9)...(27/16)..7/4.....2/1
.............0.....203.9.266.9....470.8....702.0..764.9.....905.9..968.8...1200
Eagle 53:....0.....203.8.271.7....475.5....701.9..769.8.....905.7..973.6...1200
.............0......-0.1..+4.8.....+4.7.....-0.1...+4.9......-0.2...+4.8.....0
Pythagorean..0.....203.9.270.7....474.6....702.0..768.7.....905.9..972.6...1200
.............0.......0.0..+3.8.....+3.8.......0....+3.8........0....+3.8.....0
700.5-384.0..0.....201.0.268.5....469.5....700.5..768.0.....901.5..969.0...1200
.............0......-2.9..+1.6.....-1.3.....-1.5...+3.1......-4.4...-0.2.....0
POTE:........0.....200.8.267.7....468.5....700.4..767.3.....901.2..968.1...1200
.............0......-3.1..+0.8.....-2.3.....-1.5...+2.4......-4.6...-0.7.....0
Here either the POTE or the 700.5-384.0 cent system is more accurate than the
Eagle 53 or Pythagorean implementations with ratios of 2-3-7, although notably
less accurate with 3:2 and 4:3, as well as 9:8 and 27:16.
I must mention that the POTE is also superb for the neomedieval progression
found on the tempered 7/5 step:
12/5---- 5/2
9/5 ----15/8
7/5 ---- 5/4
Here we have 7:9:12 at 0-432.7-932.3 cents, with 9:7 narrow by only 2.4 cents,
and 12:7 by only 0.8 cents, providing thus also a virtually just 7:6. The
tempered 9:8 at 200.8 cents is narrow by 3.1 cents, still more accurate than in
12n-ed2 (3.9 cents narrow), while 28:27 at 66.9 cents or 3.9 cents wide is
almost identical to Pythagorean (66.8 cents).
The POTE has the best 7/5 of any of these implementations at 584.5 cents, or
1.9 cents wide. The 7:5 interval is interesting for a configuration that occurs
at another location in this general scheme, 8/5, where a sonority of either
4:5:6:7 (1:1-5:4-3:2-7:4) or 7:6:5:4 (1:1-7:6-7:5-7:4) is available. Let us
look at the requisite notes in reference to the 8:5 step:
JI:.........1/1.....7/6...5/4.....7/5....3/2....7/4..
.............0.....266.9.386.3...582.5..702.0..968.8.
Eagle 53:....0.....271.7.384.9...588.7..701.9..973.6
.............0......+4.8..-1.4....+6.2...-0.1...+4.8
Pythagorean..0.....270.7.384,4...588.3..702.0..972.6
.............0......+3.8..-2.0....+5.8.....0....+3.8
700.5-384.0..0.....268.5.384.0...585.0..700.5..969.0
.............0......+1.6..-2.3....-2.5...-1.5...+0.2
POTE:........0.....267.7.383.6...584.5..700.4..968.1
.............0......+0.8..-2.7....-1.9...-1.5...-0.7
The POTE and 700.5-384.0 schemes appreciably although subtly compromise fifths
in order to arrive at near-just 2-3-7 intervals, with the POTE getting both 7:6
and 7:4 within 0.812 cents of just, and the 700.5-384.0 shading making 7:4 only
0.174 cents wide of just, but thereby making the impurity of 7:6 comparable to
that of the fifth. While both these systems do nicely with 7:9:12, the POTE is
superb with this sonority and its standard neomedieval resolution.
Because Eagle 53 is so close to a Pythagorean implementation of the Marveldene,
these two systems are likewise comparable, both producing ideal fifths, and
also close approximations of 5:4 and 6:5.
It is not my purpose here to focus on John's criteria for rating intervals. My
own approach is rather impressionistic. Thus I find 28:27 ideal as a small
melodic semitone, and would temper my mathematical explanations to be
consistent with this observation. And, vertically, I consider the 16:21:24:28
sonority used by LaMonte Young at least as early as the 1960's according to
Kyle Gann, and a feature of Wilson's 1-3-7-9 hexany, to make 21:16 an interval
that can ornamental a tuning in many contexts, not only in 16:21:24:28 (or
9:12:16:21, for example, but also, as George Secor pointed out to me, in other
contexts.
It is from this perspective that I am fascinated to find that the Marveldene
design includes a 1-3-7-9 hexany, although that does not seem to be the main
purpose of this mostly prime 2-3-5 system. How much one prioritizes just or
virtually just fifths and maximally accurate representations of 5:4 and 6:5, as
opposed to 7/5 and some of the prime 2-3-7 attractions to be found in a few
locations, are factors in one's choice of an implementation.
----------------------
Summary of Comparisons
----------------------
To summarize the above survey of some Marveldene implementations,
Pythagorean intonation offers just 3:2 fifths, 4:3 fourths, and 9:8 tones;
and also very accurate 5-limit approximations, with 5:4 and 6:5 impure by
the 32805/32768 schisma (1.954 cents), and likewise intervals such as the
16:15 semitone and 9:5 minor seventh. However, 7/5 is wide by 5120/5103 or
5.758 cents, being implemented as a just diminished fifth at 1024/729.
Ratios of 2-3-7 (e.g. 7:6, 9:7, 7:4) are impure by a "septimal schisma" at
3.804 cents.
The Eagle 53 tuning is generally quite similar, with its virtually just
fifths and fourths impure by only 0.068 cents. The advantage Eagle 53 offers
is that, of all the Marveldene tunings surveyed here, it provides the best
approximations of 5:4 (1.408 cents narrow) and 6:5 (1.340 cents wide). A
consequence is that 7/5 is slightly less accurate (at 6.167 cents wide), and
likewise ratios of primes 2-3-7. Thus 7:6 is 4.827 cents wide; 9:7, 4.895
cents narrow; and 7:4, 4.759 cents wide.
My impressionistic tuning with fifths at 700.5 cents (1.455 cents narrow)
and near-5:4 major thirds at 384.0 cents (2.314 cents narrow) involves a
very substantial compromise of fifths and fourths as compared with
Pythagorean or Eagle 53, and also of 5:4, which is almost a cent more impure
than in Eagle 53, and 0.36 cents more impure than the Pythagorean 8192/6561
at 1.954 cents narrow. This solution is more accurate with 7/5 (2.488 cents
narrow), and also the prime 2-3-7 ratios. Here we have 7/6 at 1.629 cents
wide; 9/7 at 3.084 cents narrow; and 7/4 at only 0.174 cents wide.
The POTE (Pure-Octave Tenney-Euclidean) optimization according to the
methods of Graham Breed, if I have presented it correctly, goes a tad
further in compromising 3:2 (700.408 cents, 1.548 cents narrow) and also 5:4
(383.638 cents, 2.676 cents narrow). The special attractions of this tuning
are yet more accuracy for 7/5 (1.940 cents wide), also overall for ratios of
primes 2-3-7. Thus we have 7/6 at 0.812 cents wide; 9/7 at 2.359 cents
narrow; and 7/4 at 0.736 cents narrow.
Margo Schulter
26 May 2017