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 www.chromaticgrouptheory.com 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.


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 an accidental network?

Each mode has a unique collection of flat and sharp accidentals relative to ionian that define that mode.  This concept was explored in the accidental abacus where we also saw that remaining diatonic required a very specific perfect fifth key center change, a jump.  In an accidental network, these jumps are modeled as arrows that connect modes to form chains of modes or networks.  And while there is a mathematical foundation to the accidental network (which can be explored at www.chromaticgrouptheory.com), for orijikan it is enough to associate an accidental network with its visualization:

AN diagram.

To start, lets consider what a key change would look like from C to G

staff notation

C ionian –P5–> G ionian
Cion + P5 = Gion

Where “–P5–>” stands for “change the key by an ascending fifth”.  We set the convention that going with the arrow is an ascending fifth while going against the arrow is an ascending fourth.

Now consider the transition from C ionian to C mxyolydian; it only involves a flattening the seventh degree (an operation we will annotate as b7 from now on).  As an accidental network, this looks like

C Ionian –b7–> C Myxolydian

Where going with the arrow means you flat the seven while going against the arrow means you sharp the seven.

If we now wish to transition from C ionian to G myxolydian and thus remain diatonic, the accidental network would look like this

C Ionian — P5 –> G ionian — b7 –> G myxolydian

As stated in the accidental abacus, we can combine the P5 and b7 operation into an accidental jump.  There is one accidental jump associated with each scale degree.  We denote a “forward jump” as one that goes with the stated arrow and corresponds to a perfect fifth key change.  We denote a “backward jump” as the opposite, going against the arrow and corresponding to a perfect fourth jump.

An open accidental network is one in which there is no link from the first mode to the last.  For example,  the accidental network in going from C ionian to G myxolydian to D dorian is:

Notice that there is no link connecting D dorian with C ionian thus making this an open network.  The links in an open network will always be accidental jumps and thus correspond to perfect fifth key change movements.  Open networks seem to be only useful for theoretical discussion and pedagogy given that the two most important structures in music theory, the scale and the circle of fifths, both have their last mode connecting with their first, i.e. closed.

There are two types of closed networks: rings and wheels.  In each case, the last mode has a link that connects it to the first mode.  The difference lay in the type of links and modes allowed.  In a ring, we only use diatonic sets of  modes but allow the links to represent any intervals.  In a wheel, we use only non-diatonic sets of modes but the links must be exclusively accidental jumps.  Let us examine several examples to better understand the different networks.  Notice that by this definition, the basic construct of the accidental network, two modes connected by one lin

Open Networks


What is a jump?

  • While a singular jump is a fifth or a fouth, by adding jumps together, a simultanous jump can represent any chromatic interval.  An accidental jump refers to an modal operation which simultaneously involves a jump key change and an accidental operation; they can also be singular or simultanous.

    Step = M2 or m2
    Skip = M3 or m3
    Jump = P5 or P4

When we speak of changing notes, the distance between the two notes is known as an interval.  Thus the distance between C and D is a whole tone or two half-steps.  There are 12 intervals each corresponding to a specific distance from the root:



It is customary to refer to the minor second and major second intervals as steps.  This gives rise to the whole-step and half-step nomenclature.  Similarly, it is customary to refer to a minor third and major third movement as a skip.  Here, we extend the step and skip definition by referring to any perfect fourth or a perfect fifth movement as a jump.  Furthermore, it will be customary to refer to a forward jump as a movement in fifth and a backward jump as a movement in fourth.

While the notion of a jump is universal and merely refers to a generic musical interval, when we combine the fact that any accidental movement also corresponds to a key change jump, then we arrive at the concept of the accidental jump.

Major network study: progression 1 and retrogression 1

The below is a study on the major network.  It consists of one piece moving in fourths, a progression and one piece moving in fifths, a retrogression.  Each “-gression” is repeated three times, inverting chords on each repetition.


There is nothing particularly unique about these major studies… they are merely diatonic chords in ascending and descending fifths.  However, many if not all of the patterns of music such as these are already present in the major scale and thus the major network.  As such, this study could represent a “diatonic harmonic template” insofar as it gives the listener a reference to what movements in fifths (jumps) sound like.  As well, the idea of going one direction and then another and in choosing closed “orbits” within the accidental network will be incorporated in all future studies.