The Triadic Networks: periodicity for the 21st century.

This is a great summary paper by Dr. Eric Scerri on the role of triads in evolution of the periodic table.

This is the paper by Dr. Alfio Zambon which inspired this work.

Here then is my contribution to chemistry: the Triadic Networks (TN), which is a general mathematical design, and the Triadic Elemental Networks (TEN), which apply that design to chemical elements.

I will be filling in more as need be but for now I’m putting up the diagrams that are necessary and sufficient to understand the concept.

I look forward to discussing this further with you and everyone else!!

The Triadic Network

This is a general math relationship based on the triadic average, where the middle numer of a link is the average of the outer two numbers.  There are many ways to do this in a set of numbers; here we see one such design inspired by Zambon’s Periodicity Tree 0 (Figure 3 in his paper).

The Hydrogen Hexanet

As the figure itself says, this draws attention to the fact that a 6-element collection of linked numbers (or elements) which we call a hexanet is responsible for half the elements that occur naturally on earth and, based on predictions, in the universe.

The Triadic Elemental Networks

Fully expanding figure 4 and linking redundancies generates this new version of the periodic table.  Here we see four layers of even and odd numbers connected by triads.  As the legend indicates, the layers are inter-connected by two linear triadic networks, 1234, and 5678, and layers 2, 3, and 4 are more strongly connected to each other than layer 1.










What are the modes of orijikan?

  • The modes currently used in orijikan come from the major, melodic minor, neapolitan major, harmonic minor, harmonic major, melodic major, neapolitan minor, hungarian, byzantine, enigmatic, whole-tone, and diminished scales.

The modes used in the weaving of orijikan are not restricted to the church modes.  In fact, any scale other than C ionian can be said to be a “mode” relative to it since only C ionian is all natural and thus all other scales must by more “accidental” than it.

“Scale permutation explanation”

Consult this page [lnk] for more information on these scales and their modes.
“Ionian as origin; everything else relative”

The most complete accidental network to date incorporates 73 modes from 12 different scales, each scale consisting of seven tones spanning one octave.  If instead we have seven tones spanning two octaves, this would be known as a heptatonic 2-scale; if spanning seven octaves, a 7-scale; and so on.

Heptatonic scales with steps only (21 modes)

(ionian, dorian, phrygian, lydian, myxolydia, aeolian, locrian)
(ion, dor, phr, lyd, myx, aeo, loc)
(Maj1, Maj2, Maj3, Maj4, Maj5, Maj6, Maj7)

Melodic Minor
(jazz, javanese, lydian augmented, lydian dominant, hindu, superlocrian, altered)
(jaz, jav, lda, ldd, hin, slo, alt)
(Jaz1, Jaz2, etc)

Neapolitan Major
(Nea1, Nea2, etc)

Heptatonic 2-scale with skips only


Heptatonic 7-scale with jumps only


Heptatonic scales with one skip (49 modes)

Harmonic Minor
(Har1, Har2, etc)

Harmonic Major
(Hrm1, Hrm2, etc)

Melodic Major
(Mlm1, Mlm2, etc)

Neapolitian Minor
(Nem1, Nem2, etc)

(Hun1, Hun2, etc)

(Byz1, Byz2, etc)

(Eng1, Eng2, etc)

Non-Heptatonic scales (3 modes)


(Whf, Hfw)

What is orijikan?

Orijikan derives from the Japanese
Ori, meaning fold, and
Jikan, meaning time.

Origami uses
folding space to create visual art;
Orijikan uses
folding time to create aural art.

The core philosophy behind orijikan is that canonical music theory relationships can be visualized using chromatic group theory.  

Here are a few items presented as evidence of the utility of this viewpoint:

  • The Accidental Abacus: this visualization of the degrees of the heptatonic scale is a convenient device to familiarize oneself with musical inverse relationships as well as  the patterns that the orijikan view of music opens up.
  • The Major Network: a chain of major modes linked by perfect fifth jumps, this example of a modal ring is a stepping stone for understanding all the other diagrams and relationships within orijikan.  The major scale upon which the major network is based has played a similar role in music theory throughout history.  What does this network sound like?
  • The Little Wheel: adding the melodic minor scale to the major network results in the little wheel.  The importance of this modal wheel lay in the fact that every other mode can be mapped to within three perfect fifth jumps of this wheel and that its two component scales are the most important scales in jazz melody and harmony.  What does this wheel sound like?

While this site is meant to discuss and disseminate the more visual and practical outcomes of my investigations, its sister site will serve a complementary purpose by focusing more on the theoretical and mathematical arguments for this viewpoint.  The reader is encouraged to participate in both websites given that they are both part of an overarching framework linking music and math.

Thank you for visiting my site and I look forward to folding time with you sooner and later.