The Economy Of Scale: The laws of diminishing intervals
By now we are aware of how the building blocks of chords - varied harmonic interactions within sets of differing intervallic structures - fit together, but what happens when we take just one variable of the input stage, and then multiply that one interval which we like the sound of, building a whole chord out of only one particular interval? Or, make a mirrored image from the current intervallic relationships? Economics.
Economics is an environment of variables. Economies of scale are environments of variables within pre-determined systems of known variables. Chords are equations of intervals based within a pre-determined system of intervals, the scale. Their environment, the song you are playing.
The conclusions that we can draw, by analyzing the intervallic relationships between the chords in the flowchart above, are that strange chord names are simply inversions of the intervals which made up the chord in the first place - exactly the same way that chord inversions are a different order of notes, a different order of intervals creates a different kind of chord. Chord types are chords which are made of similar intervallic structures - seconds, minor thirds, major thirds, fourths, fifths, sixths, minor sixths, major sevenths, dominant sevenths. Minor chords will contain the exact opposite intervallic structure as their major counterparts - each being chords built of thirds. There are two types of thirds - major and minor. There are two types of sevenths - major and dominant. There are two types of sixths - minor and dominant. There are two types of fourths - perfect fifths and fourths. There are two types of fifths - perfect fourths and fifths. There are two types of seconds - ninths and seconds.
On the guitar, because it is possible to play only six notes or less, notes must then be eliminated from upper-extension chords. In a G13 chord, the first note to be eliminated is the perfect fourth, C; the second note is the fifth, D; the third note is the root note. Then the next note in the close race to be eliminated, is the third, B. If we eliminate the right notes, 13 chords can become negative sus chords. When I, IV & V are eliminated from the chord. The group of eliminated notes - I-IV-V – are cast way into negative space, making their own chords. Suspended chords. Csus2 = Gsus4 = G-C-D = I-IV-V. While considering intervals, we can make the following observations:
(C – E – G) = C Major & (E – C – G) = C/E & (A – C – E) = A Minor & (C – E – A) = Amin/C
Major Triad = R + (Δ3 + m3) & Minor Triad = R + (m3 + Δ3)
XΔ7 = R + (Δ3 + m3 + Δ3) & Xmin7 = R + (m3 + Δ3 + m3)
R + (Δ3 + m3 + Δ3) = Δ7 chord & (Δ3 + Δ3 + m3) = Augmented Δ7 chord.
If: (A + C + E + G) = A min7 & X(min7) = R + (m3 + Δ3 + m3) & if: F#min7(b5) = (F# + A + C + E) & Xmin7(b5) = R + (m3 + m3 + Δ3)
Chords can be classified according to taxonomy:
- The chord Domain: is where the chord roams – what is the song, progression, or scale structure?
- The chord Kingdom: is where the chord rules – whom is the true, Great One?
- The chord Phylum: is where the chord serves - where does this chord lay between I and VII, and where does it best serve I?
- The chord Class: is relative to the intervals between each successive note - are they seconds, thirds, fourths, fifths, sixths, sevenths?
- The chord Order: is relative to the order of those same intervals, between each successive note - is it a major third, then a minor third? If a chord contains both major and minor intervals, inverting the order of intervals changes the musical taxonomy.
- The chord Family: is relative to the variable extensions introduced in the chord formula - is it a major seventh, or a dominant seventh?
- The chord Genus: is dependent upon its relationship with major and minor - is it a major third, or a minor third? Neither, or both?
- The chord Species: is the root note of the chord - what is the single note before any other interval or variable extensions are added?
The Devil's Chord: R + (-6–6–6) & R =√±1The diminished chord, or Co7, in common shorthand, is built of whole tones. Co - no seventh tone - is every other note in the whole tone scale. When adding the seventh tone, it gets diminished one step farther than a dominant seventh tone does. Here, the diminished seventh tone is equal to the sixth, while in a half-diminished chord, the seventh tone is diminished half as much to be a dominant seventh.
The half-diminished chord, is made up of exactly the same notes as, the minor7(flat 5) chord - the names are synonymous.
While being related to the Whole Tone Scale, it is usually thought of in relation to other scales, such as the Major Scale. It's possibly the only chord whose root structure is the same as all of its four inversions, the diminished seventh chord can be thought of from both the major and the minor perspectives. When building this chord from the minor perspective, it is built of three consecutive minor third intervals. When building from the major perspective, it is built of three consecutive major sixth intervals. In this particular, harmonically symmetrical chord, major is equal to minor. Building it from the major perspective, however, makes it more fun to think about: R + 6-6-6!
Let's rock and roll, all night long, baby!
About the Author:
Dealey is a Vancouver, Canada based guitarist, songwriter, recording engineer and producer. He is the author of the forthcoming independent book, "The Relative Nature of Chords: A Street-Smart Field Guide for Guitar." Watch for exclusive excerpts on Ultimate-Guitar! You can support his ventures by supporting his music here - or talk to him about collaborating on your project by email to: info(a.t.)aurora-studios.ca.