diminished scale theory

It is only within the system of equal temperament that the diminished color exists. In no other system of music or music associated with a particular culture globally do we find this unique and potentially important group of pitches. So where does diminished color come from within the system of equal temper? Well, depending on how one understands the theory, there are a couple of ways to easily view the organic basis of the diminished color from within the system of equal temper, and not surprisingly, they are somewhat interrelated. Please remember that the folks that discovered and designed this system were perhaps not only pretty serious players, but also some of the advanced mathematicians and scientists of their day ( 1550's or so ).

By knowing the interval formula for the diminished scale, we can easily extract its pitches from the chromatic scale. Example 1.

As one of three of our symmetrically created scales, the diminished color is unique in its repeating whole step whole step / half step motion combining to create the minor 3rd interval of the diminished arpeggio and chord. A half step / whole step would work also eh? Cool with this theory? But what key are we in? Here is the sound of the above pitches. Example 1a.

Interesting in that of the 12 pitches of the chromatic scale, that the pitches of the 4 note diminished chord become leading tones of the 4 notes not used in the creation of the diminished scale, in both the major or minor tonality.

One organic basis of the diminished scale color from within the equal temper system comes from the fully diminished arpeggio that is created from the seventh degree of the harmonic minor scale. Lets find it. Here are the pitches of the harmonic minor color by scale degree then rewritten in its arpeggiated form. Example 2.

So where is the fully diminished arpeggio or chord? Can you find three consecutive minor third intervals? See it, cool, it starts on the leading tone B, which becomes the root of the chord. Three consecutive minor thirds? Yep. Here's a chart. Example 2a.

Cool so far? Why does these pitches create a fully diminished seventh arpeggio? Well, it is based on the size of the intervals used in it's construction. What is a diminished interval? Simply a musical interval that has been reduced in size or "diminished." Lets explore. Example 2b.

Musical intervals simply measure the distance between two pitches. So from the root B, we measure a minor 3rd to D, a diminished 5th to F and a diminished seventh to Ab.

Lets transpose the arpeggio in example 1 above a up a half step to the root C, to get us back to our default tonal center, hopefully easing the transfer of the forthcoming theoretical ideas and principles. Example 2c.

From this diminished arpeggio we can create a diminished scale. The diminished scale is sometimes called a symmetrical scale, created by the division of the minor third interval, the intervalic basis of the diminished arpeggio or chord. Extending the minor third interval creates a looping of the pitches. Example 2d.

See the closure or looping in the above group of pitches? Cool. Perhaps the most common diminished scale configuration is created by simply dividing the minor third interval into a repeating a whole step ( 1 ) / half step ( 1/2 ) cell, ( whole step + half step = minor third ), thus we can apply the term "symmetrical" to its intervalic configuration based on this repeating intervalic cell. Let's build a C diminished scale. Example 2e.

See how the scale is built? The pitches of the C diminished arpeggio are colored purple above. We are simple adding a half step lower neighbor to each of the pitches of the diminished arpeggio to create the scale. D is a half step below Eb, F is a half step below Gb etc. Here is the sound of the diminished arpeggio followed by the scale we just created. Example 2f.

What's your gut reaction to the sound? Cool? Uncool? Trash? Is the diminished sound new for you? The diminished sound is tense and unstable, it seeks stability, to resolve to a more at rest or stable sound. This next idea simply resolves the diminished color to the natural minor color. Example 2g. 

Pretty smooth resolution eh? Of all of the colors organically created by our equal tempered system, the diminished sound and color could perhaps provide the strongest sense of artistic tension and tonal instability. Having no real sense of tonic center, the diminished color can quickly change not only the tonal center or key of the music, but it's tonality i.e., major / minor, as well. This next ideas runs the same C diminished group of pitches into the tonal centers of G major and Db minor tonalities. Example 2h.

Another way to approach understanding the diminished color is simply to use the interval pattern or mathematical formula described above and project this pattern from any of the 12 pitches of the chromatic scale. So 12 different diminished scales? Well, yes and no. Yes that there are 12 different diminished scales and no, because there is certain degree of "looping" with the pitches due to its symmetrical construction. Lets explore. What group of pitches would we create if we were to start on Eb ( a minor third up from C) and use only the pitches of the C diminished scale? Let's compare and create. Example 2.

Look familiar? Cool, we just created the Eb diminished group from the pitches of C diminished. You mean the pitches are exactly the same within the C and Eb diminished scales? Seeing is believing eh? How? This cool diminished theoretical property is possible due to the diminished scales symmetrical intervalic formula. Really? Ya mon. If Eb is part of the C diminished arpeggio and creates the same group of pitches when used as the root of the scale, what about the other two members of the C fully diminished arpeggio? Like Gb? Example 2a.

This is too easy! The understanding of this theoretical property of the diminished color contained in the chart above is potentially huge, depending on one's artistic directions of course. How about the diminished group built on the pitch A, the fourth member of the C fully diminished arpeggio? Any guesses as to what group of pitches is going to be created by applying the whole step / half step intervalic formula of the diminished scale to the root or fundamental pitch A? Example 2b.

What we have just done is to create four different diminished scales from the one group of pitches, originally called C diminished. These four roots C, Eb, Gb and A also form a fully diminished arpeggio or chord, created exclusively by the minor third interval. Cool huh? This little bit of theory can unlock so much resource. We could also enharmonically respell this arpeggio or the scale roots to be B#, D#, F# and Bbb if necessary, as dictated by the tonal center in which it is placed.

Let's move the above program up a half step, i.e., chromatically, create a new diminished arpeggio from Db, and see if the diminished magic discovered above is consistent from other roots. Example 3.

Here is the diminished scale created from the above fully diminished arpeggio built on the root Db. Example 3a.

What group would we create if we were to start up a minor third from Db, the pitch on E, and use the same intervalic formula? Lets build and compare. Example 3b.

Cool huh, no surprises that the same "looping magic" that we found above in working with C diminished works here also. How about creating the whole tone / half tone diminished color from the roots G and Bb? Example 3c.

As originally done above with C diminished, again we have created four "different" diminished scales from the one group originally called Db diminished. These four roots: Db, E, G and Bb form a diminished arpeggio or chord, created by the minor third interval, whose dissonance is also found in pairing up of the tritones. We could easily respell the Db to C#, the Gb to F#, the Bb to A# if necessary. Notice what pitches that fall between the Db diminished arpeggio?

Lets create a new diminished scale from D and extract its other diminished groupings as done above. Any guesses as to the results? Here we will be creating the diminished scale group from the root D. Lets start with fully diminished arpeggio created from the root D. Example 4.

Now to create the scale from the arpeggio. Example 4a.

What group would we create if we were to start on F and use only the pitches of D diminished? Example 4b. You know this by now don't cha?

Cool, the same diminished "magic" continues to work here also. How about the Ab and B diminished scales? Example 4c.

Again we have created four "different" diminished scales from the one group, originally called D diminished. These four roots: D, F, Ab and B form a diminished arpeggio or chord created by the minor third interval, whose dissonance is also found in pairing up of the tritones. We could easily respell the Ab to G# if theoretically necessary, or any of the pitches for that matter. Using enharmonic equivalents when necessary is never a problem, the idea is to stay as diatonically close to the key center being used as possible. Notice what pitches that fall between the D diminished arpeggio? Are we combining two different diminished arpeggios to create a diminished scale? Yes, that we are. Is this important? Sure is. Why? Soon we shall see ...

To continue the process started at the top of the page of building the diminished color from successive chromatic roots, ( C, Db, D,etc.) our next group would be on Eb. Any ideas what the pitches of the Eb diminished scale would be? Maybe identical results to the pitches of the C diminished scale? The roots of C and Eb are a minor third right? Let's compare. Example 5.

Look familiar? Same chart from example 2 above. By identifying the pitches of the C and Eb diminished scales being identical, we have looped the theory onto itself. We’ve created the three distinct diminished scales from the roots C, Db, and D from which we can extract the other nine groups, allowing us to create a diminished scale on each of the 12 chromatic positions. We simply reduce the 12 down to one of three possibilities and respell enharmonically as needed. A very cool, advanced and sophisticated system that could be as much as four centuries old.

What about the arpeggios from each of the three groups? Let’s build the diminished arpeggios from the above three main starting points then add them together. So, thinking minor thirds ... Example 6.

Our resulting group is again the chromatic scale, from which all equal tempered elements can evolve. Our initial discussion of the minor third discussed three possible divisions of this interval. Whole tone / half tone, which we just examined, by half step or chromatic which we just created, leaving the reverse of the whole tone / half tone cell which creates the half tone / whole tone diminished configuration. Let’s create this cool color by simply dividing the minor third by half step (1/2) then whole step (1). Example 7.

See any familiar pitches or arpeggios? There are two fully diminished arpeggios in the above configuration. As we can see from this configuration we still retain the initial C and the three pitches up in minor thirds. C, Eb, Gb, and A. Our second arpeggio is Db, E, G and Bb, which we encountered previously as the lower neighbor pitches of the Db diminished scale. Now they become "upper neighbor tones." Players generally call the above scale grouping "half tone / whole tone or the "flat nine" scale, due to its dominant dominant chord affiliations. What arpeggio do you think will "appear" if we create the above configuration on the root Db? Here is the chart. Example 8.

Well, did you guess that the D diminished arpeggio would "appear" in the above Db grouping? So we are back to where we started, with three fully diminished arpeggios from the roots C, Db and D from which we can create two unique intervalic structures for shaping the diminished sound, perhaps remembering that combining the three fully diminished arpeggios creates the chromatic scale, the source of all of our equal tempered resources.

Review:

1) The diminished color is based on the interval of a minor 3rd, the basis of the arpeggio / chord. In creating scales, we can subdivide the minor 3rd interval two ways, whole step / half step and half step / whole step.

2) We can extract all of the possible diminished colors from the three groups whose roots are C, Db and D and respell them into any key situation we need them. This is achieved by simply respelling enharmonically any of the three scales or arpeggios to fit into any of the 12 major or 12 minor tonal environments. 

3) The three essential arpeggios when combined together create the chromatic scale. These can also be respelt to accommodate any key. In regards to how a particular learner chooses to internalize the knowledge, is it possible to use other roots to identify either of the three main groupings? Of course. Any and all of the 12 pitches of the chromatic scale can be the center of one's musical universe. When we aspire towards the chromatic understanding of things, all pitches are created equal.

So, is it easier for you to learn 12 different diminished scales, each created from the 12 pitches of the chromatic scale? Or, to understand the "looping" properties of the color and how 3 of the different scale possibilities can each be reworked into 4 possible configurations, thus encompassing the 12 pitches of the chromatic scale? You make the call, whatever works for you based on how you best learn and retain information is probably the way to go eh? But please remember that perhaps part of one's artistic and spiritual growth is based on their ability to evolve in their way of thinking and understanding of the elements as the years pass along. 

Here is a chart of the diminished scale spelt from the 12 pitches of the chromatic scale. I break a few rules in the choosing of pitches within a few of the scale groups, sorry for this, but this is how I understand the pitches. Without a tonal center for reference, I usually default to spelling things as close to the relative major and minor groupings of pitches.  Applying a particular diminished scale to a certain chord within the framework of key center usually clears up any of the goofiness in naming the pitches below. Example 9.

Here is the above chart written out in standard musical notation. Each of the scales is preceded by its arpeggio. Example 10. 

Got these under your fingers? Cool. Survey the improvisation / tonal convergence and harmony / chords sections for more ideas to employ the diminished sounds. Here are a few suggestions for study of wonderful jazz standards that utilize the diminished color.

Here are links to pages discussing ideas on how the diminished color is potentially used.

Music is your own experience, your own thoughts, your wisdom. If you don't live it, it won't come out of your horn. They teach you there's a boundary line to music. But, man, there's no boundary line to art. Charlie Parker