In a post ear­li­er this year, we wrote about a draw­ing John Coltrane gave his friend and men­tor Yusef Lateef, who repro­duced it in his book Repos­i­to­ry of Scales and Melod­ic Pat­terns. The strange dia­gram con­tains the eas­i­ly rec­og­niz­able cir­cle of fifths (or cir­cle of fourths), but it illus­trates a much more sophis­ti­cat­ed scheme than basic major scale the­o­ry. Just exact­ly what that is, how­ev­er, remains a mys­tery. Like every mys­ti­cal explor­er, the work Coltrane left behind asks us to expand our con­scious­ness beyond its nar­row bound­aries. The dia­gram may well show a series of  “mul­ti­plic­i­ties,” as sax­o­phon­ist Ed Jones writes. From the way Coltrane has “grouped cer­tain pitch­es,” writes vibes play­er Corey Mwam­ba, “it’s easy to infer that Coltrane is dis­play­ing a form of chro­mat­ic mod­u­la­tion.” These obser­va­tions, how­ev­er, fail to explain why he would need such a chart. “The dia­gram,” writes Mwam­ba, “may have a the­o­ret­i­cal basis beyond that.” But does any­one know what that is?

Per­haps Coltrane cleared cer­tain things up with his “cor­rect­ed” ver­sion of the tone cir­cle, above, which Lateef also reprint­ed. From this—as pianist Matt Rat­cliffe found—one can derive Giant Steps, as well as “the Star of David or the Seal of Solomon, very pow­er­ful sym­bol­ism espe­cial­ly to ancient knowl­edge and the Afro­cen­tric and even­tu­al­ly cos­mic con­scious­ness direc­tion in which Coltrane would ulti­mate­ly lead on to with A Love Supreme.”

Sound too far out? On the oth­er side of the epis­te­mo­log­i­cal spec­trum, we have physi­cist and sax play­er Stephon Alexan­der, who writes in his book The Jazz of Physics that “the same geo­met­ric prin­ci­ple that moti­vat­ed Einstein’s the­o­ry was reflect­ed in Coltrane’s dia­gram.” Like­wise, sax­o­phon­ist Roel Hol­lan­der sees in the tone cir­cle a num­ber of math­e­mat­i­cal prin­ci­ples. But, remain­ing true to Coltrane’s syn­the­sis of spir­i­tu­al­i­ty and sci­ence, he also reads its geom­e­try accord­ing to sacred sym­bol­ism.

In a detailed explo­ration of the math in Coltrane’s music, Hol­lan­der writes, “all ton­ics of the chords used in ‘Giant Steps’ can be found back at the Cir­cle of Fifths/Fourths with­in 2 of the 4 aug­ment­ed tri­ads with­in the octave.” Exam­in­ing these inter­lock­ing shapes shows us a hexa­gram, or Star of David, with the third tri­ad sug­gest­ing a three-dimen­sion­al fig­ure, a “star tetra­he­dron,” adds Hol­lan­der, “also known as ‘Merk­a­ba,” which means “light-spir­it-body” and rep­re­sents “the inner­most law of the phys­i­cal world.” Do we actu­al­ly find such heavy mys­ti­cal archi­tec­ture in the Coltrane Circle?—a “’divine light vehi­cle’ alleged­ly used by ascend­ed mas­ters to con­nect with and reach those in tune with the high­er realms, the spirit/body sur­round­ed by counter-rotat­ing fields of light (wheels with­in wheels)”?

As the occult/magical/Kabbalist asso­ci­a­tions with­in the cir­cle increase—the numerol­o­gy, divine geom­e­try, etc.—we can begin to feel like Tarot read­ers, join­ing a col­lec­tion of ran­dom sym­bol­ic sys­tems togeth­er to pro­duce the results we like best. “That the dia­gram has to do with some­thing,” writes Mwam­ba, “is not in doubt: what it has to do with a par­tic­u­lar song is unclear.” After four posts in which he dis­sects both ver­sions of the cir­cle and pon­ders over the pieces, Mwan­da still can­not defin­i­tive­ly decide. “To ‘have an answer,’” he writes, “is to direct­ly inter­pret the dia­gram from your own view­point: there’s a chance that what you think is what John Coltrane thought, but there’s every chance that it is not what he thought.” There’s also the pos­si­bil­i­ty no one can think what Coltrane thought.

The cir­cle con­tains Coltrane’s musi­cal exper­i­ments, yet can­not be explained by them; it hints at the­o­ret­i­cal physics and the geom­e­try of musi­cal com­po­si­tion, while also mak­ing heavy allu­sion to mys­ti­cal and reli­gious sym­bol­ism. The musi­cal rela­tion­ships it con­structs seem evi­dent to those with a firm grasp of the­o­ry; yet its strange intri­ca­cies may be puz­zled over for­ev­er. “Coltrane’s cir­cle,” writes Fae­na Aleph, is a “man­dala,” express­ing “pre­cise­ly what is, at once, both para­dox­i­cal and obvi­ous.” Ulti­mate­ly, Mwam­ba con­cludes in his series on the dia­gram, “it isn’t pos­si­ble to say that Coltrane used the dia­gram at all; but explor­ing it in rela­tion to what he was say­ing at the time has led to more under­stand­ing and appre­ci­a­tion of his music and life.”

The cir­cle, that is, works like a key with which we might unlock some of the mys­ter­ies of Coltrane’s lat­er com­po­si­tions. But we may nev­er ful­ly grasp its true nature and pur­pose. What­ev­er they were, Coltrane nev­er said. But he did believe, as he tells Frank Kof­sky in the 1966 inter­view above, in music’s abil­i­ty to con­tain all things, spir­i­tu­al, phys­i­cal, and oth­er­wise. “Music,” he says, “being an expres­sion of the human heart, or of the human being itself, does express just what is hap­pen­ing. The whole of human expe­ri­ence at that par­tic­u­lar time is being expressed.”

Relat­ed Con­tent:

Coltrane’s “Giant Steps” Ani­mat­ed (Part II)

A New Mur­al Pays Trib­ute to John Coltrane in Philadel­phia

The Secret Link Between Jazz and Physics: How Ein­stein & Coltrane Shared Impro­vi­sa­tion and Intu­ition in Com­mon

The His­to­ry of Spir­i­tu­al Jazz: Hear a Tran­scen­dent 12-Hour Mix Fea­tur­ing John Coltrane, Sun Ra, Her­bie Han­cock & More

Josh Jones is a writer and musi­cian based in Durham, NC. Fol­low him at @jdmagness

“These go up to eleven,” Spinal Tap famous­ly said of the ampli­fiers that, so they claimed, took them to a high­er lev­el in rock music. But the work of Aus­tri­an com­pos­er Arnold Schoen­berg, one of the best-known fig­ures in the his­to­ry of avant-garde music, went up to twelve — twelve tones, that is. His “twelve-tone tech­nique,” invent­ed in the ear­ly 1920s and for the next few decades used most­ly by he and his col­leagues in the Sec­ond Vien­nese School such as Alban Berg, Anton Webern, and Hanns Eisler, allowed com­posers to break free of the tra­di­tion­al West­ern sys­tem of keys that lim­it­ed the notes avail­able for use in a piece, instead grant­i­ng each note the same weight and mak­ing none of them cen­tral.

This does­n’t mean that com­posers using Schoen­berg’s twelve-tone tech­nique could just use notes at ran­dom in com­plete atonal­i­ty, but that a new set of con­sid­er­a­tions would orga­nize them. “He believed that a sin­gle tonal­i­ty could include all twelve notes of the chro­mat­ic scale,” writes Brad­ford Bai­ley at The Hum, “as long as they were prop­er­ly orga­nized to be sub­or­di­nate to ton­ic (the ton­ic is the pitch upon which all oth­ers are ref­er­enced, in oth­er words the root or axis around which a piece is built).” The math­e­mat­i­cal rig­or under­ly­ing it all required some expla­na­tion, and often math­e­mat­i­cal and musi­cal con­cepts — math­e­mat­ics and music being in any case inti­mate­ly con­nect­ed — become much clear­er when ren­dered visu­al­ly.

Hence Schoen­berg’s twelve-tone wheel chart pic­tured at the top of the post, one of what Arnold Schoen­berg’s Jour­ney author Allen Shawn describes as “no few­er than twen­ty-two dif­fer­ent kinds of con­trap­tions” — includ­ing “charts, cylin­ders, book­lets, slide rules” — “for trans­pos­ing and deriv­ing twelve-tone rows” need­ed to com­pose twelve-tone music. (See the slide ruler above too.) “The dis­tinc­tion between ‘play’ and ‘work’ is already hard to draw in the case of artists,” writes Shawn, “but in Schoen­berg’s case it is espe­cial­ly hard to make since he brought dis­ci­pline, orig­i­nal­i­ty, and play­ful­ness to many of his activ­i­ties.” These also includ­ed mak­ing spe­cial play­ing cards (two of whose sets you can see here and here) and even his own ver­sion of chess.

As Shawn describes it, Koali­tion­ss­cach, or “Coali­tion Chess,” involves “the armies of four coun­tries arrayed on the four sides of the board, for which he designed and con­struct­ed the pieces him­self.” Instead of an eight-by-eight board, Coali­tion Chess uses a ten-by-ten, and the pieces on it “rep­re­sent machine guns, artillery, air­planes, sub­marines, tanks, and oth­er instru­ments of war.” The rules, which “require that the four play­ers form alliances at the out­set,” add at least a dimen­sion to the age-old stan­dard game of chess — a form that, like tra­di­tion­al West­ern music, human­i­ty will still be strug­gling to mas­ter decades and even cen­turies hence. But appar­ent­ly, for a mind like Schoen­berg’s, chess and music as he knew them weren’t near­ly chal­leng­ing enough.

Relat­ed Con­tent:

Vi Hart Uses Her Video Mag­ic to Demys­ti­fy Stravin­sky and Schoenberg’s 12-Tone Com­po­si­tions

The Avant-Garde Project: An Archive of Music by 200 Cut­ting-Edge Com­posers, Includ­ing Stravin­sky, Schoen­berg, Cage & More

Inter­views with Schoen­berg and Bartók

John Coltrane Draws a Pic­ture Illus­trat­ing the Math­e­mat­ics of Music

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities and cul­ture. He’s at work on the book The State­less City: a Walk through 21st-Cen­tu­ry Los Ange­les, the video series The City in Cin­e­ma, the crowd­fund­ed jour­nal­ism project Where Is the City of the Future?, and the Los Ange­les Review of Books’ Korea Blog. Fol­low him on Twit­ter at @colinmarshall or on Face­book.