Lydian-Dominant Theory

for

Improvisation

by

Norm Vincent

Lydian-Dominant Music Theory assumes the 12-Tone Tempered Scale andthe naturally occurring Physics of the OverTone Series. A small amount ofhigh school level Algebra is used in this treatise as Music Theory is highlyMathematical. In fact, in Plato's Scheme of things, Mathematics is derivedfrom Music! Music (i.e. organized vibrational frequencies) is Primal. Thissounds like modern physics to me.

Although one does not have to be a Mathematical Wizard to do Music,exciting new research has shown a definite link between the two. MusicalPerformance involves very high-level integrated mental processes we have onlybegun to explore in a Scientific manner. I find it regrettable that knowledgeknown to ancient peoples has become lost, suppressed, and distorted. It is myintention that this treatise be a "first step" toward the development of a trulyscientific exploration of the Domain of Music and all its ramifications. We willstart with the basic physical facts.

The OverTone Series

The OverTone Series is a naturally occurring physically demonstrable set ofFrequencies present above any given pitch. The relative mix of these upperfrequencies is different for every tone generator. This is why different musicalinstruments sound remarkably different even though they are sounding thesame pitch. The OverTone Series is infinite in extent, but in practice, only thefirst few are important to us here as the relative volume of the upper partialsgradually becomes inaudible.

OverTone # 1 - 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...

Note # 1 - 1 5 1 3 5 b7 1 2 3 #4 5 6 b7 7 1 ...

Note Name C - C G C E G Bb C D E F# G A Bb B C ...

Try this experiment on a Piano. Hold Down the Sustain Pedal. Strongly Hit andRelease a low 'C'. What do you hear? I hear all sorts of other strings vibrating.The sounding strings are not accidental, they are strictly determined by theOverTone Series. These associated frequencies are called Harmonics.

The exact single-octave Harmonic Series values are given in the next table. Itis an ordering of the Rational Numbers. These values are used the same waythe fundamental values of Sines and Cosines are used in Trigonometry. Yousimply multiply the initial pitch by these values to derive the frequency of thedesired harmonic.

OverTone # 1 - 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...

Note # 1 - 1 5 1 3 5 b7 1 2 3 #4 5 6 b7 7 1 ...

Harmonic # 1 - 1 3/2 1 5/4 3/2 7/4 1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 1 ...

A few comments on the OverTone Series relevant to Lydian-Dominant Theory.Notice the natural occurrence of the b7 and the #11. Also notice the naturaloccurrence of the Chord » {1 3 5 b7} and the Scale » {1 2 3 #4 5 6 b7 1}. Iwill refer back to these facts later on in this treatise. The OverTone Series isexplained in greater depth in my book Natural Music Theory.

The 12-Tone Tempered System

Our modern 12-Tone Tempered Scale is derived from the Pythagorean Spiralof 5ths.The 12-Tone Tempered Scale approximates the values of the PureHarmonics of the naturally occurring OverTone Series using only the ratio forthe 5th » (3/2).

What is a 5th? Briefly, what is known as a 5th is the first distinct (other thanoctave doublings) OverTone to emerge from the OverTone Series and isassociated with the number 3. Experiments on strings by ancient peopleshowed that when you take a string tuned to any starting pitch and divide itinto 2's you get octave doublings. When you divide it into 3's, you get what isknown as a perfect 5th. When you divide it into 5's, you get what is known asa Major 3rd. When you divide it into 7's, you get what is known as a Minor 7th.This process can continue to any desired level and is explained in greaterdepth in my book Natural Music Theory.

The formula for the Pythagorean Spiral of 5ths is:

p·(3/2)k k is any Integer and p is any starting Pitch.

Twelve intervals of a 5th almost closes in on itself - the "snake almostswallows its tail". The discrepancy has been known about since ancient timesand goes by various names. I call it - the Pythagorean Error Factor. Considerthe following table consisting of Twelve 5ths Up (#) and Twelve 5ths Down

(b).

0 1 2 3 4 5 6 7 8 9 10 11 12

C G D A E B F# C# G# D# A# E# B#

C F Bb Eb Ab Db Gb Cb Fb Bbb Ebb Abb Dbb

In natural occurring pure intervallic evolutions, a B# in not equivalent to a C.Likewise, a Dbb is not equivalent to a C. Both B# and Dbb are audiblydifferent from C. However, it was discovered in early classical times(European) that if you take an almost imperceptible amount (2 cents) awayfrom each 5th, you can get a Cycle of 5ths that does close in on itselfperfectly. The "snake eats its tail". Bach's Well Tempered Clavier was a greatsuccess in promoting the new system. The gain is tremendous - we now have12 different Keys to modulate to that all sound remarkably good. The cost isthat each 5th is 2 cents flat, a price that most are willing to pay for theusefulness of the system. In the 12-Tone Tempered system B#=Dbb=C. Thuswe end up with a true Pythagorean Cycle of 5ths.

To the right is a table showing thisCycle that is very concise andinformative. From it we can clearlysee each of the 15 Standard Keys andtheir relationships to each other. TheDominant relationship goes counter-clockwise. Notice the enharmonickeys. This is where the Flat Keysmerge into the Sharp Keys due toTempering.

From this information we canconstruct what is known as theChromatic Scale This Scale contains12 exactly equal intervals of a semi-tone (1/2 step).

{ C=B#, C#=Db, D, D#=Eb, E=Fb,F=E#, F#=Gb, G, G#=Ab, A, A#=Bb, B=Cb }

The exponential formula for our 12-Tone Tempered System is:

p·2(k/12) k is any Integer and p is any starting Pitch.

Why is the number 2 in this formula? Because the result multiplying ordividing any frequency by 2 is an 'octave' higher or lower. The Psycho-Acoustical perception of the same/different quality of octaves is discussed ingreat depth in my book Natural Music Theory.

The value of a chromatic interval is p·2(1/12). The accepted Modern Standardis A=440 cps, but any base pitch will do. In fact, the base pitch has beensteadily rising. It was A=432 in Beethoven's time.

The 12-Tone Tempered System is not without its problems. As opposed to thefact that 5ths and 4ths are only slightly out of tune, other intervals are grosslydistorted. In particular, the out-of-tune-ness of the Major 3rd led to what isknown as Just Intonation - the harmonic value (5/4) being used rather thanthe Pythagorean (81/64). Similar problems exist with the b7, #4, and othertheoretically important notes.

The Cosmic Quirk involving the number 12, legendary for its number mysticproperties, in evolving our common 12-Tone Tempered System and theevolution of other N-Tone Tempered Systems from Cycles different from (3/2),some of which are more exact than the 12-Tone Tempered, are developed ingreat detail in my book Natural Music Theory.

Discussion of Dominance in Music

Before we go any further, I will define Lydian-Dominant. Lydian is a wordfound in old Greek treatises on Music referring to the classical 7-note (so-called Dia-Tonic) Scale with the 4th Scale degree raised (#) a half-step. Theeasy way to remember this is by playing a Scale on a Piano starting on 'F' andpressing only the white notes. As the 4th degree of an F-Major Scale is a Bb,we clearly have a different Scale - the Lydian Scale. This Scale is a Major Scalewith a #4th degree. In the exposition that follows, I will be doing all examplesin the Key of C. The CLydian Scale is spelled: { C D E F# G A B C }.

The notion of Dominance is quite complex. Western polyphonic Multi-KeyedMusic based on the 12-Tone Tempered Scale has led to the concept of theDominant 7th Chord. It is a psycho-acoustic tension and releasephenomenon. This is how it is postulated to work in the European ClassicalMusic Theory.

The four note Chord formed on the 5th degree of the Major Scale is called theDominant 7th Chord. It is formally referred to as the V7 Chord. The presenceof the Dominant 7th (b7) in the Chord sets up a tension that needs to bereleased. Classical theory states that this tension is released by resolving tothe Key Root Chord, also known as the I Chord - G7 »»» C.

The table to the right shows an idealized form of this resolution.

What, exactly, causes this resolution to occur? Remember, we are dealing withpsycho-acoustic phenomena which is highly subjective and the topic of muchdebate down through the ages continuing to the present day. For now, let'sput politics aside. You are encouraged to do the following experiment on aPiano or Guitar and decide for yourself.

In the G7 Chord, the root (G) and fifth (D) are quite consonant, as are the

In the G Chord, the root (G) and fifth (D) are quite consonant, as are theroot (G) and Major third (B). The Majorthird (B) and the fifth (D) form aninterval of a Minor third, alsoconsidered consonant, as do the fifth(D) and the Dominant 7th (F). Theinterval between the root (G) and theDominant 7th (F) was considereddissonant in old classical theory. Mostmodern theorists are not so strict andwould consider the interval as colorfulif not downright consonant.

This leaves us with the interval B-F. This interval was actually outlawed by theMedieval Christian Church and marked with the name Intervallo Diabolo. ThisInterval spans 3 whole tones. There are many names for this interval -diminished 5th, augmented 4th, #11th, and my favorite - TriTone. A TriTone isnaturally formed between the Major 3rd and the Dominant 7th. Because of therelative consonance of all the other intervals in the G7 Chord, most, if not all,of the tension in this Chord is caused by the presence of this TriTone interval.Lydian-Dominant Theory is, literally, the study of TriTones.

In Western Classical Music Theory, thisinterval was always resolved inwardly.

We are now at the first really important place in Lydian-DominantTheory.

The TriTone interval also resolvesoutwardly as easily and as naturallyas it resolves inwardly.

You should try this out repeatedly on a Piano and let your ear be your finalarbiter. These resolutions are symmetric and, I believe neither has anyprecedence over the other. I agree with most modern theorists, that they areequivalent and neither should be preferred for any subjective reasons. Sowhat does this mean???

All students of Jazz soon discover the ubiquitous Chord progression:

II m7 » bII7 » I Maj7

Consider the Chord progression of the verse part of The Girl From Ipanema byAntonio Carlos Jobim. It goes like this:

FMaj7 G7 Gm7 Gb7 FMaj7 Gb7

What in the world is that Gb7 doing all over the place??? By classical rules,this should be a C7 as it is the Dominant 7th Chord in the Key of F. How doesthe Gb7 cause the desired resolution to the I Maj7 Chord? This is the heart ofLydian-Dominant Theory. In the table that follows I will spell out the requisitechords, identify the relevant TriTone - the rest is magic.

The operational TriTone is {E - Bb}(remember Fb=E). Each of theseDominant 7th chords has the sameTriTone !!! As stated earlier in theanalysis of the generalized Dominant 7th,it is the TriTone that causes the tensionthat gets resolved.

Notice also that C and Gb are themselvesTriTones. Consider this. It would seemthat the root (I) of the Chord and it's closely allied 5th are quite exchangeable.It is the TriTone Core of the Chord that is Invariant. We will see later justhow ambiguous TriTones can be. One can actually "get lost" aurally in animprovisation with many sequential Lydian-Dominant changes in the Chordprogression. Thus the first Postulate of the Lydian-Dominant Music Theory.

Postulate 1

Any Dominant 7th Chord can be replaced by its TriTone equivalentwith no loss of resolving power.

This postulate is the Fundamental Assertion of Lydian-Dominant Theory.Once we recognize the power of Lydian-Dominant structures and introducethem into our music, we find that the word modulation takes on an entirelynew and exciting meaning. I would also add, that along with this newfoundmodulating flexibility, a wealth of harmonic richness is also realized. Classicalmusic theory shortchanged itself terribly by banning and/or ignoring thisfundamental theoretical fact implied by the OverTone Series and realized bythe 12-Tone Tempered Scale.

Understanding and appreciating the fundamental assertion of the firstPostulate - TriTone Dominant Substitution - is but the beginning of ourjourney. Next we will study and develop the essential core elements that arethe "building blocks" of Lydian-Dominant Theory - the TriTones.

Postulate 2

There are 6 TriTone pairs

TT1 = { c - f#/gb } TT4 = { a - d#/eb }

TT2 = { g - c#/db } TT5 = { e - a#/bb }

TT3 = { d - g#/ab } TT6 = { f/e# - b/cb }

Each pair is associated with two interchangeable Dominant 7th Chords. Thatis, they may be substituted for each other to provide harmonic richness and/orchromatic movement as can readily be seen in the Chord progression snippetfrom The Girl From Ipanema used above.

The following table enumerates the 6 Dominant7 pairs and their associatedTriTones. Read this table up and down the columns - the involved TriTone is inbetween.

C7 G7 D7 A7 E7 B7/Cb7

a#/bb - e/fb f/e# - b/cb c - f#/gb g - c#/db d - g#/ab a/bbb - d#/eb

F#7/Gb7 Db7/C#7 Ab7 Eb7 Bb7 F7

Now we know why that Gb7 is there in Jobim's Song. In fact, all of his work isheavily Lydian-Dominant. Check out his compositions Wave and Desafinado tosee what I mean.

The BIG Fact is, that Jazz is heavily permeated with Lydian-Dominant ChordProgressions and Melodic development. Swing, Blues and their derivatives inthe Pop/Rock styles less so, but still Lydian-Dominant. South American formslike Samba and Bossa Nova and Tango are, again, heavily permeated withLydian-Dominant Chord Progressions and Melodic development. Likewise, theAfro-Cuban inspired Salsa forms. Certain 20th Century Classical Composershave also ventured into Lydian-Dominant, Debussy, Ravel, Stravinsky to namejust a few.

You should become aware of an odd thing with these pairs. Are theyaugmented 4ths (#11) or are they diminished 5ths (b5)? In Natural MusicTheory (pure Harmonic Series intervallic evolutions), there is a definitedifference. In the 12-Tone Tempered System there is not. The very processof Tempering obliterates any difference.

Indeed, the TriTone interval is an Artifact of the 12-Tone Tempered System -it doesn't even exist in non-tempered systems. Approximations of it do exist inpure Scale, in fact, an infinite number of them. But as the TriTone has a valueof p·2(1/2) , ( any starting pitch p times the square root of 2 ), all theHarmonic Series (which is based exclusively on rational numbers) can do is

Harmonic Series (which is based exclusively on rational numbers) can do isspit out closer and closer approximations to the TriTone. This is not at all asweird as it seems at first glance. A famous Mathematical Proof, attributed toEuclid, may be found in any high school Geometry textbook showing that:

No rational number, that is, an number of the form a/b , where a, b arenatural numbers, can equal 2. TriTones are intimately related to this numberthat caused the Pythagoreans so much trouble with ir-rational numbers. Thistopic and other related items are explored in greater depth in my book NaturalMusic Theory.

Postulate 3

There are 6 Dominant7 b5 Chords.

C7 b5 = F#7 b5 = { C E Gb/F# Bb/A# }

G7 b5 = Db7 b5 = { G/Abb B/Cb Db F }

D7 b5 = Ab7 b5 = { D/Ebb F#/Gb Ab C }

A7 b5 = Eb7 b5 = { A/Bbb C#/Db Eb G }

E7 b5 = Bb7 b5 = { E/Fb G#/Ab Bb D }

B7 b5 = F7 b5 = { F A Cb/B Eb/D# }

This is the quintessential Lydian-Dominant Chord. It is both Lydian andDominant. This Chord puts the 'A' in Take The 'A' Train, the 'Des' inDesafinado, the 'Tune' in Bernie's Tune, and that special sonic twist in somany Lydian-Dominant compositions.

The Chord is comprised of two TriTone pairs a Major 3rd apart. In the case ofthe C7 b5 - F#7 b5 pair, they are {c - f#/gb } and { e - a#/bb }. Play thisChord - listen to it. Grab the 4 notes in the C-F# pair - { f# a# c e }. Nowplay a C bass note - listen. Now play an F# bass note - listen. What do youhear? I hear the same tonality in each case. Nothing really changes except thevoicing, i.e. a particular rearrangement of notes.

For the Improviser, this is really important. The first problem encounteredwhen analyzing a particular Chord progression is figuring out what Scale(s) areimplied by which Chord(s). It doesn't matter how fast your fingers are or howgood your tone is if you're playing the wrong notes - it'll still sound bad. Thisis the major problem I have with some Improvisational Methods of listing aseemingly different Scale to each and every Chord in a progression. I find itmore confusing than helpful, especially to the novice.

The fact is, that the underlying scalar note group frequently does not changeat all ! More often than not, whole sequences of Chord changes define thesame note group. It doesn't matter which notes in a particular Scale youchoose to include in a motif, its still the same underlying tonality. This is why

Handel sounds as homogeneously boring as a lot of more modern music of allkinds - the whole song is defined by one scalar group! You might see a lotof Chord changes, but all that is really changing is which note(s) the bassplayer is currently emphasizing. For the Improviser, nothing changes at all -its same Scale throughout.

Once the student progresses up to Lydian-Dominant, they find that what lookslike wicked hard Chord changes are really not so bad at all. There are only 6Dominant7 b5 Chords, not 12 as with most other chords. This makes learningthem take half the time. All that remains is to fit them in properly. Lydian-Dominant is actually easier than it looks. Things get even simpler in the nextpostulate.

Postulate 4

There are 3 TriTone Quad Diminished Sub-Systems

DQ1 = Cdim = Ebdim = F#dim = Adim = { C Eb/D# Gb/F# A/Bbb }

DQ2 = C#dim = Edim = Gdim = Bbdim = { C#/Db E/Fb G Bb/A# }

DQ3 = Ddim = Fdim = Abdim = Bdim = { D/Ebb F Ab/G# B/Cb }

This is the infamous Diminished7 Chord. As we can easily see, the quads form3 mutually exclusive sets of 4 notes. Each group is comprised of 2 interlacedTriTones a minor 3rd apart. Notice that 4 super-imposed minor 3rds equals anoctave in the 12-Tone Tempered System. This note group is totally symmetricany way you look at it.

DQ1 = TT1 + TT4 = { c - f#/gb } + { a - d#/eb }

DQ2 = TT2 + TT5 = { g - c#/db } + { e - a#/bb }

DQ3 = TT3 + TT6 = { d - g#/ab } + { f - b/cb }

No group of notes has caused more problems for Music Theorists than this one.Just naming the intervals is problematic within the old system. Below is atechnically correct naming of a C dim7 Chord.

C - The Root - we'll see ...

Eb - A Minor 3rd above the root C - O.K.

Gb - A Diminished 5th above C - A Minor 3rd above Eb - O.K.

Bbb - What shall we call this interval???

Bbb is a Minor 3rd above Gb and it is a diminished 5th above Eb. But whatinterval is it above C??? I have heard it called a diminished 7th. In StandardMusical Nomenclature, 7ths are designated as major and minor along with2nds, 3rds, and 6ths - 4ths and 5ths are called perfect, and along with roots,can be diminished, and, augmented. So what is a diminished 7th???

I claim there is no such thing as a diminished 7th. This Chord is a 100%total Artifact of the 12-Tone Tempered System. It doesn't exist at all in anyOverTone Series derived Systems. It is an emergent property of the 12-ToneTempered System and is central to Lydian-Dominant Theory. Interestingly,other Tempered Systems have analogous structures and are discussed in depthin my book on Natural Music Theory.

Bbb/A is clearly a Major 6th (in disguise) above the root C. It acts like a 6th,it sounds like a 6th, so why not call it a 6th !!! I seriously suggest that werename this wonderfully ambiguous Lydian-Dominant note set thediminished 6th Chord - C dim6. As justification in addition to the aboveanalysis, I would point out that this Chord is remarkably close in sound andfunction to the minor 6th Chord, a Chord more commonly used in olderAmerican music, and still important in some indigenous styles like Tango. ThisChord has an ambivalent tonality and differs from the dim6 in that the 5th isperfect rather than diminished.

Cm6 = { C Eb G A }

Cdim6 = { C Eb Gb A }

Furthermore, if we invert the 6th in the C m6 thereby changing the root note toA instead of C, we derive the modern Jazz Chord, the A m7 b5 - the so-calledhalf-diminished Chord. This Chord will be discussed in depth later on in thistreatise.

The dim6 sub-systems also define 3 Lydian-Dominant Scalar entities calleddiminished scales. They will be discussed later on in this treatise. Thediminished quads are integrally involved in several other important Lydian-Dominant Chords which leads us to the next postulate.

Postulate 5

There are 3 Sets of Dominant7 b9 Chords, one for each DiminishedQuad Sub-System.

Technically speaking, there are 12 of these chords. In Lydian-Dominant realityhowever, they each fall into one of the 3 Diminished Quad Sub-Systems. Iwill show this using:

DQ1 = {c eb/d# gb/f# a/bbb }

Consider the Chord:

F7 b9 = { F A C Eb Gb }

As discussed before, most of the "action" (tension-release) in a Chord iscreated by the 3rd and 7th. In this Chord the b9 also contributes significantly.Play this Chord alternating the b9 (Gb')with the octave (F'). What does yourear think of this? We already know that in a Dominant Chord, the 3rd and 7th

are a TriTone. In this Chord, the 5th and b9th form another TriTone! Onceagain, as in the Dominant7 b5, there are two TriTone pairs in the same Chord.

But this is a property of diminished quad sub-systems - is there one lurkingwithin this Chord. Sure is. The 3rd, 5th, b7th, and b9th form a dim6 Chord!This is the substance of this postulate. The "action" in this Chord is caused byevery note but the root. This is one of the most striking aspects of the Lydian-Dominant System - that roots are frequently extraneous to the function ofa Chord. They can be exchanged in certain proscribed ways.

In this case, DQ1 contains the "action" notes for:

F7 b9 = { F + DQ1 = ( A C Eb Gb ) }

Ab7 b9 = { Ab + DQ1 = ( C Eb Gb Bbb ) }

B7 b9 = { B + DQ1 = ( D# F# A C ) }

D7 b9 = { D + DQ1 = ( F# A C Eb ) }

Notice also, that the exchangeable roots themselves form a dim6 quad !!!Grab the diminished quad on a Piano with the right hand. Now play each rootin turn and listen. Do you hear what I hear? The "action" notes are the sameno matter how you choose to voice them. Changing the root notes alters thenote set (thus the sonority changes), but the tension/resolution mechanism isinvariant. Lydian-Dominant is very cool. The same thing goes for the othertwo quads and figuring them out I leave to you as an exercise.

Don't forget - this note-group is in the Dominant7 Chord-Space and, as such,can be substituted for its TriTone equivalent! Lydian-Dominant is wicked cool.A frequent companion of the X7 b9 is the subject of the next postulate.

Postulate 6

The minor7 b5 / minor6 Chord.

As mentioned briefly above, this note group has a dual nature. It also called

the "half-diminished" Chord. This makes some sense in that it is formed byadding a b7 to a diminished triad. However, this pseudonym hides the fact ofthe dual nature of this Chord - it can be looked at as a 6th Chord or a 7thChord, dependent on other factors such as melodic leading, resolution, androoted-ness.

Cm7 b5 = { C Eb Gb Bb }

Ebm6 = { Eb Gb Bb C }

When used as a m7 b5, it is most commonly the first part of what I call aminor II-V-I:

Major II-V-I Dm7 - G7 - C M7

minor II-V-I Dm7 b5 - G7 b9 - C m9

Though this is the most common usage of this Chord, especially in Jazzcompositions, the subtle ambiguity of this note group lends itself to otheruses. It doesn't have to resolve to a I mx Chord through the V7 b9- it canjust as easily go other places though not anyplace. Check out Stella ByStarlight.

When this note group is used as a m6 Chord, it is quite common to find itused as a I Chord! There are innumerable songs that do this Remember, allthat has changed is the root note. It's the same basic tonality, butemphasizing a different bass note gives this note group a different quality.This note group is truly ambivalent in character and has power in manydifferent directions.

Actually, the m7 b5 is the OverTone Series Inverse of the Dominant7 Chordmaking it an important fundamental theoretical construct - want to knowmore? The derivation of this Chord and that of minor itself are presented indepth in my book on Natural Music Theory.

Postulate 7

There are 4 Augmented Triad Sub-Systems.

Notice that 3 super-imposed Major 3rds exactly equals an octave in the 12-Tone Tempered System. Like the dim6 sub-systems, the 4 Augmented sub-system triads are totally symmetric and form 4 mutually exclusive sets of 3notes. They are:

AT1 = Caug = Eaug = Abaug = { C E G#/Ab }

AT2 = Ebaug = Gaug = Baug = { Eb/D# G B }

AT3 = F#aug = Bbaug = Daug = { F#/Gb A#/Bb D }

AT4 = Aaug = C#aug = Faug = { A C# F/E# }

Though not properly Lydian-Dominant, the 4 augmented triads are heavilyinvolved in Lydian-Dominant Theory in at least two important ways. First, thescales that underlay this Chord are all Whole-Tone (altered) Scale variants.These scales can also underlay other important Lydian-Dominant Chords. I willhave more to say on this later in this treatise. Second, Augmented Triads areusually used as Dominant 7th or 9th Chords making them Lydian-Dominant andsubject to all the other Lydian-Dominant Postulates.

Here's where the fun begins again. These 4 augmented sub-systems imply 4corresponding Augmented7 sub-systems as well. I'll show you the T1 sub-system and leave the other three for you to do as an exercise.

C aug7 = { AT1 = ( C E G# ) + Bb}

E aug7 = { AT1 = ( E G# C ) + D }

Ab aug7 = { AT1 = ( Ab C E ) + Gb }

As with the diminished sub-systems, these augmented sub-systems are a100% total Artifact of the 12-Tone Tempered System. As shown above inPostulate 5, the X7 b9 is essentially a diminished quad plus one of 4 relatedroots, themselves forming another diminished quad. With these aug7 notegroups, we have an augmented triad plus 3 related Dominant sevenths,themselves forming another augmented triad!

The Aug7 Chord is not as common used as many other Lydian-DominantChords, but because it in the Dominant Group, it turns up in strategicpositions in many songs and must be handled properly. As mentioned brieflyabove, the augmented sub-systems are intimately connected with Whole-ToneScales which brings us to our next postulates after a short digression.

Before we get to the next postulate I want to briefly discuss the WesternClassical bias (from the Greeks) toward the 7-note (so called) Dia-TonicScale and an important bit of nomenclature

Despite the fact that we in the Western Cultures have come to enshrine"Rational Thinking" as the epitome of human evolution, and view anycontinued reliance on pre-rational systems as atavistic and downrightignorant, we have nevertheless perpetrate on each unsuspecting generationsince the "Enlightenment" a plethora of number mystic systems which areunquestionably accepted as "cosmic" Law.

Case in point - ask anyone why there are 7 days in a week and you will

usually get stunned silence and strange looks for a reply. Some willdesperately be mentally searching for a "logical" reason (there must beone) for these commonly encountered systems. You may get astraightforward "... and God rested on the 7th day." from a Religionist,and though I respect their right to their strongly held convictions, I don'tfeel that I am bound by them in any way. The point is, that there is Nocosmic reason at all why the number 7, or any other number for thatmatter, should be specially favored.

In Music Theory, we use the two terms Scale and Chord without muchdiscretion. In fact, there is no real difference between them. It onlydepends on how far we space out the intervals and even this is poorlydefined. If we space out the intervals in whole and half steps the note-group is usually called a Scale. If we super-impose Major and minor thirds,it is usually called a Chord. Problem is, some scales have intervals of amin 3rd, and some chords have intervals of whole step. I and manymodern music theorists use the term ChordScale. I also use the termnote-group. This makes more sense to me than trying to define adifference that does not exist.

Consider the following analysis:

C-Major Scale = { C D E F G A B }C Maj13 Chord = { C E G B D F A }

A better way to show this is:

F-Major Scale = { F G A Bb C D E }C13 Chord = { C E G Bb D F A }

C13 is in the Dominant7 Group in the Key of F. In both cases, the note-group is identical and the same ChordScale is defined. There happens tobe 7 notes in it and, indeed, there are a lot of 7-note scales. But, there aremany other ChordScales with a different number of elements that are justas useful and legitimate as the dia-tonic.

The number 12 (as in 12-Tone Tempered System, inches in a foot, monthsin a year, hours of day/night, and various groups of Apostles ) is alsototally bunged up with number mysticism. As usual, I discuss this issue ingreat depth in my book on Natural Music Theory.

Postulate 8

There are 2 Whole Tone Scalar Sub-Systems.

WT1 = AT1 + AT3 = TT1 + TT3 + TT5 = { C D E F#/Gb G#/Ab A#/Bb }

WT2 = AT2 + AT4 = TT2 + TT4 + TT6 = { F G A B/Cb C#/Db D#/Eb }

Share onFacebook

Share onTw itter

Share viaemail

This is a totally symmetrical Scale of 6 notes! It is constructed of nothing butWhole steps. Play them on your instrument - was your ear fooled? They areeven more fun to sing - try it. This Scale is a Lydian-Dominant Artifact of the12-Tone Tempered System. It is not found in natural OverTone Series harmonicderivations. Once again, tempering allows the "snake to eat its tail".

The Whole-Tone Scale and its altered variants underlay many Lydian-DominantChords. Basically, they fit any Chord with a diminished 5th or an augmented5th or both They can also be used when a #11 or a b13 is present. I will showhow they can be used to fit the common Lydian-Dominant Chord - theDominant7 b5.

C7 b5 = { C E Gb Bb }

CWT = WT1 = { C D E Gb Ab Bb }

Notice that we have 4 notes of WT1 already in the Chord itself! The two notes

that are missing are D and Ab. The D is easily justified as a 9th. As 9ths are,in reality, only the 2nd note of a Major Scale, and this is a Major Chord, it canalways be used in a situation like this. The Ab is more of a problem to justify.Technically, C7 is a Major Mode Chord and as such, a Major 6th should beplayed giving us an A rather than an Ab. Indeed an A can be played turningour Scale into one of the many Whole-Tone variants. However, using the Abgives us a slightly "outside" sound. In particular, it provides sonic varianceusing a non-critical note - the 6th. This is very important to the Improviser.

Next, I'll show how the Whole-Tone Scale can be used to fit anaugmented 7th Chord.

C aug7 = { C E G# Bb }

CWT = WT1 = { C D E F# G# A# }

Notice that we have 4 notes of WT1 already in the Chord itself! The two notes

that are missing are D and F#. As above the D is easily justified as a 9th.This time, the F# is the problem to justify. Strictly speaking, as this is a MajorChord, we should have an F rather than a F#. Indeed an F can be playedturning our Scale into a Whole-Tone variant. However, using the F# gives us aslightly "outside" sound. In particular, it provides sonic variance using a non-critical scalar note - the 11th. This is very important to the Improviser.

A comment on "playing outside"

Jazz players are famous for "playing outside" (i.e. playing non-chordscale

implied notes) in the course of their improvisations. Indeed, it is animportant part of the Jazz Style. I believe, however, that not all "outside"notes are justified at the theoretical level. Some "outside" notes are justplain wrong - i.e. not at all justifiable within the structure of the Chordprogression. Too often, "playing outside" is used as an excuse for playingwrong notes due to an inadequate analysis of the Chord Progressionimplied Harmonic Structure of a piece.

Postulate 9

There are 3 Diminished Scalar Sub-Systems.

DS1 = ( DQ1 + DQ3 ) = { C D Eb F F#/Gb G#/Ab A/Bbb B/Cb }

DS2 = ( DQ2 + DQ1 ) = { G A Bb/A# C C#/Db D#/Eb E/Fb F#/Gb }

DS3 = ( DQ3 + DQ2 ) = { F G Ab/G# Bb/A# B C#/Db D/Ebb E/Fb }

This is a totally symmetrical Scale of 8 notes! It is constructed of alternateWhole and Half-steps. Play them on your instrument - was your ear fooled?They are even more fun to sing - try it. This Scale is a Lydian-DominantArtifact of the 12-Tone Tempered System. It is not found in natural OverToneSeries harmonic derivations. Once again, Tempering allows the "snake to eatits tail".

The diminished Scale comes in two flavors DSwh and DShw depending on howthe diminished Scale is constructed - whole step first or half-step first. Thefollowing table shows the difference.

1 ½ 1 ½ 1 ½ 1 ½

CDIM(wh) = C D Eb F F#/Gb G#/Ab A/Bbb B/Cb C'

½ 1 ½ 1 ½ 1 ½ 1

CDIM(hw) = C C#/Db D#/Eb E/Fb F#/Gb G A Bb/A# C'

Notice that CDIM(hw) = C#DIM(wh) !!! All we do is start on a different notein scalar sub-system. This is generally true of every one of these scalesleading to the following relations.

DSwh1 = DShw2

DSwh2 = DShw3

DSwH4 = DShw1

There are many ways to use these scales. In either flavor, they remainwonderfully ambiguous and their use now and again over the proper Chord

changes, though tricky, creates much sonic richness. I will show some ways touse these scales and leave others for you to investigate as an exercise.

I will use the classic Lydian-Dominant Chord progression elaborated onextensively above:

The Lydian-Dominant II - V - I »»» IIm7 - bII7 - I Majx

Let's work in the Key of C. The Chord progression we need to fit is: Dm7 -Db7 - CMajx - the x signifying some form of Major Chord like a C6, a CMaj7,or a CMaj9.

LD II-V-I Dm7 Db7 CMaj9

DDIM(wh-up) (D E F G) (Ab Bb Cb Db) D'

* Set1 Set2 *

DDIM(wh-down) (D' C B A) (Ab Gb F Eb) D

Notice that there is an Up and a Down version of the Scale, both beingconstructed of alternating whole & half-steps. The Scale D DIM(wh) iscomposed of 2 sets (called tetrads from Greek Music Theory) of 4 notes. Inthe Up version, Set1 is the first 4 notes of a Dm Scale and Set2 is the first 4

notes of a Abm Scale. In the Down version, Set1 is the last 4 notes of a Dm(Dorian as implied by the Key of C) Scale and Set2 is the last 4 notes of the

corresponding Abm Scale.

I hope by now that you have noticed that D & Ab are TriTones !!! Thisshouldn't be a surprise to you anymore. Play the chords and the accompanyingscales on the Piano - listen. They are super-diminished every way you look atthem. Try playing them in "thirds" - in "fourths". See if you can find other waysto use these wonderfully ambiguous Scale patterns.

For now, I will conclude this treatise with an excerpt from my book on NaturalMusic Theory. It deals with the actual OverTone Series implied note-groupsthat underlie Lydian-Dominant Theory. To appreciate its simplicity one onlyhas to look carefully at the OverTone Series and list the note-groups byDoublings.

OverTone Note#

NoteName Analysis

1 F C0 Fundamental

2 1 C1Fifth

3 5 G1

4 1 C2

Dominant75 3 E2

2Dominant7

6 5 G2

7 b7 Bb2

8 1 C3

Lydian-DominantScale

9 2 D3

10 3 E3

11 #4 F#3

12 5 G3

13 6 A3

14 b7 Bb3

15 7 B3 Leading Tone

16 1 C4 Doubling

Postulate 0

The Primal Lydian-Dominant ChordScale

C7 » { C D E F# G A Bb }

This ChordScale is Legendary. It is found the world over and is usuallyassociated with the local culture's Goddess. Notable among these are theGreek Sappho of Lesbos and the Hindu Saraswati - Goddess of Music,Mathematics & the Sciences. It is a wonderful Scale and wholly derived fromthe OverTone Series generated Harmonic Series. This knowlege is Ancient!Most people today don't know that Plato, Aristotle, Euclid, Ptolemy, and who-knows-how-many others wrote extensively about Music Theory. It is writtenabout in the Vedas, the World's oldest books. It is amazing to me to beconstantly re-discovering facts known to humans so long ago and thenforgotten in the headlong rush of Civilization's March.

Discussion based on the previous OverTone Series Table:

Note the 'natural' note-group progression:

First, you produce a Pitch, any FrequencyNext, you get a Doubling (see below)Then, a 5th - the first interval created that is not a Doubling. This interval (3/2)xgenerates the 12-Tone Tempered (Pythagorean) ScaleThen, you get a Chord - C7 - The Dominant7 Chord. It is 100% naturally derivedfrom the OverTone Series.Lo and Behold, this Chord implies the Lydian-Dominant Scale - CLD, not thePure Major, nor the Myxo-Lydian as older Music Theories claim.

Pure Major, nor the Myxo-Lydian as older Music Theories claim.

A Leading-Tone into the next Doubling. I will have more to say about this conceptin the book.The process continues to Infinity with new chordscales emerging that transcendand include those already manifest. As usual, I discuss this issue in great depth inmy book on Natural Music Theory.

Notice my use of the word "Doubling" instead of "Octave". The wordoctave contains a built-in and totally unwarranted bias toward 7-notescales - it literally means the "eighth" note. It is true that there are manywonderful and important 7-note Scales, but this fact hardly justifies prioritystatus. Doubling is a Psycho/Physio-Acoustical phenomenon - it has nothingat all to do with scales.

Concluding Remarks:

As we continue our studies, I will point out Lydian-Dominant elements whereever they occur in the Songs we learn and the Improvisations we create forthem. Regularly and methodically practice the preparatory exercises that Ihave created for you to learn the Lydian-Dominant System. Your hard work anddiligence will reap great rewards as your Improvisations develop thetremendous sonic richness implicit in the Brave New World of Lydian-Dominant Music Theory.

Norm VincentNorthStar Studios - April 2000norm4991@yahoo.com