We describe how to build an exhaustive chord reference, using all of the concepts and rules of the previous sections, along with a few more concepts required to start discovering useful chords and naming them after the modes with which they are compatible.
From the General to the Specific
Early in the tutorials, we explained how to take a set of notes (say, on the interactive piano) and turn it into a set of pitch classes, represented as a colored clock, and from there, how to turn these into more and more general objects, in the following section. Equivalence Groups discussed how to go from a row in the set classes table to a playable, Octave-equivalent chord or scale object, but it glossed over the fact that we would need to know which set classes relate to which chord type.
Indeed, we would need to start with a list of chord types that happened to yield all of the set classes when we apply the concepts discussed in Grouping Clocks. Unfortunately, the author knew of no such a reference; but fortunately for the reader, he created one in Harmonious.
Getting a Formula from an OTC-Equivalent Object
Let’s introduce one more type of musical set theory object: the OTC-equivalent object. Such an object groups together all objects with the same interval classes above a given root. Here permutation or order matters (to some degree) so we are not permutation-equivalent.
We can tweak a visual representation we have seen already, the black-and-white clocks of set classes, but we add a twist: the clock does not have to be in prime form (the lowest numerical representation) but instead it must simply include zero (the top of the clock) in its list of pitch class digits. Zero becomes our moveable, “floating root.”
037 and 058: two objects that are not OTC-equivalent, but are OPTC-equivalent. In our case they represent different inversions of the minor triad, so we need to treat them as different objects. (A single OPTC-equivalent object, (037), would conflate this difference, ignoring order.)
From the section on colored clocks, we know that there are 212 or 4,096 possible pitch class sets (including silence, notes, intervals, chords, scales, and the chromatic scale itself). If we restrict this to just clocks including the pitch class zero, then there are 211 or 2,048 objects, or 1,969 of cardinality three to nine. If we ignore the 1,409 containing chromatic clusters, we are left with only 560 OTC-equivalent objects to investigate. Of these, 479 are subsets of the seventeen modes we have mentioned.
Chord Formulas Needed
The main problem with this representation is that it does not directly suggest what possible chord types might relate to each OTC-equivalent object, since each such object collects up interval classes relative to the root (around a circle), but not actual intervals with theoretically meaningful names. To get a candidate chord formula to attempt to match up with a mode, we need explicit interval names, like either ♭2 or ♭9, or either ♭5 or #4, not interval classes that represent more than one theoretical interval.
Since there is a one-to-many relationship between this representation and candidate chord formulas (which would help us in our search to produce useful and meaningful, nameable chord types) one approach might be to use the mode formulas (in the following section) that we do know, and match them up with every possible OTC-equivalent object.
Mining Mode Formulas for Chord Formulas
We ended Evenness & Clusters by pointing out that only 124 set classes were chromatic cluster-free. “The Game” is to match up our 479 cluster-free OTC-equivalent objects with the seventeen modes, so we can pull out the intervals from their formulas:
|R 3 5 7 911 13|
|R 3 5 7 9 11 13|
|R 3 5 ♭7 9 11 13|
|R ♭3 5 ♭7 9 11 13|
|R ♭3 5 ♭7 9 11 ♭13|
|R ♭3 5 ♭7 ♭9 11 ♭13|
|R ♭3 ♭5 ♭7 ♭9 11 ♭13|
|R 35 7 9 11 13|
|R ♭3 5 7 9 11 13|
|R 3 5 ♭7 9 #11 13|
|R ♭3 5 ♭7 ♭9 11 13|
|R 3 5 ♭7 9 11 ♭13|
|R ♭3 ♭5 ♭7 9 11 ♭13|
|R ♭3 3 ♭5 5 ♭7 ♭9|
|R 3 ♭5 5 ♭7 9|
|R 3 ♭5 5 ♭7 ♭9 #9 13|
|R ♭3 ♭5 #5 7 9 11 13|
A basic first pass results in around a thousand possible chord-mode combinations to investigate. See OTC-Equivalence for the complete tables, sorted by cardinality.
Why is it Called “The Game?”
The idea of an exhaustive list of chords and modes is inherently something of a boil-the-ocean endeavor. Putting this together involves leaving the concrete, mathematical realms of musical set theory and entering a realm of time-honored convention, where the ear is king and careful counting or mathematics cannot always answer every music theory question. But a century or more of jazz theory has laid the groundwork, for which we have explained the high-level concepts:
- Chord scale compatibility, focusing on the four scale types and their modes
- Extensions and avoid notes
- Leaving some notes out, and rootless or slash chords
Concepts from musical set theory include:
- A focus on just the cluster-free set classes (since they are subsets of cluster-free scales)
- Evenness as a guide to which chords and scales are the most useful (Q2) because of how they relate by voice-leading to other chords and scales (Q3)
The difficulty is that we have to apply all of the concepts and rules at the same time. Again, we refer to the tables in OTC-Equivalence and Chords where the work has been done. But this brings an end to (Q1): what are all the possible chords, their names, and how to play them? And more importantly, how would you use them? what do they do? etc.
If the high-level concepts have sunken in, then the beautiful structure of the chromatic scale of twelve-tone equal temperament has become more clear to us, and we can see the centrality of the diatonic scale without becoming limited by it. All without treating other scales as some weird corner of the musical landscape, but just as much a part of it.