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Encyclopedia of Microtonal Music Theory

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tuning terms by Marc Jones

[Joe Monzo]

A set of definitions from Marc Jones at Orphon Soul, Inc. Since it's not clear to me how Marc's terminology does or doesn't converge with currently accepted terms, i've decided to temporarily put all of Marc's definitions on one page.

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[Marc Jones, Orphon Soul, Inc., private communication with Joe Monzo]

The people at Orphon Soul would greatly appreciate being made aware if any of the concepts described herein are already in use with currently accepted terminology; please email tuning@orphonsoul.com. Definitions for concepts that appear to be new may be freely utilized.

"These microtonal definitions are those used since around 1987 by the until-recently mostly self-contained Orphon Soul, Inc., in multimedia research projects headed by Marc Jones."

Quantum Music
A study and practice which, while acknowledging “microtonality” as a fine sense of pitch, focuses equally on building music up from the smallest perceivable elements of pitch, time and volume, among other micro-operational considerations.
VHT - Very High Temperaments
Temperaments between roughly 244 and 999 notes per octave in which adjacent notes in a temperament become indistinguishable, but adjacent temperaments can be seen to still have very contrasting symmetries.
UHT - Ultra High Temperaments
Temperaments above 1000 notes per octave which are used primarily for calculation purposes, save for occasional nodes of convergent interest.
A transcendental etude, developed to cushion the impact of unfamiliar tunings, in which seven segments occur a multiple of seven times over the course of the piece.
Satanic comma
The difference between 665 fifths and 389 octaves, less than 1/10 of a cent, around 1/15878 of an octave. Marc sees this as a possible musical insight into the cryptic quote from the Book of Revelation, citing “the beast” as a man-made number which only someone with understanding could calculate: “It stands to reason that if the 665th turn of the cycle of fifths, modulo octave, is so close to unison as to be imperceptible, and deceive you as unity, then similarly the 666th turn would be able to deceive you as being a true fifth. (coined 1990, as a parody on the name of the syntonic comma.) [Editors note: Until 10 June 2020 this definition erroneously stated "359 octaves" instead of the correct "389 octaves".]
The process of taking a note in one temperament and finding the nearest note in another temperament, by dividing the note number by the original temperament, multiplying by the new temperament, and rounding to the nearest integer. (This has nothing to do with any harmonic interval or anything else either interval may be close to.) [For example,] 7/12 = 9.91667/17... Therefore, the nearest note to 7/12 in 17 is 10/17.
Cross-rounding / Resonance / Intervallic Resonance

If an interval in one temperament rounds to an interval in a second lower temperament, and that second interval rounds back to the first, the two intervals are said to be cross-rounding, and have (intervallic) resonance.

[For example,] 10/17 = 7.058824/12... Therefore, the nearest note to 10/17 in 12 is 7/12. 7/12 = 9.91667/17... Therefore, the nearest note to 7/12 in 17 is 10/17. 10/17 rounds to 7/12. 7/12 rounds back to 10/17. 10/17 and 7/12 are cross-rounding and have intervallic resonance.

12/17 rounds to 8/12. 8/12 rounds to 11/17. 12/17 and 8/12 are not cross-rounding and do not have intervallic resonance.

Since it accurately describes the errors from rounding 12-tone intervals into 7 and back, the term “accidental” is sometimes borrowed from history and extended to describe any note in any temperament which isn’t cross-rounding to another. In this case, 12/17 is an “accidental” of 11/17 in the 12-tone scale in 17.

Hence, the dominant scale in a scale class can be seen as a symptom of rounding. The 7 notes in the 7+5 scale in 12 can be gotten from rounding all 7-tone intervals into 12.

Legal Intervals

In using an equal temperament to represent a list of unequal values, say, just intervals, the “legal” intervals are those in which the difference in representative value between two notes is equal to the value of their difference. This is not to say it is against statute law to play “illegal” intervals, but in the context of LOGIC GRIDS, legal intervals are those allowed by the nature of the grid.

Given this scale, based in D, say in Pythagorean tuning: D Eb E F F# G G# A Bb B C C# D; from the root D, the minor second is Eb, making D-Eb the size of the legal minor second.

The legal minor seconds here would be: E-F, F#-G, G#-A, A-Bb, B-C, C#-D;

And the illegal minor seconds would be: Eb-E, F-F#, G-G#, Bb-B, C-C#; or seven perfect fifths.

Nested Scales
A series of three or more imbedded scale classes, say: 0 - 3 - 8 - 11 - 16 - 19, is actually 5 rounded to 7 rounded to 12 rounded to 19. (At Orphon Soul, written 19\12\7\5.)
Convergence Webs
The distribution of just intervals in equal temperaments implied from nesting scales through applications of the Brun algorithm “to infinity and beyond”. By cross-rounding, in essence to infinity and back, only a few dozen steps and back will yield a convergence to just intervals expressable in terms of addition and subtraction of the intervals used to seed the algorithm. This provides not only a system of just intervals which can be used to seed a logic grid for the temperament it represents, but also a useable, finite system of just intervals.
Preconvergence webs

Cross-rounding the Brun algorithm far enough will yield the convergence web. Not cross-rounding far enough will yield more liberal logic grids because of a lower number of chromatic intervals.

Consider the major chord convergence in the Brun algorithm:
12, 19, 31, 34, 53, 87, 118, etc.

19 has 1 chromatic interval, 1/19.
Cross-rounding through 31, 19 has two chromatic intervals.
Cross-rounding through 34, 19 has three chromatic intervals.
Cross-rounding through 53, 19 has three chromatic intervals.
Cross-rounding through 87, 19 has three chromatic intervals.
Cross-rounding through 118, 19 has four chromatic intervals.
Cross-rounding through 559, 19 has four chromatic intervals.
Cross-rounding through 796622, 19 has four chromatic intervals.

At the 31 point, the logic grid of 19 is identical to meantone, because of the intervallic paradoxes. At 34, it expands to 3. From 118, the convergence web has been defined.

Logic grids

A square array listing the notes of a temperament on the side and on the top, showing under what circumstances the notes are allowed to pass to one another. These can be either true or false grids, or layered from atonality through preconvergence and convergence webs all the way to serialism, in which, only one note is legal after the current one.

Correct me or excuse me if atonality is the wrong word, but I’m speaking of the legal system in which any note may follow any note and any note may be harmonized with any note. The logic grid for these conditions would be filled with “true”.

Consider the major chord convergence in the Brun algorithm:
12, 19, 31, 34, 53, 87, 118, etc.

19 has 1 chromatic interval, 1/19.
Cross-rounding through 31, 19 has two chromatic intervals.
Cross-rounding through 34, 19 has three chromatic intervals.
Cross-rounding through 53, 19 has three chromatic intervals.
Cross-rounding through 87, 19 has three chromatic intervals.
Cross-rounding through 118, 19 has four chromatic intervals.
Cross-rounding through 559, 19 has four chromatic intervals.
Cross-rounding through 796622, 19 has four chromatic intervals.

A layered logic grid would show:

"1" at points where an interval was illegal, save for anything-goes atonality.
"2" at points where the interval was legal in 31.
"3" at points where the interval was legal in 34 through 87.
"4" at points where the interval was legal in terms of the convergence web.
Intervallic paradox

An imagined or forced behavior, in a preconvergence web, in which the difference between the just intervals represented by two notes is different from the interval represented by their difference. Effectively, it is the condition wherein an interval is legal in a precovergence web, but illegal in the convergence web.


In the Brun algorithm, using 2:1, 5:3, and 4:3...

5/22 represents 32:27
2/22 represents 16:15
7/22 represents 5:4.

In the preconvergence web, moving from 5/22 to 7/22 is legal, because between 22 through 34 or 46, it has intervallic resonance; whereas 512:405 is not equal to 5:4. In the convergence web, this is not cross-rounding through 53.

Intervallic paradox describes the state of an interval being legal in a preconvergence but illegal in a convergence. It appears to make sense intervallically because it cross-rounds locally, but harmonically, the math is incorrect.

Diatonic Aperture

The ratio between the minor and major second, describing the "opening" of the minor second from non-existant in 5-tone, to the equi-diatonic of 7-tone.

5: zero (0 to 1)
7: one (1 to 1)
12: 1/2 (1 to 2)
17: 1/3 (1 to 3)
19: 2/3 (2 to 3)

(reference: stern brocot tree)

Cesium Tuning

The second is defined as 9,192,631,770 ticks of a Cesium atom, which is used in atomic clocks. Orphon Soul uses this as the pulse grid of the universe and bases all tunings and tempos on this 9GHz frequency.

Since Orphon Soul also uses D as a tone center, the D string on a guitar is always tuned to 136.98Hz, 24 octaves below the Cesium tick. Strangely enough, in 19, 22, 41, 60, 63, etc this tuning yields a note very close to 440Hz.

The root, or "D" tempo is 128.42 beats per minute, 32 octaves below the tick.

Equalization, sample durations, and synthesizer timings are also considered.


Orphon Soul projects use D, not C, as a root for calculating scale values, used because of the symmetry:

[scalar order:]        A B C ((D)) E F G: D is the center.

[chain-of-4ths order:] B E A ((D)) G C F: D is the center.

[with accidentals:]    B# E# A# ((D#)) G# C# F#
                       B  E  A  ((D))  G  C  F
                       Bb Eb Ab ((Db)) Gb Cb Fb


This is used to easily calculate interval names and their inversions, so as to not have to constantly bargain with the off-balance accidentals.


A term suggested by Fred Freeburg, circa 1990, equivalent to equal temperament, ED2, tET, EDO etc. originally in lieu of the fact that Marc Jones uses so many equal temperaments. After Fred being handed new fretboards all the time, the routine arose, "oh, what Jones is this now? 12 Jones per octave? 19 Jones?" instead of "tones".

Later renewed in lieu of the discovery of Marc’s multiplicity, in that instead of trying to remember who each persona is and which temperament they relate to, omniously enough by the coining, it would be easier to remember them as "17 Jones", "19 Jones", etc.

Originally abbreviated J. Also Js or Jz, parodying Hertz (Hz). Also taken as the rebus, Jones = J + one + s, "J1s". With JI as an abbrev for "just intontation", this could look a bit like "just ones"

12-tET. 12-EDO. 12-ED2. 12-J. 12 Jz. 12 Js. 12 J1s.

Among Orphon Soul peers, Jz was always the favorite.

The NULL Temperament
0-ET. The combination of a single note and the two silences; 0 Hz and Hz. Considers the silence between notes as actual notes themselves. (Hey if John Cage can stare at a piano and people call it music I better be able to get a little abstract and call it theory. It’s good meditation in any case.)
The ALL Temperament
-ET. Distinct from an infinite resolution of frequency or time duration, the All temperament is a virtual continuum as well, but with equal weight given to intervallic measure rather than time or inverse time. Playing a fretless instrument can be seen as not necessarily playing the All temperament because of linear tendencies which would suggest that the more densely “fretted,” or populated, regions aren’t being equally represented. The interval probability decreases the further you are along the neck and increases the closer you are to the nut.
Extended diatonic fret markings
Geometry-color scheme used at Orphon Soul used to quickly visualize nested scale positioning on guitar fretboards. Considered crucial in learning multiple temperaments. Interwoven vine-like decorations also suggested by John Karaami.
Zero temperament
Another name for the NULL Temperament. (q.v.)
Distinct from polymicrotonality, this describes two or more temperaments playing the same melodies and chords, with a preferred emphasis on compatibility by means of cross-rounding. Although it is possible to have several people playing the same thing in different temperaments in performance, thus far it has been primarily used in multitracking; recording and overdubbing two or more temperaments playing identical structures.

A multimicrotonal technique used as a slight chorusing effect to encourage and demystify practice and recording in a targeted scale class in a targeted temperament on guitar. This is broken into two parts: one nicknamed “shish kebab”*, because in the final mix you keep everything you put in, with an emphasis on the target, and “Chateaubriand”*, named after the steak dish in which two flank steaks are discarded after being used to cook the main steak; similarly, the cushions are discarded yielding the target. This is a bit like using “pads”, vocal pads, synth pads, etc, only temperament based.

* - It is a common practice at Orphon Soul to have FOOD nicknames for production techniques, related to the final preparation of music for consumption.


The conducive form of convergent cushioning.

Given A + B = C, where C is the target temperament and A + B is the scale class, the "cradling" effect would be by recording: Playing the A + C scale in (2A + B), Overdubbing the B + C scale in (A + 2B).


To encourage playing the 12+17 scale in 29, record the 12+29 scale in 41 against the 17+29 scale in 46, which are cross-rounding, and are BOTH cross-rounding to the target 12+17 scale in 29.

The inducive form of convergent cushioning.

Given A + B = C, where C is the target temperament and A + B is the scale class, the "stacking" effect would be by recording: Playing the A + (B - A) scale in B, Playing the (2A - B) + (B - A) scale in A .. etc.

There’s not a set pattern here. This can be more exploratory, not knowing what temperament you want to add back;


To play the 12 + 31 scale in 43:

1. Record in 12 maybe for a start.
2. Start with 19 or overdub 19 if you started with 12.
3. Overdub 43.

This version of stacking is called "climbing the tree", that is, the diatonic scale tree, when the temperaments stacked are moving directly up the tree.

All other stacking has no specific name although references have been made to monkeys, jumping around trees from branch to branch: Say building up 19 to 22 to 41 to 63. This would usually move up one level at a time, although not necessarily, and in this case, hardly straight up. In playing 19 to 27 to 46, 27 and 46 are on the same tree level. Or 32 to 39 to 71, 32 is actually on a higher level than 39.


The inducive form of divergent cushioning.

This is where chorusing effects take the most drastic shape. This involves creating a discrete network of temperaments wherein there is at least one noticeable occurance of a temperament not being able to cross-round to another temperament without first rounding through a third temperament.


If you round the 12 in 17 in 22 scale to 29, then 31, then 26, then 31, then 29, back to 22, you’re where you started. You have 22\17\12, 29\17\12, 31\19\12, and 26\19\12, each cross rounding to adjacent scales.

If you round 12 in 17 in 22 directly to 26, you don’t get the 19\12 nest. But sense the chorusing is all acting together, well, either use your imagination or try it yourself.


The conducive form of divergent cushioning.

This just encourages you to play something different by targeting scales that aren’t cross-rounding in the first place. This is the most free cushioning, basically playing "whatever" so that you can then play your targeted "whatever."

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