I blipped this song yesterday and decided I should write a post about it… LOL! Anyway…

It was in my freshman college years when I bought Joe Satriani’s Not of This Earth, and in that album there’s this one particular track that always bothered me. It’s a very strange song (with a very strange melody) called “The Enigmatic”. Click on the YouTube link below if you’ve never heard it yet.

We will not analyze the song itself here, or discuss any of those trademark whiz-bang guitar techniques, but rather focus on the scale that inspired the song. If you think our Lydian-dominant lesson from last time was quirky, wait till you hear this one!

I can feel that Joe was trying to stretch the boundaries of conventional melody and harmony in this song. The melody is carried mainly by doublestops on an E pedal point, accented by some eerie arpeggios. And what makes it sound so strange is primarily the scale he used to build the entire song. This scale is known as the “enigmatic” scale, and was originally used (at least according to Wikipedia) in the late 1800s by the Italian composer Giuseppe Verdi.

The enigmatic scale is actually an artificial or synthetic scale, being that it is not based on any of the traditional major, harmonic or melodic minor modes, and at the same time is also not rooted on any of the ancient, traditional or cultural intervals (such as the Byzantine, the bebop, or the various Japanese pentatonic scales). It is more an artificially-constructed scale built on an arbitrary interval of notes.

On C, the enigmatic scale is spelled C – D-flat – E – F# – G# – A# – B. Thus you would notice immediately that the scale lacks a dominant (the G-note) and subdominant (the F-note), and this is what primarily gives the scale its stinging character. Add to this the Phrygian-like flatted 2nd, which relative to the major 3rd, gives a wide interval of three semitones. And then add some more quirky elements like the Lydian-like raised 4th, and the raised 5th and 6th, and you’ve got one really twisted-sounding scale.

As originally used by Verdi in his “Ave Maria” of 1898, the scale ascends as is, but when descending, the raised 4th (i.e., F#) is replaced by the natural (F), which gives another three-semitone interval, this time relative to the raised 5th. In contrast to these wide intervals, there is a long four-note chromatic “section” too, which begins with the #6, going to the major 7th, and then the tonic, all the way to the minor 9th. It’s so enigmatic indeed!

Further analysis will lead you to think that the scale is somewhat a derivative of the “hexatonic” or whole-tone scale (again in C, spelled as C – D – E – F# – G# – A#), like a whole-tone scale with a major 2nd and missing a leading tone.

In a construction or soloing context, it would be difficult to think of anything traditional with the application of this scale, except for the fact that the raised dominant hints us of trying out this scale over an augmented chord progression. One different way of looking at C enigmatic is in the context of an E-major, C#-minor or an F#-major triad, but be careful about adding the chromatic notes that result when you use the scale like this. On the tonic, it is possible to use this scale over altered chords such as the major-flat-5th, the augmented-major-7th and the major-7-flat-5th.

Most of the time, I find it best to use the enigmatic scale as a substitute for an altered scale in a freer vamp setting. That is because the chromaticism that is introduced by the scale does not disturb anything in context.

I will try to come up with some audio examples when I get home, so please try to come back to this post in a few days.

Practice, persevere and conquer!

I caught this fascinating post at BoingBoing.net about a YouTube video showing how J.S. Bach’s Musical Offering (from 1747), specifically the Canon 1 a 2, could be played forwards and reversed, twisted and turned upside-down, split and played simultaneously inside-out by some crazy mathematicians, as a never-ending melodic movement. Well, if that’s a bit confusing, just watch the video below and you’ll get the idea…

J.S. Bach is perhaps the greatest composer of the Baroque era. He was widely regarded for the intellectual depth, technical facility and artistic beauty of his compositions. His output of works was prolific, and he was often commissioned to compose a lot of music for the Church during his time. He also paved the way for the next generation of giants that included Mozart, Beethoven, Robert Schumann, Felix Mendelsshon and even Frederic Chopin.

Bach was also the greatest proponent of the sophisticated art of the counterpoint, and specifically the canon and the fugue, which incidentally I first heard from the Swedish guitar virtuoso Yngwie Malmsteen’s compositions (although I had been listening to classical music too (and Bach specifically) since I was in high school).

The canon is a compositional style whose foundation is in fact the counterpoint but with some very strict and specific requirements. J.S. Bach’s Canon 1 a 2 is an example of an infinite or perpetual canon that is also a retrograde canon (and hence the nickname “Crab Canon”) because it can be played forward or backwards in time indefinitely. I will not go too in-depth about the anatomy of contrapuntal composition and the canon itself as it’s too intricate and technical. Just click on the links below if you need a deeper explanation.

Now, many mathematicians (and some others like Xantox, and John Leys who did the YouTube illustration above) became very intrigued by this melodic snippet, because aside from being a strict canon alright, and an infinite and retrograde canon at that (which means can be played endlessly and backwards too), it is also possible to play it simultaneously forwards and backwards in time. In their own words,

In each of these canons a musical line is played twice (or four times in Canon 10). The second version is always transformed with respect to the first by shifting in time, but it may also be shifted in pitch, turned upside-down, stretched, or played backwards. Each of these transformations occurs in the mathematics of elementary functions; they are examples of how new functions can be made out of old and of how a function can be tailored to fit a new situation.

If you listened intently on the amazing combination and interaction of the contrapuntal melodies, you will realize why it becomes so mind-boggling. The fact that it can be played indefinitely (as in the Möbius Strip transformation) makes it all the more appealing to mathematicians. It’s as if Bach himself composed this seemingly simple piece with the very intention of it being discovered later on (after centuries in fact!) of all these amazing possibilities.

For additional reading:

• Wikipedia: Canon
• The Canons and Fugues of J. S. Bach
• Math and the Musical Offering

Today I’d like to share with you something exotic… The Lydian-dominant is a strange-sounding scale I first heard being discussed by Joe Satriani in a Guitar World interview from about a decade ago. In that article, he showed how rock players can incorporate the scale into their soloing ideas. Being in a creative rut that time, I slowly learned it, until I was able to create endless melodic licks based purely on this scale, and substituting momentary lines into regular diatonic progressions.

The Lydian-dominant is named as such because it it is fact a Lydian scale (a major scale with a raised fourth) with a dominant seventh (like a regular dominant 7th chord).  In the key of C, it is thus spelled as C – D – E – F# (the raised 4th) – G – A – B-flat (the dominant 7th).

Theory-wise, the Lydian-dominant is in fact the fourth mode of the melodic-minor scale (which is like the regular Aeolian or natural minor scale, but with a raised 6th and 7th). The melodic-minor is a rather “happy”-sounding minor scale (as if that makes sense!), primarily because of the raised 6th and 7th intervals, but it also sounds quite unorthodox, because it ascends as is but descends like the natural minor.

When the great classical composers started to incorporate the melodic-minor into their compositions (Mozart used it quite a lot!), it opened so many new sound avenues for them. In the jazz world however, the use of the various melodic-minor modes is commonplace, most notably from bebop players like the virtuoso Charlie Parker and Sonny Rollins (of Miles Davis’ band).

My ears twitch every time I hear this scale, because for the moment the line emphasizes the Lydian sound, it gets suddenly knocked off every time the flat 7th is invoked, and vice-versa. Thus for me at least it sounds kinda edgy, like a pushing-pulling kind of thing.

To illustrate, here is a goofball E Lydian-dominant lick I recorded on acoustic guitar. I played this with a low open-E drone in order to anchor the rest of the line into a firm harmonic foundation. Please have a listen.

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As with creating lines in any other scale, the best way to present the distinct color of this scale is to highlight the raised 4th and the flat 7th, and put this in the context of moving melodies that anchor the dominant and the subdominant (which, in this case, is the B and the A# notes).

Now that you have an idea how it sounds, how do we actually use it in a song context then? Well, I find the easiest way to incorporate this scale is to substitute it for the Mixolydian in a dominant chord setting. This means that if I have a song in the key of E-major, then during the progression where the V-chord (that is, B7) is played, then I can in fact replace a B-Mixolydian line (spelled as B – C# – D# – E – F# – G# – A) with a B Lydian-dominant one (spelled as B – C# – D# – E# (or F-natural) – F# – G# – A). This in essence changes just one note, and that is the E-note (!), which creates suspension that “disturbs” the calm Mixolydian into a tension-filled F-note.

In fact, the scale can be used in any dominant chord setting. This makes sense because the (unaltered) dominant chord is spelled 1 – 3 – 5 – flat-7, and this is where Mixolydian is normally utilized.

Additional reading and resources:

• Wikipedia: Lydian dominant scale discusses the basics.
• In chrisjuergensen.com: The Lydian Dominant Mode, the author discusses some excellent soloing approaches.
• In Music Theory: Jazz Scales: Melodic Minor Harmony: Lydian Dominant Scale, the possibility of interpreting and using the scale as a 7-flat-5 chord rather than a 7-raised-fourth (or 7-raised-11th) is explored.
• In Lydian-Dominant Theory for Improvisation, the possibilities for using the scale are expanded into the “freer” no-holds-barred territory of the symmetrical whole-tone and 12-tone row. (Warning: This page contains some hardcore music theory stuff!)

Try it yourself. Start slowly. I guess it gets a little disorienting at first, but eventually your ears will learn to become familiar with it. Practice, persevere and conquer!