Author Archives: Richard

About version 2 of Harmonic Explorer

The essence of what Harmonic Explorer presents is a tool for exploring musical invariance as it is today and as it probably emerged, in an age when numeracy was about the properties of number rather than, what you could calculate using them. If the modal scales had been discovered in the past by innovating the mixing of powers of three and five (i.e. Fifths and Thirds) then surely the scales upon the mountains of flood heros and others should be considered as informing the cultures having such tales, and their search for a practical yet often sacred music.

One simple observation is that symmetrical twin scales either rise up to the upper register by the major third or descend into the minor thirds. It is this that led equal tempered music to relegate the significance of modal scale names, since the scales could now be played in any key and their defining character was chosen to designate via the key followed by major or minor.

  1. Dorian can be seen to both rise and fall in its sequencing because Palendromic – self symmetrical.
  2. Before there were modes, the modern Dorian was the Pythagorean Diatonic (864), which required another power of three to the opening Calendar Constant (720), and includes the Tyrant number (729).
  3. The calendar (720) can only express five modes: self-symmetric Dorian, Mixolydian-Aeolian, Ionian-Phrygian. Doubling to 1440 generates 729 but has a cornerstone of 1024 but no increase in scales.
  4. Doubling to 2880 generates twin peak 1875 and also adds the Lydian-Locrian twin scales, the scales that require the 12th tone of a-flat/g-sharp to be symmetric – a condition where the whetted area is a rhomboid, called by McClain the Bed of Ishtar.

Scales within higher limits

It is now possible, in the developing version 2 of HE, to go to higher limits and have the scales play around that D = limit. The tones will always be in the same octave 360:720 Hz but the bricks will sequence visually and circled in the tone circle. The backdrop of the higher limit numbers (though never going far from the key three rows and five columns around D for that limit), it is now possible to fully play the Pythagorean heptatonic scale, or modern Dorian.

Pythagorean Scale a.k.a. Dorian

Comparing the Just intoned and Pythagorean Dorian gives little difference to the ear, since C and E are changed by only the syntonic comma of 81/80. However, it is important to see the Pythagorean on higher limits where it appears (such as 720 times 12 = 8640) since that was the ONLY diatonic scale before different ancient cultures record the killing of the serpent by Indra (Vritra), Marduk (Tiamat), Apollo (Python) and even Zeus (Typhon), perhaps to keep precedence.

New Sonic Version of Harmonic Explorer

Following on from some dialogue with Pete Dello earlier this year and some catching up of Sumerian tuning systems, I was able to satisfy my question “What role do modal scales play in the harmonic mountains and tone circles of Ernest McClain?”

I shall be publishing a number of diagrams which show where the scales are within those mountains, forming patterns like knotwork based upon the basing three vectors within the mountains, these being the Just intervals tone = 9/8, tone = 10/9 and semitone = 16/15.

To provide some audible distance learning aids for McClain and Pythagorean tuning knowledge in general, I strayed upon a crude sound system within web pages using waveform arrays, but then was informed of HTML 5 Audio components which are more reliable, efficient and easy to employ with Harmonic Explorers existing code.

v1.13 Xperiential

I therefore have a prototype version at with the following features:

  1. You can click or touch bricks to make them sing. For this function stay in the limit 720.
  2. You can click on the seven buttons with the names of modal scales. These will play one second per note. The checkbox “ASCENDING” (default is descending) can be checked/unchecked to play a given mode up or down.
  3. The bricks sounding turn GREEN and the tone circle shows the usual circle for it as if you are hovering over the brick.

Harmonic Explorer Update 1.12

Two improvements have been made after years of leaving the application as was after the death of Ernest G McClain, for whom it was written. The look and feel are very similar and I expect more changes to come without the look changing.

  1. The control bar at the top is rationalised into buttons to increase/ decrease limiting numbers by primes or major products of 2, 3 and 5, are now grouped vertically for clarity and the layout is simpler. In the future it would be nice if scaling the graphic using zoom out for large numbers did not affect the control bar.
  2. The text box where limits can be entered manually also presents a list of known harmonic limits used in the ancient world, which can be clicked to see that mountain’s limit. But now, the hover text for each limit, giving brief references, now appears below the statistics for the mountain. Thus one can now see the mountain and have some inkling as to ancient meanings for that limit. In the future it would be nice to expand on this feature by clicking on a web page for that limit. I also want to put extra anaysis features such as number or rows, with numbers shown, and number of columns (on base, i.e. powers of three), and other aspects used by EGM in his own analysis and fiddly to keep counting.

Try it out here:  and this version 1.12 opens with limit = 720, the Calendar Constant. To open Harmonic explorer to another limit, on launch, use a hash symbol after the URL and add your own number in the location as per which will resolve to Bibal group members can feedback any issues, and other can use the contact form at


How Twelveness creates Harmony


12-ness is the product of 3 and 4, three being a prime number and 4 being the first square number, of two squared. One of twelve’s most profound manifestations is found in musical harmony because there is, to the human hearing of an octave, a natural sense of 12 semitones organised as 5 tones and two semitone intervals. The semitones can fall in different places within the octave of a given reference tone, commonly called do, to define the mode (as in C Major and D flat etc) and in different modes the semitones are always separated from each other by at least two tones or not more than three tones, when the octave is viewed as a tone circle. When ancient and modern tuning methods are considered*, these all reveal a unique role for 12 but then as the ratio 3 over 4 or 4 over 3 where the denominator times the numerator causes the notes to fall according to a 12 fold modulus because 3 times 4 equals 12. However, it is the relationship of the octave to the fifth (1:2 to 2:3) which enforces 12-ness in Harmony.

* the cycle of fifths, diatonic, and its embellishment, Just intonation.

Tuning by Fifths

When viewed numerically, the technique of tuning by fifths naturally projects a series of new tones, upwards or downwards, in a way that will exceed or decede the range of a single octave.

The rising tone interval of a fifth is 3 over 2 so that downwards will be the falling fourth of 4 over 3. But in practice the fifth must often be 3 over 4 to stay within the single octave model* just as the falling fourth must sometimes be 3 over 2, the octaval number 2 being applied to translate tones back into the octave.

* Two out of three applications of the fifth =9/4 to be reduced to
9/8, the whole tone difference between two successive fifths.
The reverse is inevitably true of successive fourths which
require 4/9 to be reset as 8/9, effectively making 4/3 into 2/3.

If an octave is populated with fifths opposed fourths, up to 13 semitones appear in symmetry about the starting point, between 12 tones, to obtain Pythagorean chromatism. The Pythagorean heptatonic requires three ascending intervals (3/2 x 3/4 x 3/4) and descending intervals (4/3 x 4/3 x 2/3) either side the reference tone. This means that octaval resetting is DEFINED within such tuning schemes by the modulus of 3 x 4 equal to mod(TWELVE). When 3 is divided by 4 there is a common unit of 1 over 12 dividing the turning circle and therefore a modulus of 12 is inherent but it is not quite clear that is really the real cause.

Just Intonation

The Modulo 12 effect continues to operate even when the interval ratios employ the prime number 5 and hence the tuning operator 5 over 4 (major third) is used, instead of the fifth and fourth, and from this usage new differential intervals, a Just tone and semitone, emerge, deviating very slightly from the same fundamental pattern of 12 semitones. From a modal perspective, this gives rise to 5 tones + 2 semitones. It is because the resulting tones differ little from the Pythagorean tones (by just 81/80), that the modulo 12 operates with Just tones, then approximating 2 ET units and semitones approximating 1 ET unit.

Equal Temperament

It is therefore no surprise that equal temperament has chosen the 12th root of 2, the octave, to create a perfect compromise between historical tuning methods, so that the ET units just mentioned are perfectly followed and hence the modulo 12 equal semitones are to be seen (geometrically), equal in the (logarithmic) tone circle used in Ernest G McClain’s work and hence in Harmonic Explorer’s display. The circle has the circumference of one octave, perceived logarithmically as from a log(do1) equal 1 to a log2(do2) equal to 1200 cents and each semitone 100 cents.

The 12 ET semitones were designed around the cycle of fifths and other tuning systems adapted from it such as Just intonation, which employs the number 5 to better approach equal divisions of the octave into five notes and two semitones.

Harmony within the Earliest Numbers

The unique nature of the fifth (3/2), employing the earliest two numbers, equates (as stated) to 3 over 4 in practice, when employed to suit the simplest interval of all, the octave (2/1). Ernst Levy, called the first six numbers (1:2:3:4:5:6) Senarius since they define the five primary intervals of octave: fifth: fourth: major third: minor third. These can be seen clearly seen in Harmonic Explorer’s (HE’s) tone circle for limit=60, where 5 first allows fifth and fourth to be accompanied by Just b and f.

The outer graticule of HE is twelve fold, indicating therefore equal temperament’s twelve semitone intervals. Parallel to this is the number of cents given to each tonal brick in the brick pile organised in tuning orders: horizontal being tuning as if* by fifths and upwards and right, tuning as if by major thirds. The fifth and fourth of G-D-A can be seen to be the first creation of tuning by fifths, first seen at all when the limit is 12, to yield Plato’s World Soul.

* I say “as if” because McClain realised that larger tonal matrices
had to have been the result of a purely arithmetical calculational process,
and that a number of alternative methods evolved in different regions of the ancient world.

The number of cents in A (=3/2) is 702, just 2 greater than seven ET semitones, and it is this which creates the modularity of the octave.

One can study modularity by dividing a circle into N equal portions, and HE has its tone circle displaying 12 divisions. Moving to A one can move a further fifth (3/2) then bringing the E produced back into the octave with 3/4 instead. The deviance from ET is then 1404 cents minus 1200 (the octave) so as to be E = 204 cents (see limit = 864, the Pythagorean heptatonic). Twelve intervals will return to D as 24 cents, a sharp excess called the Pythagorean comma, which has become audible, yet the twelvefold pattern of fifths is a dominant pattern, unbreakable exactly because of the proximity of the fifth to the ET semitone of 700 cents.

It is the proximity of the fifth (3/2) to seven twelfth roots of 2,
which makes the octave (1:2) home to
the modular Twelveness found within Harmony.

Once this pattern is established, between the two (numerically) simplest intervals, Just intonation is able to mitigate the accumulating comma inherent within successive fifths by clearing the accumulation of threes and arriving at new tones 10/9 and 16/15 from the Pythagorean tones, both of these intervals dividing by 3, twice or once, and giving us the familiar Just intonation found in Greek modal music. In the melodic view, a scale is a journey which must end in doubling or halving using seven intervals, five of which must be twice as large as the other two, i.e. 12 units. One must double or halve using 12 units, which is perhaps why J.G. Bennett sometimes gave the number twelve the systemic attribute of Autocracy.

Adventures with HARMONIC EXPLORER (a.k.a. ‘Pythagoras,’ HE of the golden thigh)

“Self-limitation” in Pythagorean allegory

Ernest G. McClain
March 2012

Richard Heath’s interactive website called is a brilliant introduction to the cosmology of our ancestors, East and West, made accessible to children of all ages by way of our inherited harmonical mythology. The subtitle of one of Heath’s books, HOW STORIES CREATE THE WORLD, is my guiding light here.

HARMONIC EXPLORER (hereafter simply HE as acronym for Pythagoras, imperturbable Greek know-it-all who faked every experiment,) opens with a default setting of 8,640 that helps Socrates tell Plato’s story in the fourth century BCE. Trusting vision as most reliable of the senses, HE opens with the Chaos that Plato deliberately bequeathed to Greece science upon his death in 349 BCE. ‘Nature likes to hide,’ so the Pre-Socratics liked to remind each other, and it is their mystery-mongering that he plumbs to its depths. Plato buries earlier tales under his own ironically cryptic abstraction of logic presented with a new mythology of his own. HE effortlessly decodes the ancient Egyptian arithmetic that Plato posits as origin of all Greek science, and so frees our concern from technology for immediate consideration of meaning. The result is proposed as a reconstruction of Aristotle’s lost book on “Pythagoreanism,” the first ever written. Its suppression seems likely to have been deliberate, for the clues have survived, colored with deceit.

  • ‘Trinitarian’ analysis in three perspectives

HE’s default setting of 8,640 is “upper bound” of a minimal numerology within the octave 2:1 imagined as 8640:4320 when the limit is halved. This range of 4,320 units has descended within the Platonic tradition [c.f. John Adams] to emerge in the 20th century as the ‘period of the phoenix,’ traditional “bird of paradise” in various cultures. The result is pictured here in three different ways as a Trinity that can be judged equally sacred or profane, as we please.


1.1)  Factor analysis of partial products of “regular” primes 2, 3, and 5 (the first three) is keyed to “a tonal algebra” for 263251 meaning 2x2x2x2x2x2=64, times 3×3=9, times 5. The result, 8,640, in retrospect becomes a Platonic ‘chaos’ framed historically in smallest integers.  This ‘bird’ of apparently 4,320 linear ‘distances’ contains only 30 that define useful ‘building bricks,’ and only 19 that matter at the moment for musical cosmology–elegantly compacted. If these ideas are new, let yourself be amazed; explanation follows; understanding comes last, and in its own ways to yourself.

1.2)  Pattern is concretized in a ‘holy mountain’ of 30 ‘baked bricks’ numbered within Pythagorean ‘Ten-ness’ (integers 0-1-2-3-4-5-6-7-8-9) become Plato’s  “form” numbers in defining the larger integers restricted to tonal significance, now visible on the bricks. They correlate with seven alphabetical tone names (A-B-C-D-E-F-G) in ways that identify 19 elements–their darkened bricks now lie within the first three rows as 6+7+6 that Plato reads as “brothers and sisters,” sprung to life within in the same octave-double, 2:1.

1.3)  Resultant tonal meaning in “cyclic permutation” is graphed as “vector analysis” by radial lines in a “tone circle” that ensures perfect cyclic coincidence at every numerical doubling. Think of them as emanating from the ‘middle of the sun’ from which fallen hero ER describes the universe at the end of REPUBLIC (more accurately translated as POLITY), for the books, as considering the meaning of “Justice” that Pythagoreans considered to be “Four.” Thus ‘2’ as its ‘square root’ and smallest prime becomes both easiest to use and most powerful for understanding.  (To the initiated, division by 2 instantly reveals the deepest secrets.)

[In harmonic analysis modern “exponents” conveniently behave as ancient “god powers.” Root meanings of “joint, proportion, concord,” fitting together as if ‘naturally’ is our aim in cooking and carpentry. HE honors Plato by studying ‘philosophy mixed with music’ as he contrives paradox upon paradox to awaken reflection during the darkest period of Athenian history—the early fourth century BCE, when Athens was shamed by the complicity of Plato’s own family in political disaster.

Default limit 8,640: toolbar technology for ‘Pythagorean’ conflict

2.1)  Within the infinite reach of the integers (our natural “counting numbers”) any convenient limit becomes our best friend. Click the cursor on its window one once to highlight, and a drop down window then offers many alternatives. Choose another by clicking on it, then activate with a click on the check mark A to its right. Whenever confused (and it will happen often), simply overtype your own choice; or close the program and reopen with default 8,640.


2.2)  Test the first ten toolbar windows that follow with a click on division (÷) or multiplication (x) by 2, 3, 5, 10 and 60 to witness simultaneous changes in all three displays. Alternate clicks to change and restore defaults to your pleasure. Keep a ‘playful attitude’ until you acquire control. Feeling counts.

[Useful trick: Hold down CTRL while rotating the mouse wheel slightly forward or back to enlarge or decrease type size, also affecting the toolbar; it is often most comfortable for eyes when the bottom row of bricks almost touches the circle, but larger limits may require adjustment or separate display.)

2.3)  Toggle cents to transform the referent 12 hours into 1200 modern logarithmic cents–as if each equally tempered semitone is subdivided into a hundred micro intervals, each too small to be noticed by human ears. (A dog does better, and Pythagoras considered it a “philosopher” for greeting a friend or stranger appropriately.)

[Toggling wings us past 5000 years of ‘spiritual’ warfare (and billions of archaic ‘dust’ particles) into the rarefied mind-space of metaphysics where only false pride can be injured. Attention to ‘12 hours’ leads naturally to the zodiac, counting days by solar sunrises, but within periods determined by the wayward moon. HE is designed to accommodate Plato’s whims, and those of scribes in other cultures.]

2.4)  Toggle “brick pile” to see it disappear and return, then toggle “notes” to see their names and circle do the same, and then toggle “circle” to see it disappear by itself. These controls enlarge or decrease both displays and printing. HE assumes that musical “wholetones” are double hours, 6 to a cosmic cycle; as if 3 “watches for the night,” and another 3 for the day, meaning that the chromatics scale of 12 semitones correlate symbolically with the 12 signs of zodiacs, each sign considered as extending to 30 degrees in a circle. Thus explanations employ the odd verbalism of “semitone hours” purely for numerical convenience.

2.5)  But HARMONIC EXPLORER has a mind of its own—favoring products of 2, 3 and 5 in the Neolithic 4th millennium BCE, 5 or 6 thousand years ago. When we divide numbers with larger prime factors, HE answers as expected until their multiples are exhausted; from that point on it responds with nearest approximations among its own “regular” numbers (products of 2, 3, and 5).

Explanation: Biblical Jews chafed for over four hundred years ‘making bricks for Pharaoh’ under restrictions (that perhaps were Sumerian in origin) before ‘approximately 600,000’ rebel under the leadership of Moses. HE helps us share their pain. The factors of 26=64 in default 8,640 descend from glyphs representing the right eye of Horus, the hawk guarding the throne of Egypt, now reduced to Platonic “nursemaids” (indistinguishable from mothers, daughters, and sisters) as ‘empty vessels’ for “male seeds” (as in Hippocratic genetic theory). “Trinitarian” upper caste factors of 32=9 descend with Philo’s later Jewish blessing as “the most warlike of numbers” in the Bible and (when looking at 1x3x3 = 9 become examples of ‘Cretan bull-leaping’ in Homeric metaphor. All of us are created naturally as Plato’s ‘auxiliaries,’ meaning lower caste “fivers.” This default setting of 8,640 cryptically correlates all of worst number problems in classical Greek mathematics into a perverse example of “smallest integers” that I suggest we accept as Plato’s ‘phoenix,’ in the sense of the Egyptian Benu bird.

2.6)  With the “pointing” fingers of both hands on the screen at pitch class “D” (at 12:00 o’clock ,“our ‘zero hour’ on American time tables), move them simultaneously outward around the perimeter to count nine pairs of ‘Platonic twins” mirrored on each side of an imaginary “plumb line” from heaven. Let’s pretend that line is Alice in Wonderland’s first of 3 magic mirrors by which  ancient “mythography” can be read with Plato’s famed stereoptical double vision but set in motion as dialectics, meaning in dialogue, most successful when some Other mind is on the opposite end of your log. This is 3-way conversation about past, present and future, (Aristotle’s beginning, middle, and end, the best advice he could append).

3) Plato’s heptatonic World Soul abstracted as default 864


3.1)  With default limit 8,640 still showing, click once on ÷10 to flee Greek confusion for 864 as Plato’s notion of heaven embodied in the ‘octave’ cycle of 8 Sirens mounted on rims of  planets in the “spindle” of his World Soul, each singing a single tone. The 8th as ‘boundary marker’ is hidden within the lower bound of 432 as 864/2, halving the referent “D” that HE always “enthrones” for immediate comparison. Within the simpler self-symmetries of “octave double” of 864:432  a heptatonic 7–tone limit of F C G D A E B lies in tuning order among the bricks as paired from the middle. (Place the pointing fingers of both hands again on brick “D,” and move them simultaneously outwards on G:A as ‘wholetones’ or ‘major 2nds,’ then on to  C:E as ‘major 3rds,’ then on to F:B as “minor 3rds,” always counted from D to simulate this bi-lateral symmetry associated specifically with Apollo. Athena is “Justice” not only as “4” but also associated with 7 (united with 4 in 4-3-2-1-2-3-4) when viewed from the middle of the sun of Platonic insight. We are leaving childhood behind to think of ourselves as the vertical embodiment of bi-lateral symmetry, wherever we find it.

3.2)  Now focus eyes and fingers on the tone circle and move your paired fingers in opposite directions around its circumference to recognize the same tones realigned into scale order as if intended for sounding on the Greek 7-string lyre—rising to the right and falling to the left as far as voice and instruments can let us sing or play.

3.3)  Next, with one finger only, start on any tone, rotating to the 8th in either direction, to meet the same pitch class again as both upper and lower boundary markers of a modal octave that consist of 5 wholetones and two semitones, always at B:C and E:F, distinctly closer together, never again in the middle of tetrachords. Thus heptatonic modal octaves never enjoy the “Edenic” bliss of an extended pentatonic system. The result is seven distinctly different modal patterns, differentiated by careful attention to semitones. But notice also the growing semantic confusion as musicians employ the same words for quite different meanings that only context clarifies. (This can sound like gibberish for the uninitiated.) Music is practiced as an art of the particular, the concrete, often with great concern for detail. But the Muses had long been loving sisters, rapidly become estranged in new Greek science.

3.4) Our sophisticated naming system employs 7 letters modified upwards or downwards by a semitone with sharps (S) and flats (F) to become 12, plus 10 that accept double sharps (##) and double flats (FF), printed and in both upper and lower case letters, and giving rise to further trivial distinctions known as “commas.” Such algebraic nicety makes it easier to follow common European practice for the last thousand years by singing only the solfeggio symbols do-re-mi-fa-sol-la-si-do with the rising scale from C as Plato’s ‘true Hellenic mode, the Greek Dorian—or the falling scale from E as mi-re-do-si-la-sol- fa-mi that need the same arithmetic for contrary locations of their semitones. Greek ratio numbers and proper pitch names belong only to theory, not to musical performance. We are free either to heal or split our own souls over this area. Pythagorean issues must be left to those who find them interesting.

Comment: This falling E mode was considered more authentic for Plato’s Greek Dorian until Francis Cornford pointed out that our own rising C mode was assumed as implied by Plato’s paired reciprocals. He exercises future “guardians” of his model cities by ‘wrestling naked in the gymnasia’ in ‘paired teams’ of  2 to 10 each. Here we see two ‘3-man teams’ around their ‘referee.’ Any seven consecutive elements in any row of such a ‘matrix’ (i.e., mother) enjoys equivalent worth in some context.

3.5)  We have arrived at the Platonic entrance to a welter of heptatonic 7-tone confusion that today is severely strained by modern 12-tone chromaticism. Each tone of the 12 is a possible “tonic” reference for all seven of the Greek diatonic modes (among very many other structures), translatable through 12 different keys producing—on paper—7×12=84 “mode keys” that are recognizable only among about one-percent of the population that enjoys “absolute pitch” memory. English habits recognize that the rest of us possess, at best, only a sense of relative pitch, thus favoring seven solfeggio syllables for all purposes. Pitch class “D” sung as ‘do’ (from Latin dominus, Lord) thus  makes a delightful bridge to Plato’s ancient habits that, in turn, function as a bridge to a musicology emerged in the late Neolithic before the invention of writing.

3.6)  With default 864 displayed, click once each on ÷2 and ÷3 (using their partial results) to divide by 6 [now available] into 144 for further study of these first two primes.

4) Default 144: Plato’s “human soul” descends from stars (not planets)


4.1)  With division by 6 we reach Egypt with default 144 as sexually ambiguous “humanoid fivers” (displayed visually as 2+3) with greater distances of “minor thirds” A:C and E:G imposed between ‘feet’ and ‘hands.’ We appear articulated happily “at the waist” like an insect, near the diaphragm where “spirit” as the “wind of God” originates.

4.2)  Only 5 bricks (darkened in the foundational base) among 12 “regular” candidates can be given tone names as twinned Platonically because they constitute the lower border (lacking opposites below). All 12 are required later for calendar and zodiac. What these 5 mean tonally is experienced easily by singing do re mi sol la (rising from left to right, or falling in reverse as mi re do la sol. These elements are best absorbed unconsciously very early in children’s songs and dances. Sing them while embodying them personally to link your own tonal feeling with the history of ideas: here is Platonic philosophy mixed with music. A mysterious sense of tonality likely will guide your pitch if you’ve been properly loved—for we are long conditioned harmonically by the natural resonances in our own voice and of those closest to us. Wholetones do-re-mi  rise to the right in patterns of 3s, while sol-la continues in pairs, and all fall to the left in reverse as la-sol-mi-re-do,  repeating at the octave 2:1 on any sixth tone (a contradictory verbal locution that proves convenient). On keyboards HE’s enthroned “D” correlates with the middle of three black digitals now named either A-FLAT or G-SHARP, but this confusion is avoided in Plato’s models by his exclusion of  ‘enharmonic genera,’ eventually very widely exploited. (In the arts, whatever proves “worst” proves useful for some contrary purpose.)

4.3)  Glide the cursor across these five favored bricks (white labels help them stand out) while noticing that pitch names are encircled in the cycle, alerting us that numbers no longer signify abstract “points of no dimension” as Platonic ‘boundary markers,’ but have become “embodied” pitches for ears that require spatial extension  over some radial portion of the cycle, assumed to be about a quartertone. As a result, an ‘octave’ theoretically of six wholetones (difficult to sing accurately) and often referred to more simply as tones, invites further semantic confusion) while permitting practical halving into 12 semitones that never were standardized until Equal Temperament was recently imposed (for commercial reasons). Plato’s social theory goes no further than necessary to document Socrates’ concern with “What twelve is.” Here we see 3 wholetones and 2 semitones filling the space reserved for twelve intervals whose perfect equality matters only to astronomers and calendar makers, thinking of a lunar zodiac rather than ‘good enough” pitches. Thus Plato’s “numbers in motion,” by themselves and at our convenience (an idea still roundly ridiculed) serve very different sister Muses.

5) Platonic technology—from the one to the many

5.1)  Plato’s arithmetic is disciplined teleologically by the end he has in mind, described in S. 546 as “3 multiplied by 4 and 5, and raised to the fourth power” as 3x4x5=601,2,3,4 =12,960,000– decomposed (when Socrates’ pretentious jesting is penetrated) into the square of 3,600 and/or a rectangle of  2,700×4,800.  Since understanding comes last to a learner, let’s “saltate” (leap) to the end as he does–to the middle of the sun itself–and watch its ‘radiance’ grow as if “seeded” here by 34=81 in default 144 as fifth tone in the pentatonic subset.  A click on ÷3 eliminates ‘arms’ and reduces the default to 48, with 36 and 27 showing to its right as a preview of their later roles in his analysis.  Another click on ÷2  eliminates ‘legs’ and reduces default bricks to “the One itself” as the modern zero power all numbers, justifying the verbal insistence of some ancient authors that “creation proceeded from nothing,” although philosophers disagreed.



5.2)  Now click 4 times on X60 to watch Plato’s explosive overview of harmonic theory, radiating as if from the middle of the sun. Our referent D now is 5th among 9 symmetries in the fifth row—an ennead traceable to 34=81 among 144 being reciprocated. This basic Mesopotamian tuning theory was published only in 1976 by Kilmer, Crocker and Brown as inferred from cuneiform documents from the early 2nd millennium BCE, 4000 years ago—when adapted to its 9-string lyre, with strings 1 and 8 both tuned in octaves, and with the 2nd and 9th tuned similarly. Pythagorean economy knew only the 7-string version, with pentatonic C G D A E extended by F and B; it is now doubly extended by Bb on the left and F# on the right. (Pythagorean Ten-ness is served by a lonely C#, still unnamed on the far right. “Citizens” older than 10 are excluded from foundational theory.)


5.3) Click twice on ÷60 to see the “diminished light” in 603 = 216,000HE_3600

and 602 = 3,600, and then once more  to see the reduced model of 601 itself as “big One” in the new arithmetic that Sumer is credited with inventing in the fourth millennium BCE near the end of the Neolithic period, before verbal writing appeared.



This version of pentatonic symmetry (with drooping ‘arms’) may look dejected, but head and feet are ideally located to expose the first paired symmetries.  G and A are Plato’s “guardians of the highest property class” (meaning as musical 5th and/or 4ths in opposite directions from referent D,  always enthroned). They now enjoy help from b and f (in lower case type face) as ‘auxiliary guardians’ of the second class (or caste). The ratio 40:50 integrates 4:5 and 45:36 integrates its reciprocal as 5:4. New symmetries of 5:6 as 50:60 also pair inversely with 36:30 (lower bound as half of 60). We are gazing on a revolution in arithmetic that happened before any stories were written, for base 60 cuneiform notation henceforth permitted division by 2, 3, and 5 to infinity to be simulated  by merely rotating the matrix 180 degrees, half-a-turn, thus validating reading from both 12:00 and 6:00 o’clock (i.e., from 1200 and 600 cents.)   When 50 is set aside (for chromatic extension), the remaining seven constitute a “Just” alternative to Spiral 5ths heptatonic tuning, falling D-c-bF-A, G-f-eF-D  when the brick pile is upright, but rising D-e-fS-G, A-b-cS-D when it is inverted. The fixed limits of these pairs of similar “tetrachords” remain A-D-G as upper caste “guardians” among the bricks, but are realigned as G-D-A with the scale to accommodate 2 pairs of “auxiliaries” as “moveable sounds.” We can only wonder how closely Plato thought tones and planets behaved.

5.4)  This revolution in “musicology” that happened long before Plato warrants expanding typeface and bricks with CTRL and the mouse roller, and then rotating the printed copy in our own hands.  Cultures that preferred working in base 10 for its simplicity never needed to rotate “brick walls.” It was always assumed that, “As Above, So Below.” Our modern writing habits encourage working upward and to the right from the cornerstone. Scribes often could imagine the superimposed patterns without computing new reciprocals. But the underlying pattern reduces to that for only 3×5, so that for many purposes no doubling proves necessary. Plato provides for a tonal calendar that integrates reciprocals under a new limit. Lets exit here with default 60 to watch him integrate Time.

6) Plato’s 2-year Tonal Calendar as Heath’s “Lunar Zodiac”

6.1)  With default 60 showing, click the cursor on 45 to see a surprising result as 45 rescues the “cornerstone 32” as a skewed pendulum from the throne. This proves essential to Semitic musicology and to the Bible’s, but not for Plato, who refuses to recognize approximation, however valuable. Heath has cleverly contrived this one exception to naming only Platonic twins; the extension to lonely small a-flat in “Just” tunings is given a dashed red vector not quite aligned with a ‘plumb line’ from the throne; this single a-symmetry achieves a remarkable digital economy. He is not as opposed to sensation as he sometimes appears to be.


6.2)  With default 45 displayed, click slowly four times on X2 to see familiar “degrees” at right angles of 45-90-180-360, embraced within its double at 2×360=720 “days plus nights.” Although this concept of “degrees” is formalized only in the 2nd century of the Common Era (CE) by Claudius Ptolemy (assuming that 720 half-degrees map nearly the same area as a circle with a figure having that many equal sides, it is apparent that this coincidence with ancient practice became habitual with some scribes far earlier. Each hour on HE’s circle for 720 can be imagined as divided into 720/24=30 parts within the 2:1 octave as 60:30 with accurate ratio divisions at 30-32-36-40-45-48-54-60. Thus astronomical measures projected on this tone-circle in the 2:1 octave read as 720:360 would have approximated accuracy reasonably well while naked eye observation was being improve. Oddly, we know that for centuries after Plato’s death, Alexandrian Platonists tried to correct his tuning to fit better celestial observations.

But we can only guess how tone circles might have been viewed before producing “god numbers” for imperial tyrants.

HARMONIC EXPLORER - Ancient Musicology by Limiting Numbers (2)

6.3)  Click now twice more on default 720 to reach default 2880 where small a-flat gains its twin as small g-sharp. This increase in numerosity by four times is obviated by rotation of the table. These twins function as Horus and Seth in Egyptian mythology, vying for father Pharaoh’s throne, and inspiring many tales that Plato tells in his own way.  At 3 places in the circle (2:00, 6:00, and 10:00 o’clock)  Plato places the “three fates” as “daughters of Necessity” (now enthroned at 12:00). These 3 poor girls, severely disciplined to spin out our solar fates, must constantly adjust his circles to keep them moving at a steady pace. Two of them (presumably at 10:00 and 2:00) reach out occasionally—one with the right hand, the other with the left–to adjust the speed of  Plato’s “Spindle of Necessity,” carrying his 8 Sirens, but the third, presumably at 6:00, occasionally must use both hands. Under these slaving conditions we can appreciate why Plato postulates “Chance” as also a deity.



6.4)  Now with default 2880 showing, click once on x3 to see default 8,640 reappear with both “Horus and Seth” surrounding 600 cents at 6:00 o’clock. Our pentatonic Egyptian model is displayed with purple radials, and the seven tones of the World Soul are displayed in upper case capitals as Platonic guardians, vs his auxiliaries in lower case.

6.4)  One click on ÷2 vanquishes “bad twin Seth” as g-sharp to display (in my imagination) the Platonic “bird of Paradise” as 4,320 reported by James Adam, streamlined elegantly for action as the Egyptian Bennu. (Visit Wikipedia for a view). Then restore Seth to the default by a click on X2 to leave 8,640 still showing.

7) Platonic origins in the third millennium BCE: 8,640,000,000 as the universal flood

7.1) From default 8,640 multiply by millions with six successive clicks on X10 to extend the “tail” of the phoenix to represent the beastly Imdugud, of ancient Mesopotamia,  winged with a “spiral of 5th and 4ths” reaching to the 21st feather as 320 = 3,486,784,401, and with a “peak” at 514=6,103,515,625. [Use Ctrl and mouse roller to adjust display to view limits.]


This vulgar (view also on Wikipedia) broke a compact with the serpent to “feed each other’s children” by traitorously swallowing those in the serpent’s nest. Now toggle cents to see 598 at the peak, and 602 as seventh in the base, meaning within 2/100th of a semitone when viewed from the throne as fourth in the eighth row. They are very near coincidences to the square root of 2, a point of central interest in Platonic mathematics, and the subject of his MENO where the untutored ‘slave boy’ (under Socrates questioning) proves wiser than his master. This picture had inspired not only Noah’s flood but hundreds of other stories throughout the ancient world. We shall return to it often.


7.2) Toggle the brick pile out of your way to view ‘Horus and Seth’ now doubly represented near 6:00, for the “flood twins,” far more accurate, have no practical role except to inspire wonder and reflection. Here, by the grace of Richard Heath’s skill as a web developer, mathematician, gracious author, and geometer, we see for ourselves the ancient foundations of Plato’s science in ways he probably never witnessed himself, and was trying to re-invent from traveller’s tales.

Here I must stop to take a breath.

And after 6 pages of being led through HE with an “Egyptian nose-rope” the reader probably will welcome a recess also.


Frequencies or Wavelengths, Pitch or String Lengths?


Ernest G. McClain’s Holy Mountains represent an ascending octave integer set as ascending pitch classes (frequencies) yet Greek scales were described as notes descending in pitch class with increasing string length. Such mountains were often “reciprocated” and visually inverted, pivoted on the octaval D, and if frequencies were being inverted, then the inverted mountain would then represent string lengths. It is these contradictions which need to be resolved, probably by accepting two different views with the familiar as McClain’s notation.


In Myth of Invariance (McClain, 1976)[1] the Samkhya term Purusha (cosmic principle or self) is used to describe an inverted mountain pivoted about D and Prakriti (material principle) is compared to the holy mountain itself, as it stands “upon the earth”. It seems clear that the material principle behind tuning is the instrumentality, such as a set of string lengths, whilst the more abstract principle is our modern term frequency, known only recently as the reciprocal of string length. But the modern notion of frequency seems conflated with how antiquity must have managed its own musical tuning theory, if and when it had one.

[1] (see “The Holy Mountain” page 50-53)

I remember being told (in Physics) that Pythagoras discovered musical ratios through experimenting with string lengths, this to introduce the (then) modern schoolboy to the inverse proportionality of wavelength (a.k.a. string length) and frequency. But how did the ancients understand the lower pitch produced by a lengthened string? – except as being directly proportional, the opposite of inversely proportional.

Our term frequency is only inversely proportional to wavelength because of the ascending directionality eventually chosen for frequency. String length and wavelength are more or less synonymous, and the modern interest in frequency only came to the fore after the logarithmic relationship of frequency to wavelength was discovered, a discovery related to the discovery of logarithms, the theory of functions and modern experimental equipment.


Figure 1 Part of Chart 11 in Myth of Invariance illustrates how the scale order of the holy mountains was interpreted in the sense of increasing frequency rather than string length.


If the ancient world thought only of string length and descending pitch, then an inverted holy mountain only appears to contain frequencies via the inverse law relating wavelength and frequency. When the inverted string numbers overlay the non-inverted string lengths, Jay Kapraff calls the mirroring of wavelength and frequency within an octave, a Tetrapolarity (Kapraff, 2002): where symmetry in string lengths across the axis of symmetry of an octave is matched by a symmetry in the corresponding string frequencies across the same axis of symmetry.

The earlier use made by McClain of an inverted holy mountain pivoted about D, was in fact to visually mask out tones which are not found in both the holy mountain and its pivoted inverse. One can then, through this visualisation, identify those tones symmetrically connected to both low and high D, as Plato’s “ambidextrous pairs of fighting men” which, of necessity, are those tones whose frequencies happen to be doubly connected to low and high D. The inverted mask, not of frequency but of symmetrical tones within a limit, is only a simple way to visually present these symmetrical twin tones as string lengths, so that,

  1. One must forget about “frequency” in an ancient context.
  2. the integer matrix in an ancient context should be read as of string lengths and
  3. higher numbers should generate a lower octave classes (note letters).

McClain showed the increasing numerical values (unfortunately called tone numbers) as higher in pitch so that the Pythagorean “tuning order” of FCGDAEB is a view based on ascension whilst the string length view would tune 3/2 from high D as G (rather than 3/2 from low D as A). If right, this means that the holy mountains have been notated incorrectly. Figure 2 shows the re-notated mountain for a limit of 720, indicating the significant changes when tone classes are allocated according to string length.



Figure 2 The pivoted mountain for limit D = 720, shown with tone classes allocated according to string lengths rather than frequency


It seems the holy mountain should not have been confounded with frequencies for the purposes of ancient musicology, or its attendant symbolism, as the mountain stands upon the earth and the idea of frequency has only been abstracted recently by modern science, as an ascending ordinality. The inverted mountain is actually showing the special relationship within an octave of one realised string length, across an octave’s axis of symmetry to another “twin”, a string whose length is similarly and “ambidextrously” connected, by intervals summing to two, to both low and high D.

A trial version of Harmonic Explorer can be tried prior to a “string length/pitch” checkbox being added later in the year to the standard version.

Works Cited

Kapraff, Jay. 2002. Beyond Measure. s.l. : World Scientific, 2002.

McClain, Ernest G. 1976. Myth of Invariance. New York : Nicolas Hays, 1976.

Mantles of Radiance and the 118th Root of Two

Ernest McClain wrote a short note called Mantles of Radiance as assumed, computed, and interpreted. In it he compares the three situations found in the logarithmic tone circle for equal temperament, the cycle of pure fifths and Just intonation. The tone circle as shown in Harmonic Explorer by default beside the holy mountain (aka matrix) for each limit, covers the invariant tones generated by the limit, when these are symmetrically paired, left and right – these also shown on the mountain as “wetted” bricks.

Mantles of Radiance as assumed, computed, and interpreted

12 equal tones in tone circleA) The cycle (circle) as 12 in both circumference and area, not by computation but by cultural assumption. To Plato this meant that our “Equal Temperament” model was anticipated as convenient “boundary markers” for the cultural norm.





B) The serpentine “spiral” of musical fifths and fourths naturally approximates the cultural assumption. Only on the 13th pitch class is a cumulative excess in musical fifths discovered, with a consequent defect in fourths, by the overlap or gap of a “comma”.



C) An alternate “triadic” Just tuning correlates consonant major thirds of 5:4 with resulting semitones of three different sizes, leaving a defect of a comma. This tuning correlates directly with clock and calendar within the limit of 720 “days plus nights.”

If you were able to detect it, each Just tone is 81/80 different with each of the cyclic fifths, leading to double lines in Ernest’s Holy mountains, generated using a single limiting number as high D for an octave.


D) Correlation of the two systems displays commas at nine different locations surrounding the assumed perfection. They are Plato’s metaphorical “no man’s land” and locus of eternal conflict, resolved only by moderation in demanding “exactly what is owed.”

McClain goes on to say;

Sumerian affection for “mantles of radiance” assigned to the gods assured close scribal attention to music as an easy manipulable model for the heavens, algebraically and geometrically. This level of computing accuracy is displayed in the Shuruppak “grain-constant” of 1,152,000 units surviving on tablets from two different houses. When the city was burned c. 2600 BC the soft clay tablets of this “problem text” were baked for posterity. Working with symbols only for 1, 10, 60, 600, 3600, and 36000 students were required to discover the whole from its division into 164,571 thousand portions of 7 each, with a remainder of only 3. The answer, correct on one of the tablets, required 32 symbols for 36,000. Platonic decomposition of 1,152,000 exposes a brilliant assembly of harmonic formulas, “sealed” forever.

Analysis of the “Mantles of Radiance”

There is more to these mantles of radiance than meets the eye for in larger limits one can see that the mantles all fall according to some inherent granularity of spacing within this logarithmic world. I was intrigued, whilst Ernest was alive, to add a graticule to Harmonic Explorer which could match the granularity observed between the tones we call harmonic.

The solution at a simple level is to give each graticule an angle so as to divide the circle into 118 parts. One then sees that the tones all fall upon the graticule and hence fall into discrete units of the 118th root of two.

Dividing the logarithmic world of the octave tone circle into 118 equal parts means that every tone lands upon one of the divisions



Dividing the logarithmic world of the octave tone circle into 118 equal parts
means that every tone lands upon one of the divisions

The 118th root of two is 1.005891415 which equals 171/170 = (9 x 19) / (17 x 10) to high accuracy. If we call this ratio A then

  1. A^2 approximately equals 81/80 but
  2. A^7 = 25/24
  3. A^9 = 254/243
  4. A^11 = 16/15
  5. A^13 = 27/25
  6. A^18 = 10/9
  7. A^20 = 9/8 (2 x 3 above)
  8. A^31 = 6/5
  9. A^38 = 5/4
  10. A^49 = 4/3
  11. V^69 = 3/2 where 49 + 69 = 118

and so on. The explanation in the world of number arises from the nature of the ratio 171/170 in that it can be decomposed into two ratios, 9/10 and 19/17 which are very close to each other and when multiplied yield the 118th root two. So 171/170 = 9/10 times 19/17 = the ratio for the 118th root of two which divides the tone circle into 118 parts. The resulting octave is slightly short at 1.997874 rather than 2.


This cross product is interesting because the first ratio different by two instead of one (numerator to denominator), equal to 10/9, has been found as a near ratio – which need not have given the 118th root of two. In practical terms, a right triangle with hypotenuse 10 and base 9 can be extended at that invariant angle until, at hypotenuse 19 and base 17 the near approximation to whole number lengths, with the same angle between, is 19/17.

To see the graticule of 118 in action, use this link to a development version of Harmonic Explorer.