## 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:

diagram
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:

P1:
m2
M2

m7
M7
P8:

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.