Lydian-Dominant Theory for Improvisation

Lydian-Dominant Theory for

Improvisation by

Norm Vincent

Lydian-Dominant Music Theory assumes the 12-Tone Tempered Scale and the naturally occurring Physics of the OverTone Series. A small amount of high school level Algebra is used in this treatise as Music Theory is highly Mathematical. In fact, in Plato's Scheme of things, Mathematics is derived from Music! Music (i.e. organized vibrational frequencies) is Primal. This sounds 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. Musical Performance involves very high-level integrated mental processes we have only begun to explore in a Scientific manner. I find it regrettable that knowledge known to ancient peoples has become lost, suppressed, and distorted. It is my intention that this treatise be a "first step" toward the development of a truly scientific exploration of the Domain of Music and all its ramifications. We will start with the basic physical facts.

The OverTone Series The OverTone Series is a naturally occurring physically demonstrable set of Frequencies present above any given pitch. The relative mix of these upper frequencies is different for every tone generator. This is why different musical instruments sound remarkably different even though they are sounding the same pitch. The OverTone Series is infinite in extent, but in practice, only the first few are important to us here as the relative volume of the upper partials gradually 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 and Release 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 the OverTone Series. These associated frequencies are called Harmonics. The exact single-octave Harmonic Series values are given in the next table. It is an ordering of the Rational Numbers. These values are used the same way the fundamental values of Sines and Cosines are used in Trigonometry. You simply multiply the initial pitch by these values to derive the frequency of the desired harmonic. OverTone # 1 - 2 3 4 5

6

Note #

5

1 - 1 5 1 3

7 8 9 b7 1

2

10

11

12

13

14

15

16 ...

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 natural occurrence of the Chord » {1 3 5 b7} and the Scale » {1 2 3 #4 5 6 b7 1}. I will refer back to these facts later on in this treatise. The OverTone Series is explained 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 Spiral of 5ths.The 12-Tone Tempered Scale approximates the values of the Pure Harmonics of the naturally occurring OverTone Series using only the ratio for the 5th » (3/2). What is a 5th? Briefly, what is known as a 5th is the first distinct (other than octave doublings) OverTone to emerge from the OverTone Series and is associated with the number 3. Experiments on strings by ancient people showed that when you take a string tuned to any starting pitch and divide it into 2's you get octave doublings. When you divide it into 3's, you get what is known as a perfect 5th. When you divide it into 5's, you get what is known as a 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 greater depth 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 almost swallows its tail". The discrepancy has been known about since ancient times and goes by various names. I call it - the Pythagorean Error Factor. Consider the 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 audibly different from C. However, it was discovered in early classical times (European) that if you take an almost imperceptible amount (2 cents) away from each 5th, you can get a Cycle of 5ths that does close in on itself perfectly. The "snake eats its tail". Bach's Well Tempered Clavier was a great success in promoting the new system. The gain is tremendous - we now have 12 different Keys to modulate to that all sound remarkably good. The cost is that each 5th is 2 cents flat, a price that most are willing to pay for the usefulness of the system. In the 12-Tone Tempered system B#=Dbb=C. Thus we end up with a true Pythagorean Cycle of 5ths. To the right is a table showing this Cycle that is very concise and informative. From it we can clearly see each of the 15 Standard Keys and their relationships to each other. The Dominant relationship goes counterclockwise. Notice the enharmonic keys. This is where the Flat Keys merge into the Sharp Keys due to Tempering. From this information we can construct what is known as the Chromatic Scale This Scale contains 12 exactly equal intervals of a semitone (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 or dividing any frequency by 2 is an 'octave' higher or lower. The PsychoAcoustical perception of the same/different quality of octaves is discussed in great depth in my book Natural Music Theory.

The value of a chromatic interval is p·2(1/12). The accepted Modern Standard is A=440 cps, but any base pitch will do. In fact, the base pitch has been steadily rising. It was A=432 in Beethoven's time. The 12-Tone Tempered System is not without its problems. As opposed to the fact that 5ths and 4ths are only slightly out of tune, other intervals are grossly distorted. In particular, the out-of-tune-ness of the Major 3rd led to what is known as Just Intonation - the harmonic value (5/4) being used rather than the Pythagorean (81/64). Similar problems exist with the b7, #4, and other theoretically important notes. The Cosmic Quirk involving the number 12, legendary for its number mystic properties, in evolving our common 12-Tone Tempered System and the evolution 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 in great 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 word found in old Greek treatises on Music referring to the classical 7-note (socalled Dia-Tonic) Scale with the 4th Scale degree raised (#) a half-step. The easy way to remember this is by playing a Scale on a Piano starting on 'F' and pressing 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 Scale with a #4th degree. In the exposition that follows, I will be doing all examples in 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-Keyed Music based on the 12-Tone Tempered Scale has led to the concept of the Dominant 7th Chord. It is a psycho-acoustic tension and release phenomenon. This is how it is postulated to work in the European Classical Music Theory. The four note Chord formed on the 5th degree of the Major Scale is called the Dominant 7th Chord. It is formally referred to as the V7 Chord. The presence of the Dominant 7th (b7) in the Chord sets up a tension that needs to be released. Classical theory states that this tension is released by resolving to the 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 with psycho-acoustic phenomena which is highly subjective and the topic of much debate down through the ages continuing to the present day. For now, let's put politics aside. You are encouraged to do the following experiment on a Piano 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 the root (G) and Major third (B). The Major third (B) and the fifth (D) form an interval of a Minor third, also considered consonant, as do the fifth (D) and the Dominant 7th (F). The interval between the root (G) and the Dominant 7th (F) was considered dissonant in old classical theory. Most modern theorists are not so strict and would consider the interval as colorful if not downright consonant. This leaves us with the interval B-F. This interval was actually outlawed by the Medieval Christian Church and marked with the name Intervallo Diabolo. This Interval spans 3 whole tones. There are many names for this interval diminished 5th, augmented 4th, #11th, and my favorite - TriTone. A TriTone is naturally formed between the Major 3rd and the Dominant 7th. Because of the relative 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, this interval was always resolved inwardly.

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

The TriTone interval also resolves outwardly as easily and as naturally as it resolves inwardly.

You should try this out repeatedly on a Piano and let your ear be your final arbiter. These resolutions are symmetric and, I believe neither has any precedence over the other. I agree with most modern theorists, that they are equivalent and neither should be preferred for any subjective reasons. So what 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 by Antonio 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 does the Gb7 cause the desired resolution to the I Maj7 Chord? This is the heart of Lydian-Dominant Theory. In the table that follows I will spell out the requisite chords, identify the relevant TriTone - the rest is magic. The operational TriTone is {E - Bb} (remember Fb=E). Each of these Dominant 7th chords has the same TriTone !!! As stated earlier in the analysis of the generalized Dominant 7th, it is the TriTone that causes the tension that gets resolved. Notice also that C and Gb are themselves TriTones. Consider this. It would seem that 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 just how ambiguous TriTones can be. One can actually "get lost" aurally in an improvisation with many sequential Lydian-Dominant changes in the Chord progression. Thus the first Postulate of the Lydian-Dominant Music Theory.

Postulate 1 Any Dominant 7th Chord can be replaced by its TriTone equivalent with 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 introduce them into our music, we find that the word modulation takes on an entirely new and exciting meaning. I would also add, that along with this newfound modulating flexibility, a wealth of harmonic richness is also realized. Classical music theory shortchanged itself terribly by banning and/or ignoring this fundamental theoretical fact implied by the OverTone Series and realized by the 12-Tone Tempered Scale. Understanding and appreciating the fundamental assertion of the first Postulate - TriTone Dominant Substitution - is but the beginning of our journey. Next we will study and develop the essential core elements that are the "building blocks" of Lydian-Dominant Theory - the TriTones.

Postulate 2 There are 6 TriTone pairs TT 1 = { c - f#/gb }

TT 4 = { a - d#/eb }

TT 2 = { g - c#/db }

TT 5 = { e - a#/bb }

TT 3 = { d - g#/ab } TT 6 = { f/e# - b/cb } Each pair is associated with two interchangeable Dominant 7th Chords. That is, they may be substituted for each other to provide harmonic richness and/or chromatic movement as can readily be seen in the Chord progression snippet from The Girl From Ipanema used above. The following table enumerates the 6 Dominant7 pairs and their associated TriTones. Read this table up and down the columns - the involved TriTone is in between. 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#/e b 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 is heavily Lydian-Dominant. Check out his compositions Wave and Desafinado to see what I mean. The BIG Fact is, that Jazz is heavily permeated with Lydian-Dominant Chord Progressions and Melodic development. Swing, Blues and their derivatives in the Pop/Rock styles less so, but still Lydian-Dominant. South American forms like Samba and Bossa Nova and Tango are, again, heavily permeated with Lydian-Dominant Chord Progressions and Melodic development. Likewise, the Afro-Cuban inspired Salsa forms. Certain 20th Century Classical Composers have also ventured into Lydian-Dominant, Debussy, Ravel, Stravinsky to name just a few. You should become aware of an odd thing with these pairs. Are they augmented 4ths (#11) or are they diminished 5ths (b5)? In Natural Music Theory (pure Harmonic Series intervallic evolutions), there is a definite difference. In the 12-Tone Tempered System there is not. The very process of 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 in pure Scale, in fact, an infinite number of them. But as the TriTone has a value of p·2(1/2) , ( any starting pitch p times the square root of 2 ), all the Harmonic Series (which is based exclusively on rational numbers) can do is

Harmonic Series (which is based exclusively on rational numbers) can do is spit out closer and closer approximations to the TriTone. This is not at all as weird as it seems at first glance. A famous Mathematical Proof, attributed to Euclid, 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 are natural numbers, can equal  2. TriTones are intimately related to this number that caused the Pythagoreans so much trouble with ir-rational numbers. This topic and other related items are explored in greater depth in my book Natural Music 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 and Dominant. This Chord puts the 'A' in Take The 'A' Train, the 'Des' in Desafinado, the 'Tune' in Bernie's Tune, and that special sonic twist in so many Lydian-Dominant compositions. The Chord is comprised of two TriTone pairs a Major 3rd apart. In the case of the C7 b5 - F#7 b5 pair, they are {c - f#/gb } and { e - a#/bb }. Play this Chord - listen to it. Grab the 4 notes in the C-F# pair - { f# a # c e }. Now play a C bass note - listen. Now play an F# bass note - listen. What do you hear? I hear the same tonality in each case. Nothing really changes except the voicing, i.e. a particular rearrangement of notes. For the Improviser, this is really important. The first problem encountered when analyzing a particular Chord progression is figuring out what Scale(s) are implied by which Chord(s). It doesn't matter how fast your fingers are or how good your tone is if you're playing the wrong notes - it'll still sound bad. This is the major problem I have with some Improvisational Methods of listing a seemingly different Scale to each and every Chord in a progression. I find it more confusing than helpful, especially to the novice. The fact is, that the underlying scalar note group frequently does not change at all ! More often than not, whole sequences of Chord changes define the same note group. It doesn't matter which notes in a particular Scale you choose 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 all kinds - the whole song is defined by one scalar group! You might see a lot of Chord changes, but all that is really changing is which note(s) the bass player 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 looks like wicked hard Chord changes are really not so bad at all. There are only 6 Dominant7 b5 Chords, not 12 as with most other chords. This makes learning them take half the time. All that remains is to fit them in properly. LydianDominant is actually easier than it looks. Things get even simpler in the next postulate.

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 form 3 mutually exclusive sets of 4 notes. Each group is comprised of 2 interlaced TriTones a minor 3rd apart. Notice that 4 super-imposed minor 3rds equals an octave in the 12-Tone Tempered System. This note group is totally symmetric any way you look at it. DQ1 = TT 1 + TT 4 = { c - f#/gb } + { a - d#/eb } DQ2 = TT 2 + TT 5 = { g - c#/db } + { e - a#/bb } DQ3 = TT 3 + TT 6 = { 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 a technically 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 what interval is it above C??? I have heard it called a diminished 7th. In Standard Musical Nomenclature, 7ths are designated as major and minor along with 2nds, 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 any OverTone Series derived Systems. It is an emergent property of the 12-Tone Tempered System and is central to Lydian-Dominant Theory. Interestingly, other Tempered Systems have analogous structures and are discussed in depth in 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 we rename this wonderfully ambiguous Lydian-Dominant note set the diminished 6th Chord - C dim6. As justification in addition to the above analysis, I would point out that this Chord is remarkably close in sound and function to the minor 6th Chord, a Chord more commonly used in older American music, and still important in some indigenous styles like Tango. This Chord has an ambivalent tonality and differs from the dim6 in that the 5th is perfect rather than diminished.

C m6 = { C Eb G A } C dim6 = { C Eb Gb A } Furthermore, if we invert the 6th in the C m6 thereby changing the root note to A instead of C, we derive the modern Jazz Chord, the A m7 b5 - the so-called half-diminished Chord. This Chord will be discussed in depth later on in this treatise. The dim6 sub-systems also define 3 Lydian-Dominant Scalar entities called diminished scales. They will be discussed later on in this treatise. The diminished quads are integrally involved in several other important LydianDominant Chords which leads us to the next postulate.

Postulate 5 There are 3 Sets of Dominant7 b9 Chords, one for each Diminished Quad Sub-System. Technically speaking, there are 12 of these chords. In Lydian-Dominant reality however, they each fall into one of the 3 Diminished Quad Sub-Systems. I will 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 is created 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 your ear 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! Once again, 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 lurking within 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 by every note but the root. This is one of the most striking aspects of the LydianDominant System - that roots are frequently extraneous to the function of a 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 root in turn and listen. Do you hear what I hear? The "action" notes are the same no matter how you choose to voice them. Changing the root notes alters the note set (thus the sonority changes), but the tension/resolution mechanism is invariant. Lydian-Dominant is very cool. The same thing goes for the other two 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 by adding a b7 to a diminished triad. However, this pseudonym hides the fact of the dual nature of this Chord - it can be looked at as a 6th Chord or a 7th Chord, dependent on other factors such as melodic leading, resolution, and rooted-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 a minor 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 Jazz compositions, the subtle ambiguity of this note group lends itself to other x 7 b9 uses. It doesn't have to resolve to a I m Chord through the V - it can just as easily go other places though not anyplace. Check out Stella By Starlight.

When this note group is used as a m6 Chord, it is quite common to find it used as a I Chord! There are innumerable songs that do this Remember, all that has changed is the root note. It's the same basic tonality, but emphasizing a different bass note gives this note group a different quality. This note group is truly ambivalent in character and has power in many different directions. Actually, the m7 b5 is the OverTone Series Inverse of the Dominant7 Chord making it an important fundamental theoretical construct - want to know more? The derivation of this Chord and that of minor itself are presented in depth 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 12Tone Tempered System. Like the dim6 sub-systems, the 4 Augmented subsystem triads are totally symmetric and form 4 mutually exclusive sets of 3 notes. They are: AT 1 = Caug = Eaug = Abaug = { C E G#/Ab }

AT 2 = Ebaug = Gaug = Baug = { Eb/D# G B } AT 3 = F#aug = Bbaug = Daug = { F#/Gb A#/Bb D } AT 4 = Aaug = C#aug = Faug = { A C# F/E# } Though not properly Lydian-Dominant, the 4 augmented triads are heavily involved in Lydian-Dominant Theory in at least two important ways. First, the scales that underlay this Chord are all Whole-Tone (altered) Scale variants. These scales can also underlay other important Lydian-Dominant Chords. I will have more to say on this later in this treatise. Second, Augmented Triads are usually used as Dominant 7th or 9th Chords making them Lydian-Dominant and subject to all the other Lydian-Dominant Postulates. Here's where the fun begins again. These 4 augmented sub-systems imply 4 corresponding Augmented7 sub-systems as well. I'll show you the T 1 subsystem and leave the other three for you to do as an exercise. C aug7 = { AT 1 = ( C E G# ) + Bb} E aug7 = { AT 1 = ( E G# C ) + D } Ab aug7 = { AT 1 = ( Ab C E ) + Gb } As with the diminished sub-systems, these augmented sub-systems are a 100% total Artifact of the 12-Tone Tempered System. As shown above in Postulate 5, the X7 b9 is essentially a diminished quad plus one of 4 related roots, themselves forming another diminished quad. With these aug7 note groups, 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-Dominant Chords, but because it in the Dominant Group, it turns up in strategic positions in many songs and must be handled properly. As mentioned briefly above, the augmented sub-systems are intimately connected with Whole-Tone Scales which brings us to our next postulates after a short digression. Before we get to the next postulate I want to briefly discuss the Western Classical bias (from the Greeks) toward the 7-note (so called) Dia-Tonic Scale 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 any continued reliance on pre-rational systems as atavistic and downright ignorant, we have nevertheless perpetrate on each unsuspecting generation since the "Enlightenment" a plethora of number mystic systems which are unquestionably 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 will desperately be mentally searching for a "logical" reason (there must be one) for these commonly encountered systems. You may get a straightforward "... and God rested on the 7th day." from a Religionist, and though I respect their right to their strongly held convictions, I don't feel that I am bound by them in any way. The point is, that there is No cosmic reason at all why the number 7, or any other number for that matter, should be specially favored. In Music Theory, we use the two terms Scale and Chord without much discretion. In fact, there is no real difference between them. It only depends on how far we space out the intervals and even this is poorly defined. If we space out the intervals in whole and half steps the notegroup 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 a min 3rd, and some chords have intervals of whole step. Share I and many Share via Share on on modern music theorists use the term ChordScale. use Facebook I also Tw itter the term email note-group. This makes more sense to me than trying to define a difference 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 notegroup is identical and the same ChordScale is defined. There happens to be 7 notes in it and, indeed, there are a lot of 7-note scales. But, there are many other ChordScales with a different number of elements that are just as useful and legitimate as the dia-tonic. The number 12 (as in 12-Tone Tempered System, inches in a foot, months in a year, hours of day/night, and various groups of Apostles ) is also totally bunged up with number mysticism. As usual, I discuss this issue in great depth in my book on Natural Music Theory.

Postulate 8 There are 2 Whole Tone Scalar Sub-Systems. WT 1 = AT 1 + AT 3 = TT 1 + TT 3 + TT 5 = { C D E F#/Gb G#/Ab A#/Bb } WT 2 = AT 2 + AT 4 = TT 2 + TT 4 + TT 6 = { F G A B/Cb C#/Db D#/Eb }

This is a totally symmetrical Scale of 6 notes! It is constructed of nothing but Whole steps. Play them on your instrument - was your ear fooled? They are even more fun to sing - try it. This Scale is a Lydian-Dominant Artifact of the 12-Tone Tempered System. It is not found in natural OverTone Series harmonic derivations. Once again, tempering allows the "snake to eat its tail". The Whole-Tone Scale and its altered variants underlay many Lydian-Dominant Chords. Basically, they fit any Chord with a diminished 5th or an augmented 5th or both They can also be used when a #11 or a b13 is present. I will show how they can be used to fit the common Lydian-Dominant Chord - the Dominant7 b5. C7 b5 = { C E Gb Bb } CWT = WT 1 = { C D E Gb Ab Bb } Notice that we have 4 notes of WT 1 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 can always 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 be played giving us an A rather than an Ab. Indeed an A can be played turning our Scale into one of the many Whole-Tone variants. However, using the Ab gives us a slightly "outside" sound. In particular, it provides sonic variance using 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 an augmented 7th Chord. C aug7 = { C E G# Bb } CWT = WT 1 = { C D E F# G# A# } Notice that we have 4 notes of WT 1 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 Major Chord, we should have an F rather than a F#. Indeed an F can be played turning our Scale into a Whole-Tone variant. However, using the F# gives us a slightly "outside" sound. In particular, it provides sonic variance using a noncritical 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 an important part of the Jazz Style. I believe, however, that not all "outside" notes are justified at the theoretical level. Some "outside" notes are just plain wrong - i.e. not at all justifiable within the structure of the Chord progression. Too often, "playing outside" is used as an excuse for playing wrong notes due to an inadequate analysis of the Chord Progression implied 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 alternate Whole 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-Dominant Artifact of the 12-Tone Tempered System. It is not found in natural OverTone Series harmonic derivations. Once again, Tempering allows the "snake to eat its tail". The diminished Scale comes in two flavors DSwh and DShw depending on how the diminished Scale is constructed - whole step first or half-step first. The following table shows the difference. 1

CDIM(wh) = C

½

Eb

D ½

1

1

½

F ½

1

½

1

½

F#/Gb G#/Ab A/Bbb B/Cb C' 1

½

CDIM(hw) = C C#/Db D#/Eb E/Fb F#/Gb

1

G

½

A

1

Bb/A# C'

Notice that CDIM(hw) = C #DIM(wh) !!! All we do is start on a different note in scalar sub-system. This is generally true of every one of these scales leading to the following relations. DSwh1 = DShw2 DSwh2 = DShw3 DSwH4 = DShw1 There are many ways to use these scales. In either flavor, they remain wonderfully ambiguous and their use now and again over the proper Chord

changes, though tricky, creates much sonic richness. I will show some ways to use these scales and leave others for you to investigate as an exercise. I will use the classic Lydian-Dominant Chord progression elaborated on extensively above: 7 b 7 x The Lydian-Dominant II - V - I »»» IIm - II - I Maj

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. Dm7

LD II-V-I

Db7

CMaj9

(D E F G) (Ab Bb Cb Db)

DDIM(wh-up)

Set1

*

DDIM(wh-down) (D' C B A)

D'

Set2

*

(Ab Gb F Eb)

D

Notice that there is an Up and a Down version of the Scale, both being constructed of alternating whole & half-steps. The Scale D DIM(wh) is composed of 2 sets (called tetrads from Greek Music Theory) of 4 notes. In the 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 !!! This shouldn't be a surprise to you anymore. Play the chords and the accompanying scales on the Piano - listen. They are super-diminished every way you look at them. Try playing them in "thirds" - in "fourths". See if you can find other ways to use these wonderfully ambiguous Scale patterns. For now, I will conclude this treatise with an excerpt from my book on Natural Music Theory. It deals with the actual OverTone Series implied note-groups that underlie Lydian-Dominant Theory. To appreciate its simplicity one only has to look carefully at the OverTone Series and list the note-groups by Doublings. OverTone

Note #

Note Name

Analysis

1

F

C0

Fundamental

2

1

C1

3

5

G1

4

1

C2

5

3

E2

Fifth

Dominant 7

2

Dominant 7

6

5

G2

7

b7

Bb2

8

1

C3

9

2

D3

10

3

E3

11

#

F#3

12

5

G3

13

6

A3

14

b7

Bb3

15

7

B3

Leading Tone

16

1

C4

Doubling

4

Lydian-Dominant Sc ale

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 usually associated with the local culture's Goddess. Notable among these are the Greek Sappho of Lesbos and the Hindu Saraswati - Goddess of Music, Mathematics & the Sciences. It is a wonderful Scale and wholly derived from the OverTone Series generated Harmonic Series. This knowlege is Ancient! Most people today don't know that Plato, Aristotle, Euclid, Ptolemy, and whoknows-how-many others wrote extensively about Music Theory. It is written about in the Vedas, the World's oldest books. It is amazing to me to be constantly re-discovering facts known to humans so long ago and then forgotten 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 Frequency Next, you get a Doubling (see below) Then, a 5th - the first interval created that is not a Doubling. This interval (3/2)x generates the 12-Tone Tempered (Pythagorean) Scale Then, you get a Chord - C7 - The Dominant7 Chord. It is 100% naturally derived from the OverTone Series. Lo and Behold, this Chord implies the Lydian-Dominant Scale - CLD, not the Pure 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 concept in the book. The process continues to Infinity with new chordscales emerging that transcend and include those already manifest. As usual, I discuss this issue in great depth in my book on Natural Music Theory.

Notice my use of the word "Doubling" instead of "Octave". The word octave contains a built-in and totally unwarranted bias toward 7-note scales - it literally means the "eighth" note. It is true that there are many wonderful and important 7-note Scales, but this fact hardly justifies priority status. Doubling is a Psycho/Physio-Acoustical phenomenon - it has nothing at all to do with scales. Concluding Remarks: As we continue our studies, I will point out Lydian-Dominant elements where ever they occur in the Songs we learn and the Improvisations we create for them. Regularly and methodically practice the preparatory exercises that I have created for you to learn the Lydian-Dominant System. Your hard work and diligence will reap great rewards as your Improvisations develop the tremendous sonic richness implicit in the Brave New World of LydianDominant Music Theory.

Norm Vincent NorthStar Studios - April 2000 [email protected]