Lately I've been trying to learn a little bit about music notation and theory (both for the sake of better understanding Brian's work and just in general). It's been pretty cool learning about Pythagoras, the fact that note intervals are determined by ratios of prime numbers and that there's no perfect or universal tuning system. From the different types of diatonic scales (like Phrygian and Locrian) to the alternatives to the chromatic scale's equal division of the octave (EDO) like 19, 31 and 53 step scales.
I got it in my head one day that if a person only had access to all 4 of these "perfect" sclaes (12, 19, 31, 53) at once, in a convenient singular instrument, they'd have the power to make any kind of music they wanted, unlimited by the ergonomics our ancestor's settled for when 12-notes became the standard. So that led me to "invent" a new instrument and I was wondering if others with more musical training than me think it's a good idea or not.
NEW INSTRUMENT
In this vein, I was thinking it might be interesting to design a "super-keyboard" with 4 rows of keys, in ascending order: 96 keys of 12-EDO (8 octaves), 95 keys of 19-EDO (5 octaves), 93 keys of 31-EDO (3 octaves), then 106 keys of 53-EDO (2 octaves). The 53-EDO keys would be presented in 3 tessellating "sub-rows" of hexagon-shapes (as opposed to the usual rectangles) of 35/36/35. Since this would realistically be an electronic/digital instrument, there's no reason the octave-range of the higher three sets can't be shifted between uses (or have any assortment of keys mapped to whatever frequencies you want, really) but ergonomically this layout makes sense and I like how you lose 3, then 2, then 1 octave as you go up to the more complicated scales. (Although, I've also thought maybe the top 53-EDO keyboard might be expanded to as many as 5-octaves in rows of 88, 89, 88 if they'd fit in a roughly even space, but I worry this may be *too* complicated for the player.)
ASIDE: There's a noticeable relationship between these scales. 53x0.585= 31.005, 31x0.585= 18.14, 19x0.585= 11.12, 12x0.585= 7.02, 7x0.585= 4.095 (close enough to round up to 4.1). If there's a non-0 value in the tenths place, it's significant enough of a figure to round up to the next whole number. (All these values are simplified.) There's a near relationship with electron orbitals as well (pentatonic scale is three more than the K shell (or s subshell), diatonic is one more than the p subshell and one less than the L shell. Then the chromatic is two more than the d subshell OR K+L shells, 19-EDO is one more than the M shell, 31-EDO is one less than the N shell, 53-EDO is three more than the O shell. This pattern would completely break if it continued just one more time, where the electron formula 2(n)2 would equal 72 while 53/0.585 would demand a ~91-EDO scale, which surprised me because you never hear of such a thing. In fact, besides 72-EDO, the next one with even its own wikipedia page is all the way at 96 notes.) The keys could all be colored, with "rainbows" repeating within each octave to help musicians keep track at a glance between 4 sets of keys how everything "lines up." The "white keys" might go from one "major color" to the other (so, on 12-EDO with 7 "whites" per octave, this becomes ROYGBIV in that order) while the "black keys" now present as an "off-rainbow" going the other direction (12-EDO with 5 "blacks" is now blue-violet/turquoise/chartreuse/yellow-orange/orange-red in that order). In 19 through 53-EDO, the intervals of colors get finer and more specific, but the "main hues" still line up roughly so one can ascertain the enharmonic (or at least nearest equivalent) keys between rows. Perhaps each new octave in each individual row might increase in tint or shade (with "brighter" versions of the same hue representing higher octaves and darker keys in lower octaves).
The entire "console" could be programmed and reprogrammed either natively or via computer hookup to map keys with digitized instruments and allow the user to tune systems for exotic xenharmonic scales (see below). There could be a mode to highlight (illuminate) the enharmonic keys across all four rows as a guide, so users only familiar with 12-EDO can intuitively learn the other systems. Perhaps the console could be programmed to automatically play enharmonics across the four rows every time the user presses any key or, in higher-order scales, it could play the nearest bordering keys to the pressed-key's frequency, creating a "shimmering" effect. (So the operator plays a chromatic note and rather than the exact enharmonic on 53-EDO, the computer plays the adjacent ascending and descending notes to "enhance" the sound in a cross-scale chord). Maybe pre-set chords could be specified, so playing a single note in one scale automatically triggers a pleasing chordal grouping in the other scales. This would mean "single-finger chords" that simplify the efforts of the player and free their remaining digits for increasingly stacked chord structures. (There might even be a "always play the opposite color from the color wheel" or "always create 'net-purple'" settings that prioritize key combos by their color mapping than any established musical relationship. Or always preferring certain letter-groupings or whatever random stipulation.)
You could call this instrument the "Telechord" or "Chromachord" or "Parachord." (It looks like each of these names is used for some obscure thing, ought to be easy enough to overshadow and just use whichever is least established...or some other name it doesn't matter.) Maybe add -ion to the end of any of these words if they're trademarked and it must be done. If it has to be something completely totally unique, call it the "Theochord" and be done with it.
NEW HARMONIC SYSTEMS & POSSIBILITIES FOR CHORDS
Then we can use this tool to look for the music in mathematical and physical reality.
1) Perhaps one might express transcendental numbers like Pi and e in a musical notation system where the number is written in base-12, base 19, base-31 and base-53 and each digit is mapped to a note and played in sequence (allowing liberties to be taken with rhythm, clumping subsequent digits into chords vs single notes, as well as interval-length for what sounds good...if anything even does). Or transcendental numbers (and equations) might be expressed musically by rewriting every individual integer (or even every digit) in terms of its prime factorization and using these ratios to determine what notes should be played.
2) Maybe instead of prime factors, ratios determining the frequencies of notes could be built around that of a triangular number's sum to its product (3:2, 5:6, 10:24. 15:120, 21:720, etc), or one triangular number to another (1:3, 3:6, 6:10, 10:15, 15:21, 21:28, 28:36 etc). Or Pell numbers (1:1, 3:2, 7:5, 17:12, 41:29, 99:70, 239:169...) which converge on the square root of 2, (musically on the 600-cent tritone). Another might be the pancake numbers (4:7, 7:11, 11:16, 16:22, 22:29, 29:37, 37:46, 46:56, 56:67, 67:79, 79:92, 92:106). Less certain might be tetrahedral numbers, Fibonacci numbers or Hilbert primes. Other methods for deriving alternate scales could involve the revolution speeds of the planets (one ratioed to its neighbor(s)) to make a new diatonic scale, maybe one scale starting with Mercury:Venus and another Neptune:Uranus. Or a new chromatic scale of the ratios of edges between the 13 Archimedean solids. Or the ratios of Ptolemaic stars in each of the zodiac constellations (looping back around to give the 12th note ratio). And different key rows could have different systems, with say the bottom pre-set 12EDO row now using a "planetary scale" while the previously 19-EDO is using a "zodiac scale" and what was once 31-EDO is now a "hexagonal scale," etc. This would provide a way of reaching across different divisions of the octave to find new "superchords." Since the colors of the keys would no longer natively indicate what is enharmonic across rows, the "key highlight/illuminate" feature would be especially helpful here.
3) In the tradition of Pythagoreanism, a system could be developed where the user plays and notates new chords out of mapped polygons in the Circle of Fifths (or Circles of Fourths, Thirds, whatever). Sometimes these might be the same polygon in the same position across all used rows, others the same shape rotated or reflected in some way when played in different rows, perhaps combinations of two or more different polygons entirely. Regular polygons may go as high as decagons, the most complex face used in the Archimedean solids, or else the Isosceles/Scalene triangles and rhombi/kites of the Catalans. (Just to start anyway, then irregular shapes can be tried.)
4) One could use the trigonometric and hyperbolic functions (sine, cosine, tangent, secant, cosecant, cotangent, arcsin, arccos, arctan, arcsec, arccsc, arctan) to graph note/chord progressions across the staves of sheet music. Such as:
a) a repeating rising/falling scale for sin/cos,
b) rising OR falling melodic flourishes for arcsin/arccos and the same with more dramatic slopes with sec/csc,
c) oscillations around a note that's never quite reached with arccsc/arcsin,
d) possibly longer notes sustained then quick semi/quarter tone ascending/descending scales concluding on higher/lower sustained notes with arctan/arccot,
e) rising-cresting-falling counter-melodies with csc/sec,
6) all the hyperbolic offshoots of these. Perhaps chord structures that aren't polygons mapped onto the circles of fifths (or thirds, etc) could be expressed in these terms, reverse-engineering mathematical values from pleasing musical harmonies.
5) The Archimedean spiral and other 2D graphs, patterns or complex numbers might be created by arbitrarily assigning numbers falling in the 4 quadrants ( ++, -+, --, +-) to the different keyboards, or using wave functions (including those in 3D like the Schrodinger equation) to determine the beat (rate of change over time) and pitch-range (in up to three dimensions) of a section of music. These are just rough examples, and the ultimate prize would be Quaternions or 4-part complex numbers (the real plus i, j and k components) that plot 3D shapes with each separate component of the equation/function pertaining to the four keyboards. (Perhaps one user has real-12EDO, i-19, j-31, k-52 or the reverse or something else.)
ULTIMATE PERFORMANCE
Realistically, this instrument has to be digital/electronic or else it would be insanely expensive, spacious and a huge pain to keep in tune. But since this is all a fantasy anyway, what if we built an analog model and constructed a giant cathedral around it as a
Temple of All Harmony (Panharmonicon)? I'm not sure what built-in instruments the four rows of keys would best represent but for now I was thinking organs (vibrating pipes) for 12-EDO, harpsichord (plucked strings) for 19-EDO, piano (struck strings) for 31-EDO and clavichord (bowed strings) for 53-EDO. I figure organs must be more space-intensive and hard to maintain so make the fewest of those, renaissance music tended to use 19-EDO to my knowledge so use a "renaissance-sounding" instrument for that, 31 is considered the perfect scale by many musicologists and it's not like there's many (any?) 31-note pianos out there so I think it's about time, then I figure the "gliding/glimmery/smooth" sound of bowed strings would fit the fine/delicate divisions of 53-EDO well.
In homage to the tetractys, patrons of the Panharmonicon would hear one symphony in two ears/hemispheres, played by triads of "Theochords" with four different "instruments" at their disposal. Shape-chords can't exceed decagons (10) and there's twenty-four total scales (possibly with some tuned to different xenharmonic systems) as the ingredients. One begets two begets three begets four (and all the other numbers you can make from adding/multiplying them) as Pythagoras taught us. The patrons would sit in between two trinities of musicians playing slightly-to-completely different music at once, in order to create stereophonic sound in person. In front of the audience and/or above (like an imax dome) there could be a lightshow created by the sounds. Like, playing a key causes its color-code (remember from earlier) to flash on the "screen" as long as it's pressed. Or if the musicians are playing "polygon chords" those shapes are projected overlapping across the dome. Maybe the sheet music is projected and as notes are played they "explode" and their color expands while diminishing proportionally as its diameter increases. If the musicians are playing equations, those mathematical formulas could flash on the screen. These are just some ideas.
Maybe the dome could have stained glass windows of tessellating shape patterns (pure equilateral triangles/squares/hexagons in some, or penrose tiling, Pythagorean tiling, Pentagonal tiling, Truchet tiles, Girih, Pinwheel tiling, Gilbert tessellation and zellige among others...) that are illuminated from the outside shining in. Hanging from the ceiling are two sets each of the Platonic+Archimedean+Catalan solids and spheres (64 shapes total) as mobiles, with one set mirrored and the other prismatic. These are constantly rising and falling, rotating and revolving (clockwise or counter) to the music as keys are pressed. If this is too limiting for all songs/movements within the set/concerto, it could be limited for just one section and otherwise the shapes move in a predetermined manner or remain stationary.
Every crazy idea for musical composition I've thrown out (new scales/tuning, "playing" equations through a preset number-to-note cipher or prime factorization, quaternions and whatever combinations thereof) could be worked into a wider symphony, either as distinct pieces each following their own unique rules, or as recurrent overlapping leitmotifs in one big concerto. This setup for the ultimate audio-visual extravaganza could be made available to the wider public, for aspiring composer/performers to see what they can do with it. There'd be reigning champions who made the best use of these incredible tools, overthrown only when a more impressive composition comes along. Some visionaries might make use of more impressive math, or keep the 64 mobiles moving by note the entire time in the most delicate dance (where a foul note or one too many causes them to smash into each other), or sound the best or have the best lightshow. It'd be up to the public to determine what combination of these impressive feats they admire most and dub that artist the Master of Harmony. This could be our civilization's version of the sword in the stone (the cantata in the chaos?) and (s)he that is talented, creative, spatially aware, mathematically competent enough to make the best use of the Temple is deemed worthiest to lead (or delegate that task and become their court composer).
This would be the worthiest tribute to Pythagoras one can dream of, and he deserves it as one of the earliest great thinkers in the Western canon. (He's pre-Socrates, pre-Zeno, roughly contemporary with Zoroaster and Buddha, beaten only by Solomon, Anaximander and the Seven Sages of Greece). He was arguably the first to imbue Westerners with a reverence for math and music as the twin languages of God. Music is special as the bridge between "pure" numeric logic and "primal" emotional resonance, the unique combination of which makes us human. As sentience is an emergent property of chemical bonds, symphony is an emergent property of numerical ratios. Amen.
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