The Charm of Impossibilities 2 : Sympathy for the Devil’s Interval

The musical interval of a tritone (augmented 4th, or diminished 5th) is the most enigmatic of the intervals, and as such, has been given a bum rap, in my opinion. In the days of “early music” pre-Renaissance, the tritone was referred to as the “Devil’s interval”, and was forbidden for a time, at least in church music. Nowadays the interval is well accepted and used, but it is still interesting to explore why on earth anyone would have ever associated this interval with “the devil”.

But right there, we should take note. If the church didn’t like it, it must be magic. There must be some power in it, otherwise they wouldn’t bother condemning it.

And i do notice also, that music that contains a lot of tritones doesn’t usually tend to become as popular as music that keeps the tritone in its “proper position”, so to speak, in other words, relegated to the role of the tension/release paradigm, and nothing more.

So what is the magic of the tritone?

another access point to world beyond the mirror

Well, in keeping with the theme of the Charm of Impossibilities, which includes the mirror and the palindromic aspects of music, let’s check the tritone. One thing we notice right away, is that it is “non-invertible”. It is the only interval, (besides the octave) that is the same if you invert it. An inversion in music is simply when you move one of the tones of an interval or a chord up or down an octave. So if you change either tone of the tritone up or down an octave, it is still a tritone. None of the other intervals except the octave have this quality. Seconds become sevenths, thirds become sixths, and fourths become fifths, when inverted.

Olivier Messiaen, the 20th century French composer, who coined the term “Charm of Impossibilities”, used scales that he called “modes of limited transposition”, as the harmonic and melodic part of his vision, at least in his earlier work. The idea is that certain scales can only be transposed a limited number of times, and because of that, there is an inherent charm in them, a kind of magic. And, as it turns out, all of those scales contain multiple tritones, unlike the diatonic scales, which each only contain one.

So, of course, an interval that is impossible to invert can also carry the charm of impossibilities.

the Heyokah interval

But before we get into these scales let’s just investigate the interval itself, the tritone. First, it divides the octave exactly in half, so it is really the balance point of the octave, and secondly, it cannot be inverted. The charm of impossibilities. It is the only interval in which there is no power struggle, or dominance of one tone over the other. It is completely balanced. it cannot be turned upside down. According to the German composer Paul Hindemith, in his book on musical composition, every interval EXCEPT the tritone has a root, a note that has more weight than the other. For example, the root of the perfect fifth is the bottom tone (C in the fifth of C to G), but the root of the fourth (G to C) is the top note (C).

So in that sense, we can also connect the tritone with the infinite mirror, which we looked at in the first Charm of Impossibilities article. My feeling is that the tritone invites the infinite into the equation, and that that is what the church didn’t like about it. Church is imitation infinity. Direct access to infinity is not allowed, because one who has direct access cannot be controlled.

So rather than call it the devil’s interval, we could call it the shaman’s interval, or the trickster’s interval, or perhaps the sorcerer’s interval. The heyokah interval.

Or how about the sage’s interval. Here is the one, without a root, without a home, but without whom changes cannot be made. Modulations to different keys, for example, require the input of the tritone, much as society moves to new eras and new ways of living together and cooperating through the input of the sages. And one of the tritone’s greatest contributions in the “modulation department”, is that it facilitates modulations in keys that are far away from each other.

the Chess of Music

For a long time i have been enjoying free-associating the analogy of diatonic music with the game of chess. For me, the diatonic system of 12 tones fits this model in many interesting ways, not exactly literally, but there is a lesson there. Please bear with me, this is nothing to take too seriously, but might provide some helpful insights or springboards to new ways of hearing and working with music.

In music, whenever a tone sounds, a series of harmonics sounds along with it, which determine the timbre of the tone. It is a hierarchical setup, just like chess. The naturally occurring harmonic series contains octaves, fifths, and other intervals, related to the root tone. In a way, a scale is a collapsed version of this multi-octave pile of harmonics into the space of one octave, though not exactly. But because certain tones, like the fifth especially, occur in the harmonic series close to the root tone, they become more important in the scale. The fifth is called the “dominant”, because after the octave, it occurs next in the harmonic series. Also the interval of a fifth contains the relationship of the Golden Mean, 3:2, which gives it a special place in the hierarchy. It is the king.

So then the dominant chord, the V7 chord, becomes a prevalent force. Especially in the major scale. And why? because the tritone is part of the dominant 7th chord in the major scale. Does the shaman/sage want to be in the service of a king?

Here is my music/chess analogy:

The rooks, or castles, in chess, represent the tonic key, or the kingdom. According to some sources on chess, these rooks are also movable, in the battlefield, almost like the Trojan Horse. The tonic key is moveable, by modulation.

Inside the rooks are the knights, or horses. This represents the army, the enforcer of the boundaries of the kingdom. In the scale, these would be the 2nd and the 7th, always tending to resolve outward, to the rooks, the “roots” of the tonic key, to reinforce the boundary of the kingdom, or the supremacy, the “Dominance” of the tonic key. And these are also the most “dissonant” intervals. War is never very harmonious, is it.

Then, within those, we have the bishops, or the religious power in the kingdom. In the scale these would be the 3rd and 6th. These consonant tones function to preserve the harmony, and strengthen the feeling of the home key, or kingdom.

The King is the fifth, the “dominant”, and the Queen is the 4th, the “subdominant”. Sometimes, in chess, the queen refers more to the position of a minister, rather than a queen, according to some sources on chess.

And then there are the pawns, all the “black notes” (in C), or all the notes that are outside of the scale. They are the “expendable” tones, the chromatic passing tones, the ones that are passed through quickly and discarded, like the foot soldiers on the battlefield. Even the ways the pawns are allowed to move in chess are very reminiscent of the ways chromatic tones can be used, as passing tones, neighbour tones, approach tones, from above and below, and so forth. Pawns in chess can move one step in any direction, including diagonal, except backwards.

Location, location, location

So then i was noticing that the location of the tritone in a scale, or mode, has more to do with whether the tritone interval itself sounds dissonant or not, than just the sound of the interval itself. By itself, it is not really dissonant.

Let’s look at the most commonly used scale, the “Ionian” mode, or major scale. In the major scale, the tritone appears between the 4 and the 7 (in the key of C, that is F and B). With one of the tritone’s notes on the 4, we have a dissonant relationship with the third (the 4th is called the “avoid note” in jazz, really the only dissonant tone in the major scale.). The other tone, the 7 (or B in the key of C) is dissonant with the root, or C. So, in the major scale, the tritone is used to reinforce the “kingdom”, whatever key the music is in. Whenever it sounds, we have the feeling that it wants to resolve. It always is in the position of creating tension that needs to be released, sort of like the devil’s “job” in society.

But if we check some of the other diatonic modes, we will see that the tritone appears in a different position in each. We can quickly see and hear that in those cases, the tritone does not have the same kind of dissonant quality that wants to be resolved.

Perhaps we can see, through this, how what we call demonic is really just entrapped elemental, natural energies, as the Tibetan Buddhists have pointed out. It is by repression that natural forces become “negative”. So, does our inculcated way of hearing music help to cut the magic of music, by forcing it into a certain framework? We could even think of each scale as a kind of group dynamic, or community.

Check out this particular community, from the Amazon:

“the Tsimané (pronounced ‘chee-mah-nay’)—an Amazonian society with barely any exposure to Western music. And they, to McDermott’s surprise, don’t care about consonance or dissonance. They can tell the difference between the two kinds of sounds, but they rate both as being equally pleasant.”

http://www.theatlantic.com/science/archive/2016/07/music-to-our-western-ears/491081/

the Lydian Chromatic Concept
In George Russell’s Lydian Chromatic Concept, he argues the case for the Lydian mode, as being a more natural version of the major scale. This mode has the raised 4th, which IS the tritone (F# in the key of C), and not the 4th, which is really the dissonant note in the Ionian, or major scale. In the major scale the “natural” 4th is the dissonant one, actually. hmmm… the queen? or, the minister?

So in the Lydian scale, one of the tones of the tritone is the root. If we are in the key of C. we have a C, and an F# forming the only tritone in that scale, because in Lydian, the 4 is raised to a #4. So one of the tones is the root, but the other is just one of the pawns. BUT, this pawn happens to be the court jester as well (F#)! In this scale the Lydian, he is in his natural place.

Also, in the rules of chess, the pawns, once they have reached the opposite side of the chess board, are able to become any of the other pieces, except the king. So this wild card shaman pawn has found his way into the court, and sits between the king and the queen, or minister. Perhaps the raised 4th of the Lydian scale could represent the enlightened minister, or benevolent compassionate queen, one who had risen “above” the dissonant position of the “subdominant” one, who is always trying subtly to subvert the dominant one. And perhaps this uplifted minister is someone who has been all the way through the battle, and has transformed himself into whatever was necessary.

So in the Lydian scale we could perhaps say that the court jester, the heyokah spirit, or shaman, actually reinforces the community, by bringing in the spontaneous and infinite element. The tension-release paradigm is not so strong. And in this scale, there is no “avoid note”. Notice that when we play a chord such as a C major 7, and we add an F# as an upper extension, (C major 7 #11) there is no need to resolve anything. The chord floats, but is stable too. Whereas, if we play the same chord, C Major 7, but add an F as an upper extension, well, we never want to do it! (at least, I don’t). There is a horrible dissonance, between the F, and the E, which is the 3rd of the chord. And, if we move that F down in octaves, it eventually becomes the root, when it is played below the C major chord, because C major over an F in the bass really is an F major 7/9 chord. A musical coup d’etat!

the Modes and the Assemblage Point

If we look at where the tritone is placed in the other diatonic modes, we can see how its placement causes a different dynamic in each mode. But in every other mode except the natural minor, we will find that one of the notes of the tritone is either the root, third, or fifth of the “parent” 1 chord of the mode. For example in D Dorian, the tritone is between the third of D minor, the parent chord, and the 6th of the mode. No need for resolution there. Or, in E Phrygian, the top note of the tritone (B) is the 5th of the parent chord. The F pushes for resolution down to E, but the B is stable, as it is the 5th of the parent chord, E Minor.

In the following chart we can see all the diatonic modes. The half notes show the placement of the tritone in the scale. Please ignore the time signature (the 10/4), i just couldn’t figure out how to get rid of it. 🙂

So, only the major and minor scales put the tritone in the position of causing tension that needs to resolve. And the tension is greatest in the major scale. An interesting thing to notice. What does it mean? Well, nothing literal, but it’s worth noticing at the very least. Could it be that the major scale is the “devil’s scale”, actually? It is reminiscent of how modern culture tends to demonise the shamanic by suppressing it. And certainly we have a great history of military music in the “major” (which is also a military word) scale.

In the Castaneda teachings of Don Juan, he talks a lot about shifting the “assemblage point”, which is an invisible point apparently behind the right shoulder, and has to do with how one sees the world. And the sorcerer, or shaman, can shift the assemblage point in a way that one perceives the universe in an entirely different way, or even an entirely different universe altogether. A parallel reality.

Perhaps in music, this shaman is the tritone, and where he is placed in the scale, in the kingdom, whether he is “honored” or not, has a lot to do with how the world (the music) manifests. So the different modes could represent a kind of shifting of the assemblage point. Certainly various cultures favor different scales and modes in their musics, that perhaps correspond to their vastly different perspectives of the world.

Duendes

The point is, all the intervals are like living beings. They each have a particular, and very unique quality. Almost like a personality, or spirit, a “duende”. And, we saw already in the previous Charm of Impossibilities article, how each interval is really a unique palindromic rhythm, which has been sped up, and has shifted from the world of rhythm to the world of tone. Perhaps this inherent palindrome that each interval has is the mirror window to this particular being’s version of the universe.

And how these intervals (particularly the tritone) are placed in relation to other tones or intervals can change their meaning, function, and effect in many interesting and powerful ways, just like enlightened individuals within the society can effect changes in a myriad of ways.

Stay tuned for more explorations about these curious beings, the intervals!

The Cycle of Fifths

Another interesting “intrusion” of the tritone into the language of music theory occurs in the cycle of fifths. If we consider C as the mirror point, and move in fifths in either direction, we get the cycle of fifths. As we move up in fifths, we get the “sharp keys”, and if we move down in fifths we get the “flat keys”. What this means is that each fifth we move up, we add a sharp to the key signature, and each fifth we move down we add a flat to the key signature.

So, for example, we start at the key of C, which has no sharps or flats. From there we move up a fifth to G major, whose key signature contains one sharp. Then D major, a fifth up from G, has two sharps in the key signature, and so forth. A major has 3 sharps, E major has 4 sharps, etc.

So, if we continue moving up in fifths, after E (4 sharps) we get B (5 sharps), and then F# (6 sharps), which is a fifth above B. Wait! F# ! the tritone! hmm…

This chart shows how if we go up in fifths from C at the bottom, we reach F# at the top. And don’t forget to notice, by the way, that if we collapse all these notes into one octave, we get the C Lydian scale.

ok, let’s wait there for a moment, and see what happens when we move DOWN in fifths from C.

One fifth down from C is F major, and the key signature has one flat. Two fifths down from C is Bb major, and the key signature has 2 flats. Then Eb major, which has 3 flats, then Ab major, which has 4 flats, then Db major with 5 flats, and the Gb major with 6 flats.

Wait! Gb major! hmm, that’s the enharmonic equivalent of F#! And when we were going up, F# major’s key signature had 6 sharps.

This chart shows how going down in fifths from C at the top, we arrive at Gb at the bottom.

So when we do a mirror image movement in fifths from C, we end up meeting at F#/Gb, the tritone! With 6 sharps (F# major) in the key signature from one side, and 6 flats (Gb major) in the key signature from the other. It is like a node on a vibrating string, where the wave crosses to the other side.

Then if we keep going from there in fifths in both directions, the next step is C# and Cb. 7 sharps in C# major, and 7 flats in Cb major. Now we don’t usually use these keys, because they translate better to enharmonic keys with less sharps and flats, (C# = Db, and Cb = B) which makes it easier to conceive while playing. But what happens is, the sharp keys have basically turned into flat keys, after meeting at the F#, and the flat keys have changed to sharp keys after meeting there at Gb. Technically, to keep naming them as they would naturally progress, we would have to employ double flats, or double sharps. For example, after C flat, we would get F flat (which is really E, 4 sharps), then Bbb, (B double flat) which is really A (3 sharps), then Ebb, which is really D (two sharps), then Abb, which is really G (one sharp), and then Dbb, which is really C (no sharps or flats).

Notice also, that our first move of fifths in either direction from C yields F on the down side, and G on the up side. The queen and king. And right between them? the F#. And the first move of fifths in both directions after the meeting at F#/Gb, yields C# and Cb, just on either side of the C. So immediately we receive a telegraphed message about where we are going.

And one other interesting thing to notice concerning the F, and its place in the hierarchy: In order to get to the fourth, an F in the key of C, by going up in fifths, we are going to have to go through the whole cycle and won’t reach F until the very last one. Because F is the first fifth down from C, it is actually the last fifth up from C. This reinforces the argument for the Lydian mode, with the #4, as a more natural scale, because the #4 is closer in the cycle of fifths than the “natural” 4.

And, since C is the dominant of F, the presence of F in the scale always tends to try to almost force a modulation to it (F) as the tonal center. Almost like the minister who wants to take over the kingdom.

Try this on the piano to see for yourself. These two directions of fifths, meet at C, then F# (which is also Gb), the tritone, and then back again at C. Like a spiral, or even a bit like the DNA strands. It is even torus-like in its shape, or we could also think of the infinity symbol.

The tritone saves the day. This is the unpredictable balanced being that brings infinity into the mix, and disrupts the status quo. Because the hierarchy doesn’t want the status quo interrupted, the tritone has been marginalised. So how can we bring more of the tritone’s influence into the music?

Whole Tone Scale

So let’s take a look at some other scales. The whole tone scale is a six note scale, which contains only whole steps. This is one of the “modes of limited transposition” that Olivier Messaien mentions in his book “The Technique of my Musical Language”. Because the scale has only two transpositions, before once again it is the same scale, it carries what Messaien referred to as the “Charm of Impossibilities”.

But another thing that we immediately notice, is that every tone in the whole tone scale is part of a tritone. There are SIX tritones in the scale, not one, as in diatonic scales.

So does this make the whole tone scale more evil? Because with all those “devil’s intervals” in there, it MUST be, right? Well, take a listen to this excerpt from Debussy’s piano music, and see what you think. Does it sound “evil” to you?

To me, not at all. Debussy was well known for using the whole tone scale a lot, it was part of his “impressionistic” signature sound. As you can hear, the music simply floats, like clouds floating by in the sky. It has an ethereal, beautiful sound, not at all dark.

Diminished Scale (Messiaen’s Mode 2)

Now, Messiaen didn’t want to use the whole tone scale, even though it is a mode of limited transposition, because it had already been used so much by Debussy and his contemporaries. Messiaen’s favorite was what we now call the diminished scale, which is an 8 tone scale that has three transpositions. It is much more flexible and interesting than the whole tone scale for many reasons, AND it contains lots of tritones as well. 8 to be exact, because, like the whole tone scale, every note is part of a tritone. But also, every other note has a perfect fifth above it, so it contains more stability than the whole tone scale which has no perfect fifths at all.

Also, like the whole tone scale, it is symmetric, as both scales contain the tritone at the middle as a balance point. This is an 8 tone scale, and the whole tone scale is a 6 tone scale.

The diminished scale alternates whole steps and half steps, and gives rise to many different triads as well. It can really be thought of as a scale with 4 tonal centers, not one. For example, on the diminished scale starting on C, (the version beginning with a half step), either C, Eb, Gb, or A can be the tonal center.

So we can easily see that this is not your standard hierarchical model as in the diatonic scale.

Also this scale contains a lot of major and minor triads, unlike the whole tone scale, which has none. This scale contains C major and minor, Eb major and minor, F# major and minor, and A major and minor. These and their inversions can be combined in some many beautiful and unusual ways, that are definitely “thinking outside the box”, so to speak.

And, in jazz, this scale is widely used whenever a dominant 7th chord appears. The possibilities of tonal material in this particular context are huge. And when we combine this with blues, which uses mostly dominant 7th chords, we find ourselves in a whole new territory. Miles Davis was one of the first to start exploring the diminished scale over blues changes, and there we can hear a very open and flexible sense of tonality, that also still contains plenty of the iconoclastic and funky tritone, whose roots in American music are planted in the original African based blues of the deep south. And the infamous “blue note”, which basically is what makes blues what it is…? You guessed it, the tritone.

And by the way, you will notice too, that ALL funky music contains the tritone. The minute you play the chords with tritones, especially without resolving them, but rather just hanging on and repeating them (mostly the dominant 7th, but also the minor 6th chord, used by James Brown a lot), everybody wants to dance. So he is not just an ethereal character, the tritone. He can activate the lower chakras like no other interval. He is very familiar with the street, this guy.

Messiaen’s Mode 3

Another of Messaien’s favourite modes of limited transposition is a scale based on dividing the octave into tree equal parts: 3 major thirds. That makes an augmented triad, Built on C it would be C, E, Ab. And then within these 3 equal parts we have several options, but the scale he favored went in the order of whole step/half step, in each of these three partitions of the octave. hence the scale is C, D, Eb, then E, F#, G, then Ab Bb, B. Notice that this is as if you started a minor scale on each of the three tones of the augmented triad. And this is a scale of 4 transpositions, before returning to the same sequence. Although it doesn’t contain as many instances of the tritone as the diminished scale, it contains enough (6 to be exact) to make a very unusual sound. It also has fifths over every other tone, contains major and minor triads, and is a 9 note scale.

Just for fun i created a little compositional sketch using this scale, you can hear it here. You can hear immediately that it doesn’t sound so familiar…but certainly not “evil” 🙂 I simply created some scale tone chords from this scale, shown here, and them moved them around a bit, with all melodic content also drawn from the scale, even in the bass.

Just in case you are running out of melodic material or ideas, check out this interesting explanation and exploration by Mark Hannaford of Messiaen’s 3rd Mode, used in relation to the John Coltrane jazz classic Giant Steps:

To see the complete list of Messiaen’s modes of limited transposition click here:

notice that all of these modes feature the tritone at the center.

https://en.wikipedia.org/wiki/Modes_of_limited_transposition#Messiaen.27s_list

Here you can listen to some of Messiaen’s sublime piano music, Preludes for Piano, in which all of these modes are employed, mixed in different ways, and in different transpositions:

Has the diatonic become diabolic?

So, why am I writing about this? Well, I am just hoping to spark people’s imaginations to look deeper into possibilities in music. Or perhaps into the impossibilities.

It seems, from listening to music today, that we’ve run out of possibilities. We hear the same melodies, chords, and rhythms rehashed in a million different ways, but often with the overwhelming sense that we are stuck in some kind of loop, which yields nothing new. Have all the notes been played already? This dilemma also translates to our world. Have we run out of possibilities here? Are we doomed? Has the diatonic become diabolic?

Perhaps we need to reassess just what we believe is possible, and what we believe is impossible. Perhaps we haven’t yet explored enough what we consider to be impossible. This reminds me of a moment, during the lunch break of a Tibetan Buddhist teaching i was at years ago. The llama went into a little junk shop next to the hall where the event was, and as he was looking through all the stuff there, he said: “there must be something good here, we’re just not looking hard enough!” For some reason, that moment stuck with me.

With rhythm it is the same thing. Have all the rhythms been played already? Are we really hearing anything truly NEW these days? I usually get the sense that there are a few limited rhythms that show up in almost everything, when there are actually SO many other possibilities that haven’t been explored yet. With Fractal Rhythm Theory i want to open new ways of exploring rhythm, and how to apply rhythm to not only percussion, but every level of the musical language, as in the rhythms used in melodies, the harmonic rhythm of chord progressions, and even the rhythmic proportions of interval placement in chords.

Of course, any combination of tones, even the most overused ones, can sound new, based on the presence of the musician. But we can expand our tonal palette as well. We can begin to notice how our ears have been conditioned to hear some combinations as dissonant, or “bad”, and others as consonant, or “beautiful”. To keep the hearing open is the art of the audience. When the listener has a well-developed sense of harmony, it really does help the musician to play and sound much better, and sometimes musicians are even surprised at what comes through because of the interaction with a good audience. As listeners let’s not underestimate our part of the equation. Without good listeners, the music suffers. And when the ears receiving are wide open and present, the music soars. We are all connected.

With an open mind we can devote a new kind of curiosity into the sacred geometry of music, where we can discover the infinite fractal quality that comes from the endless creation of Source. As we are each a unique fractal of Source, we each have our own continuous access to this fountain of creation. But we have to drop out of the mechanical repetition paradigm, the world of ego, the world of the “doer”, and explore new ways of allowing things to happen. These treasures open when we allow the Divine Composer to take over the creative process. And that is what is going to open up a sorely-needed new level of creativity and originality in music.

So let’s show the tritone some love !

Disclaimer: there is nothing at all scientific about all this. This is just an imaginal mytho-poetic attempt at expanding the musical vista into a living world. We can call it “musicalizing existence”. Maybe someone will find it helpful. I hope so.