------------------------------------------------------------------- Eagle 53: John O'Sullivan's Marveldene Tuning and Some Alternatives ------------------------------------------------------------------- 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