--------------------------------------------------- The Noble Mediant: Complex ratios and metastable musical intervals --------------------------------------------------- by Margo Schulter and David Keenan --------------------------------------------------- Posted to the tuning list 17-Sep-2000 Last updated 12-Oct-2007 One feature of European music of the 13th and 14th centuries using Pythagorean tuning with pure fifths, is a predilection for thirds and sixths with complex ratios. This is also a feature of "neo-Gothic" temperaments in the most characteristic range between Pythagorean and 17-tET, with fifths somewhat wider than pure. Pythagorean major and minor thirds at the large integer ratios of 81:64 (407 cents) and 32:27 (294 cents), and also neo-Gothic thirds at or near 14:11 (417 cents) and 13:11 (289 cents), are prized for their complex and active quality.[1] They often serve as points of directed tension or instability, resolving to stable 3-limit sonorities with pure or near-pure fifths and fourths. Since for many modern readers, as opposed to medieval or neo-Gothic theorists, an integer ratio may imply an interval simple enough to be tuned by ear through a "locking-in" of the partials, we emphasize that no such implications necessarily attach to the larger integer ratios mentioned in this paper.[2] In a dialogue on the role of these intervals in Gothic and neo-Gothic music, we were both intrigued by the concept that they may draw their appeal precisely from their complexity. We conceive of major thirds in regular tunings between Pythagorean and 17-tET as located on a gently rounded plateau region between the notch-like valleys of 5:4 and 9:7. Minor thirds are likewise situated on a plateau of complexity between the simpler 7:6 and 6:5. The central portions of such plateaux might be called regions of maximal "harmonic entropy" (Paul Erlich) or "sensory dissonance" (William Sethares) or "complexity" [M.S.].[3] Even while we were engaged in an absorbing dialogue on these plateau regions and their mathematical and musical nature, Keenan Pepper, in a delightful synchronicity, posted an article to the Tuning List[4] on the application of the Golden Ratio or Phi to another area of music: the generation of scales with particular relationships between scale steps. One of us recognized that Pepper's Phi-related function could also be applied to the problem of finding the region of maximum complexity between two simpler ratios, providing a shortcut to the longer process of successive approximation by iterating mediants. In what follows, we show how this function, here termed the "Noble Mediant", can be used to locate regions of maximum complexity. Section 1 presents the classic mediant function, and Section 2 the process of finding closer and closer approximations to such regions of maximum complexity. Section 3 shows how the Noble Mediant can simplify this process, yielding results in general agreement with the local maxima of Paul Erlich's "harmonic entropy"[5] or William Sethares' "dissonance"[6]. Both of these must be calculated using complicated computer algorithms and do not admit of a simple closed-form expression like the Noble Mediant. Section 4 briefly considers the classic mediant and Noble Mediant in relation to the tuning of certain common unstable intervals along the most characteristic portion of the neo-Gothic spectrum from Pythagorean to 17-tone equal temperament (17-tET). While our perspectives may somewhat differ, we strongly agree that the Noble Mediant provides new cardinal points of orientation in exploring the subtle shadings of neo-Gothic intervals and tunings. At the same time, we emphasize that the problem of locating plateau regions may be of interest for various musics, and that we welcome the application of the concepts here described, to a variety of styles. --------------------------------------------------- 1. The classic mediant and plateaux of complexity --------------------------------------------------- In locating a region of complexity between two simpler ratios such as 5:4 and 9:7, one useful index is the _mediant_ of the two ratios. For any two ratios i:j and m:n, this mediant is defined as the sum of the two numerators over the sum of the two denominators: (i + m) ------- (j + n) To illustrate this mediant formula, let us apply it to 5:4 and 9:7, finding a size of major third which may be close to the central plateau region of equal remoteness from both valleys: (5 + 9) ------- (4 + 7) This value, as it happens, is identical with the favored 14:11 ratio for a major third in neo-Gothic theory, giving this ratio a new mathematical significance fitting its intriguing musical qualities of instability and complexity in Gothic or neo-Gothic styles. This mediant is very closely approximated, for example, by 46-tET. Finding the mediant of 6:5 and 7:6, two simple ratios or valleys for minor thirds, gives a similar meaning to another favored neo-Gothic ratio: (6 + 7) ------- (5 + 6) This mediant is identical to the neo-Gothic 13:11, closely approximated for example by 29-tET, a leading neo-Gothic tuning. If we wish to make these mediant relationships of complexity explicit, or avoid the assumption that these are to be considered as relative _consonances_ of say an 11-limit or 13-limit just intonation, we can leave 14:11 written as (5+9):(4+7), and likewise 13:11 as (6+7):(5+6). Here we shall refer to this mediant of (i+m):(j+n) as the "classic mediant" to distinguish it from the Phi-based "Noble Mediant" we shall describe below. ----------------------------------------------- 2. Refining our estimates: the Fibonacci series ----------------------------------------------- From one viewpoint, the point of maximum complexity between two simple interval ratios is like the highest point of a gently rounded plateau between valleys, a familiar metaphor in relation to Paul Erlich's and William Sethares' studies and charts. From another viewpoint, it might be compared to the point of equal gravitational attraction between two planets or planetlike bodies such as the Earth and Moon. In physics, if an object were placed at such a point it would be said to be in a _metastable_ state. This is understood to be a special kind of _un_stable state, one which may persist for a very long time, but not forever, since the slightest perturbation of the object will see it eventually tumble all the way to one side or the other. We think the term metastable may also be descriptive of the quality of the corresponding musical intervals. The planetary metaphor suggests a refinement in our process of approximating the point of metastability. Since the Earth is larger than the Moon, and exerts a greater gravitational attraction, we find that the point of equal attraction is actually located somewhat closer to the Moon than to the Earth, roughly at about 3/4 of the way from the Earth to the Moon. Similarly, while both 4:3 and 5:4 are simple or "planetlike" ratios, the 4:3 has a greater degree of simplicity or attraction, so that we might expect the point of maximum complexity or ambiguity to be somewhat closer to 5:4. The classic mediant already gives us this result to some degree. (4+5):(3+4) (or 9:7) is indeed closer to 5:4 than to 4:3. However, in this case we find that 9:7 is itself simple enough to be a weak attractor and greater complexity can be obtained by taking the mediant of 9:7 with the less simple of its predecessors, giving (5+9):(4+7). This is the complex major third known in neo-Gothic theory as the 14:11, at around 417.508 cents. It is instructive to note that this interval is about 31 cents wider than 5:4 (386.3 cents), and about 18 cents narrower than 9:7 (435.1 cents). This position of the classic mediant somewhat closer to the less simple 9:7 fits our intuitive expectation that the region of rough gravitational equality should be closer to the less powerful attractor or "planet." One might feel justified in stopping when the resulting ratio is too complex to be considered an attractor, but if we want the _most_ complex ratio we think the process should be continued. The mediant of 9:7 and 14:11, is known in neo-Gothic theory as 23:18, and is located around 424.4 cents, or about 38 cents from 5:4 and 11 cents from 9:7. The major third of 17-tET, at 423.5 cents, is quite close to this intermediate ratio. As we progress through the successive mediants, our approximations gradually converge toward a limit about which they oscillate more and more closely. At this stage we will drop the 4:3 and consider the series to have begun with the last two attractors to appear, 5:4 and 9:7, and we look at the pattern of successive mediants. ---------------------------------------------------------------------- Mediant Ratio Cents Dist from: 5:4 9:7 ---------------------------------------------------------------------- (5+9):(4+7) 14:11 417.5 +31.2 -17.6 (9+14):(7+11) 23:18 424.4 +38.1 -10.7 (14+23):(11+18) 37:29 421.8 +35.5 -13.3 (23+37):(18+29) 60:47 422.8 +36.4 -12.3 (37+60):(29+47) 97:76 422.4 +36.1 -12.7 (60+97):(47+76) 157:123 422.53 +36.21 -12.55 (97+157):(76+123) 254:199 422.47 +36.15 -12.61 . . . . . ---------------------------------------------------------------------- Curiously, as it happens, the region between 5:4 and 9:7 seems to resemble the Earth-Moon system in that the point of gravitational parity appears to be situated about 3/4 of the way (in a logarithmic sense) from the more powerful to the less powerful attractor. As we progress through this series of approximations, our values for this central plateau region of maximal complexity approach convergence at around 422.5 cents, or about 1 cent narrower than the 17-tET major third. If the sides of the ratios are considered separately, each may be seen to be a series of integers where every number after the second is the sum of the preceding two numbers. We say they are Fibonacci-like. Originally designed as a model for the reproduction of rabbits, the Fibonacci series begins with the first two numbers 1, 1 -- each new member of the series then being equal to the sum of the previous two members: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 ... This famous number series was interestingly described by Leonard Fibonacci of Pisa around 1200, the same era in which the composer Perotin and his colleagues made composition for three and four voices a regular practice. In his _Liber Abaci_, Fibonacci also introduced the decimal system of Arabic numerals to Gothic Europe. To apply this famous series to our problem of finding the region of maximum complexity or gravitational balance between two simpler intervals, we begin with a "weighted" version of the formula for the classic mediant (Section 1.1), where i:j is the simpler ratio and m:n the less simple ratio or less strong "attractor": (xi + ym) --------- (xj + yn) Here x and y are weights for the two ratios, with the second or "y" term being applied to the _less simple_ ratio. We show what happens when the weights are chosen to be successive members of the Fibonacci series. This will be easier to understand if we apply it to our problem of the zone of gravitational parity between 5:4 and 9:7, stepping through the first few approximations near the beginning of the Fibonacci series. Our first two Fibonacci numbers are 1, 1, so that x=1 and y=1. This gives a result identical to the classic mediant: (1*5 + 1*9) (5 + 9) ----------- = ------- = 14:11 (1*4 + 1*7) (4 + 7) Our next pair of Fibonacci numbers is 1, 2, giving us x=1, y=2. Note that the larger value for y gives the less simple ratio more "weight," bringing our estimate of the point of maximum complexity somewhat closer to 9:7. (1*5 + 2*9) (5 + 18) ----------- = -------- = 23:18 (1*4 + 2*7) (4 + 14) Our next Fibonacci pair of 2, 3 gives us x=2, y=3: (2*5 + 3*9) (10 + 27) ----------- = --------- = 37:29 (2*4 + 3*7) (8 + 21) As we progress through the successive Fibonacci pairs for our weights x and y, we reproduce exactly the same successive approximations as obtained above by iterating the mediant. If we apply the same series of Fibonacci weights to the 6:5 (315.6 cents) and 7:6 (266.9 cents), the 6:5 having the more powerful gravitational attraction (or being situated in a deeper "valley"), we get these results, starting with the classic mediant (6+7:5+6) where x=1, y=1: ---------------------------------------------------------------------- Fibonacci weights Wtd mediant Cents Dist from: 6:5 7:6 ---------------------------------------------------------------------- x=1, y=1 (6+7):(5+6) 289.210 -26.432 +22.339 x=1, y=2 (6+14):(5+12) 281.358 -34.283 +14.487 x=2, y=3 (12+21):(10+18) 284.447 -31.194 +17.576 x=3, y=5 (18+35):(15+30) 283.281 -32.360 +16.410 x=5, y=8 (30+56):(25+48) 283.728 -31.913 +16.857 x=8, y=13 (48+91):(40+78) 283.557 -32.084 +16.687 x=13, y=21 (78+147):(65+126) 283.623 -32.018 +16.752 . . . . . ---------------------------------------------------------------------- Here the terms appear to be converging on a region around 283.6 cents, or roughly 2/3 of the way from the wider 6:5 to the narrower 7:6. The central plateau of maximum complexity or gravitational parity is again closer to the shallower valley, or the planet with the less powerful attraction. On the spectrum of regular neo-Gothic tunings, this region of maximal complexity for the minor third is located between 46-tET (287.0 cents) and 17-tET (282.4 cents). ---------------------------------------------------------------- 3. A new index of complexity: the Noble Mediant ---------------------------------------------------------------- As we have seen, the Fibonacci series of values for x and y beginning with the classic mediant (x=1, y=1) offers us closer and closer approximations converging on a limit which may indicate the region of maximal complexity between two simpler intervals. We can simplify the process by directly finding this limit itself, here termed the "Noble Mediant," using a function like that applied to the different area of scale generation by Keenan Pepper (see note 4). As terms of the Fibonacci series grow larger and larger, the ratio between any two successive terms converges on a value known as Phi, or the Golden Ratio. Phi has the property: 1 ----- = Phi - 1 or Phi^2 - Phi = 1 Phi As a solution of the above quadratic one finds that Phi is (sqrt(5)+1)/2 or approximately 1.61803398874989484820459. Thus we can find the Noble Mediant for the region of maximum complexity between two simpler intervals by setting x=1, y=Phi. For two such interval ratios i:j and m:n where i:j is the simpler ratio: (i + Phi * m) NobleMediant(i/j, m/n) = ------------- (j + Phi * n) For a maximally complex major third between 5:4 and 9:7, or a maximally complex minor third between 6:5 and 7:6, our new Phi-based function yields these results: ---------------------------------------------------------------------- Intervals Noble Mediant Cents Dist from: i:j m:n i:j m:n (i + m Phi):(j + n Phi) ---------------------------------------------------------------------- 5:4 9:7 (5 + 9 Phi):(4 + 7 Phi) 422.5 +36.2 -12.6 ---------------------------------------------------------------------- 6:5 7:6 (6 + 7 Phi):(5 + 6 Phi) 283.6 -32.0 +16.7 ---------------------------------------------------------------------- It is interesting to compare these results, and some others of relevance to neo-Gothic music, with Paul Erlich's values to the nearest cent for the regions of "maximum entropy" or complexity between these interval pairs: ---------------------------------------------------------------------- Intervals Measure Cents Dist from: i:j m:n i:j m:n ---------------------------------------------------------------------- 5:4 9:7 classic mediant 417.5 +31.2 -17.6 Noble Mediant 422.5 +36.2 -12.6 Erlich 423 +37 -12 ---------------------------------------------------------------------- 6:5 7:6 classic mediant 289.2 -26.4 +22.3 Noble Mediant 283.6 -32.0 +16.7 Erlich 285 -31 +18 ---------------------------------------------------------------------- 5:3 12:7 classic mediant 918.6 +34.3 -14.5 Noble Mediant 923.0 +38.7 -10.1 Erlich 924 +40 -9 ---------------------------------------------------------------------- 7:4 9:5 classic mediant 996.1 +27.3 -21.5 Noble Mediant 1001.6 +32.8 -16.0 Erlich 999 +30 -19 ---------------------------------------------------------------------- 5:4 6:5 classic mediant 347.5 -38.9 +31.8 Noble Mediant 339.3 -47.0 +23.7 Erlich 348 -38 +32 ---------------------------------------------------------------------- 9:7 4:3 classic mediant 454.2 -43.8 +19.1 Noble mediant 448.5 -49.6 +13.4 Erlich 457 -41 +22 ---------------------------------------------------------------------- For the first three pairs of "valley" or "planet" intervals, separated by the ratio of 36:35 (5:4-9:7; 6:5-7:6; 5:3-12:7), the Noble Mediant and the maxima of Erlich's harmonic entropy coincide within 1-2 cents. For the similar pair 7:4-9:5, they differ by about 3 cents, with Erlich's point of maximum entropy about midway between the classic mediant (the Pythagorean minor seventh at 16:9) and the Noble Mediant. For the pair 5:4-6:5, where a zone of maximum complexity or "ambiguity" might be expected to fall around the 11:9 "neutral third" (the classic mediant), Erlich's 348 cents virtually coincides with this mediant, while the Noble Mediant at 339.344 cents is decidedly closer to the less simple 6:5 ratio. For the pair 9:7-4:3, where 13:10 is the classic mediant, in an area of complexity or ambiguity where large major thirds begin to shade toward narrow fourths, the Noble Mediant leans more toward the less simple 9:7, while Erlich's 457 cents leans more toward the fourth. Intonationally complex intervals of all these varieties may occur in neo-Gothic styles. The regular thirds and sixths, as in historical Gothic music, play vital cadential and coloristic roles in various unstable sonorities. Minor sevenths at or near 16:9, conceived of not as especially "complex" intervals but rather as comparatively simple ones derived from two pure or near-pure 4:3 fourths, may play similar roles. The two remaining interval types are less conventional "special effects" categories. Regular neo-Gothic tunings in the range from around 29-tET to 17-tET feature diminished fourths or alternative major thirds (372.4-352.9 cents) and augmented seconds or alternative minor thirds (331.0-352.9 cents) offering various intermediate shadings converging on the "neutral third" of 17-tET. In 29-tET, the interval of the "wide major third" at 11/29 octave, 455.2 cents, is also a "special effects" interval in the zone of ambiguity where such thirds approach the region of narrow fourths; this interval is close to the classic mediant or Erlich's "entropy" maximum. In addition to providing these comparisons and inviting readers to perform their own listening tests, we suggest that Erlich's algorithm might usefully be modified to use the Noble Mediant where it currently uses the classic mediant or his more recent "limit-weighted midpoint". Of course, having put Noble Mediants into the algorithm, we should not then be surprised if we obtain Noble Mediants out of it. In the case of Sethares' algorithm we find that the position of the local maxima are too dependent on the parameters of the model to permit any detailed comparison. In particular, one can (and must) specify the timbre being used. Like Erlich's model, ours (which is a rule-of-thumb rather than a model) is intended to apply only to "typical" harmonic timbres. In such cases we expect we would be in general agreement with Sethares. ----------------------------------------------------- 4. Shadings of complexity and the neo-Gothic spectrum ----------------------------------------------------- Both the historical Gothic music of Europe based on a pure Pythagorean tuning, and also neo-Gothic temperaments in the most characteristic range of Pythagorean to 17-tET, feature fifths at or reasonably close to pure and complex thirds and sixths. As the historian Carl Dahlhaus has written, such complexity in a Gothic setting fits the role of thirds and sixths with their "factor of instability."[7] Using the classic mediant and Noble Mediant together with Erlich's values for regions of maximum entropy, we can briefly survey the subtle shadings of complexity along the spectrum from Pythagorean to 17-tET. Here we sample three categories of unstable intervals: major and minor thirds, and major sixths. --------------------------------------------------------------------- Interval Classic Mediant Noble Mediant Erlich --------------------------------------------------------------------- M3 (5:4-9:7) 417.5 422.5 423 ..................................................................... Pythagorean 407.8 - 9.7 -14.7 -15 29-tET 413.8 - 3.7 - 8.7 - 9 46-tET 417.4 - 0.1 - 5.1 - 6 17-tET 423.5 + 6.0 + 1.0 0.5 --------------------------------------------------------------------- m3 (6:5-7:6) 289.2 283.6 285 ..................................................................... Pythagorean 294.1 + 4.9 +10.9 + 9 29-tET 289.7 + 0.4 + 6.4 + 5 46-tET 287.0 - 2.3 + 3.7 + 2 17-tET 282.4 - 6.9 - 0.9 - 3 --------------------------------------------------------------------- M6 (5:3-12:7) 918.6 923.0 924 ..................................................................... Pythagorean 905.9 -12.8 -17.2 -18 29-tET 910.3 - 8.3 -12.7 -14 46-tET 913.0 - 5.6 -10.0 -9 17-tET 917.6 - 1.0 - 5.4 -6 --------------------------------------------------------------------- For the major and minor thirds, Pythagorean intervals are located on the portion of a plateau with a shading somewhat closer to that of the simpler or more strongly attracting "valley" or "planet": the 4:5 or 5:6 rather than the 7:9 or 6:7. In the especially characteristic portion of the neo-Gothic range from around 29-tET to slightly beyond 46-tET, these intervals are at or near their classic mediant values, (9+5):(7+4) and (6+7):(5+6), the celebrated 14:11 and 13:11 of neo-Gothic theory. Around 17-tET, these intervals approach the point of maximum complexity as defined either by the Noble Mediant or by Erlich's statistical model. For major sixths, we remain on the portion of the plateau somewhat closer to 3:5 than the classic mediant until around 17-tET, and to reach the Noble Mediant or Erlich's region of maximal entropy, we would need to temper the fifth by 5.7 cents. This is almost exactly the fifth of 39-tET (707.7 cents), a tuning in what is termed the "far neo-Gothic" zone beyond the characteristic range of Pythagorean to 17- tET. Fine distinctions of shading within a plateau region may be reflected in descriptions of 29-tET as "gentle," and tunings around 46-tET also as "mild" in comparison to the "stronger" or more "avant-garde" 17- tET.[8] Since major and minor thirds in 29-tET or 46-tET are close to the classic mediants, while 17-tET thirds closely approximate the Noble Mediants or Erlich's regions of maximal entropy, it would appear that the choice between shades of complexity is a matter of musical discretion and taste. --------------------- Update (October 2007) --------------------- After the initial publication of this article, Kraig Grady kindly referred us to some earlier work by Lorne Temes [9], and the diagram by Erv Wilson showing the noble numbers in relation to the Stern- Brocot tree [10], and more recently, some comments by Erv Wilson on Lorne Temes' Phi neutral sixth (833.1 cents). "... it is the worstest of the worst -- and yet somehow with divinity imbued, Lord have mercy!" [11] One of the authors [M.S.] finds the Phi-sixth, a kind of "superminor sixth" in a neo-Gothic setting, to be not so much "dissonant" as often charmingly "vague" or "diffuse," evoking a certain association with Debussy (e.g. _Nuages_) and Impressionism as they might be realized in some parallel universe of intonation. ----------------------- Update (September 2022) ----------------------- It is important to note that the noble mediant is not simply a phi- weighted mediant. There are two important differences: 1. The order of the arguments is irrelevant in the case of the noble mediant. The phi weight always goes to the more complex ratio. 2. The noble mediant is only defined for pairs of ratios where, if you cross-multiply and subtract, you obtain +-1. The result of a phi- weighted mediant is not a noble number unless this condition is satisfied. This has been called the "unimodular" requirement, as the calculation is like the determinant of a 2x2 matrix, and a matrix is called unimodular if its determinant is +-1. To summarise: When i/j is the simpler ratio, NobleMediant(m/n, i/j) = / (i + Phi * m) | ------------- if |in - jm| = 1; NobleMediant(i/j, m/n) = < (j + Phi * n) | \ undefined otherwise. ----- Notes ----- 1. In the interests of familiarity, D.K. has agreed to use, in this paper, the convention of placing the larger number first in ratios for musical intervals, despite his objections as outlined in https://dkeenan.com/Music/ANoteOnNotation.htm. 2. On the basis of experiment, D.K. asserts that, under ordinary conditions, the bare dyads 14:11 and 13:11, and likewise the Pythagorean 81:64 and 32:27, are not directly recognizable or tuneable by ear, any more than are tempered intervals at nearby locations on the continuum. This is in contrast to simpler ratios such as 5:4, 6:5, 9:7 and 7:6, and possibly such as 11:7 and 11:8. 3. The term complexity is used in this paper to mean both (a) the complexity of the ratio as given (e.g.) by the product of its two sides when in lowest terms, and (b) the way an interval sounds to us. We must point out that these do not always correspond, as Paul Erlich's example of 3001:2001 makes clear. 4. Keenan Pepper, "The Other Noble Fifth," Tuning Digest [TD] 794:8, 10 September 2000. 5. See, for example, http://www.tonalsoft.com/sonic-arts/td/erlich/entropy.htm, including a table of "entropy maxima" quoted in this article. Erlich's model proposes that there is a kind of probability curve that a listener's auditory system will perceive any given interval as fitting one of a set of more or less simple integer ratios. Thus in the historical meantone range near 5:4, a major third is very likely to be perceived as this ratio; around 9:7, as in some of Erlich's music based on 22-tET, recognition is also likely, although this is a shallower "valley," or less strongly attractive "planet." On the neo- Gothic plateau between these valleys or planets, such identifications would seem very problematic, giving major thirds a complex and intriguing quality. 6. William Sethares, _Relating Timbre and Tuning_, https://eceserv0.ece.wisc.edu/~sethares/consemi.html 7. Carl Dahlhaus, _Studies on the Origin of Harmonic Tonality_ (trans. Robert O. Gjerdingen), Princeton: Princeton University Press, 1990, p. 187. Dahlhaus specifically mentions "the complicated Pythagorean proportions 64:81 and 27:32" for major and minor thirds, emphasizing that this tuning "should ... be understood as a musical phenomenon rather than as a mathematically motivated acoustical defect," ibid. p. 178. 8. This subjective contrast may reflect not only the tuning of regular intervals, but the differing qualities of diminished fourths and augmented seconds. In the more "avant-garde" 17-tET, these are identical neutral thirds, with cadential resolutions differing radically from a usual Gothic flavor; the spectrum of intermediate thirds from around 29-tET to 46-tET may involve a less dramatic contrast with other elements of intonation and style. 9. Lorne Temes, "Golden Tones?" in a letter to Erv Wilson, 4 January 1970. https://anaphoria.com/temes.PDF 10. Erv Wilson, "Scale Tree", 1994. https://anaphoria.com/sctree.PDF 11. Erv Wilson, "Radical Just-Unjust property of 72-ET", 28 June 2002. https://anaphoria.com/tres.PDF (second last page)