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In Part I of "Redefining the 12 Tone System of Harmony" this article will provide proof that the Diminished scale and its' resulting harmonies are not a 4 tonic system as frequently taught in Jazz theory and Jazz vernacular. Second, this article will provide proof that the augmented scale and it's resulting harmonies are not a 3 tonic system. Third, this article will replace this information with the truth.
Fact: The HW Diminshed scale has eight transpositions within the scale that are the same set of notes. These transpositions occur at minor 3rd intervals from any point in the scale. This property allows the same scale to be used for D,F, Ab and B HW Diminished scales or Eb,Gb,A and C WH Diminished scales. Either way, it's the same set of notes. However, these eight transpositions are within the same set of notes and are not tonics of diffent scales but modes of the same scale..
Fact: All Diminished Scales are a combination of two of the following chords that are interchanged Do7, Go7 or Co7.
Do7 and Co7 creates the DHW Diminished Scale. Go7 and Do7 creates the GHW Diminished Scale and Co7 and Go7 create the CHW Diminished Scale. All other diminished scales are a transpositions of these three scales i.e. three tonics of the diminished scale. Each of the three diminished scale has four modes that transpose within the same scale. i.e. the same scale with the same interval structure within the same scale. Therefore the D HW Diminished Scale covers the D, F, Ab, and B Diminished Scales, GHW Diminished covers the G, Bb, Db, and E HW Diminished Scales and CHW Diminished Covers the C, Eb, Gb, and A HW Diminished Scales. Based on this information, the diminished system is a 3 tonic system with 4 transpositions and is not a 4 tonic system.
The four transpositions of a the DHW Diminished scale D,F,Ab and B , astonishingly, are tonics of four distinct Augmented scales. This is a well overlooked fact regarding the Diminished scale. Now, it is a well known fact that the HW Diminished scale only has three distinct transpositions in which the notes of the scale are not the exact same set of notes. The tonics of those scales are D, F# and A# HW Diminished Scale or Do7, Co7 and G07 as described above. Note that the transposition to different Diminished scales transpose in Major thirds with the tonics or roots of the chord outlining and Augmented triad while the transpositions to diffent Augmented scales transpose in Minor thirds and outline a D Diminished 7th chord. Check it out for yourself.
On the other hand, the Augmented scale has similar transposing qualities as the diminished scale. The minor third half step augmented scale transposes it's intervals in major thirds intervals from any note within the same scale. This property allows the same scale to be used for D, F# and A# Minor third Half step augmented scales or F, A and C# Half step Minor third augmented scales. Either way, it's the same set of notes. However, these six transpositions are within the same set of notes and are not tonics of diffent scales but modes of the same scale.
Conclusion, in the 12 tone system, each tonic or key has a distict set of note intervals which are transposed thereby created 12 different Major/Minor scales. The true 3 tonic system is the Diminished Scale because it can only be transposed into three distict scales and the true 4 tonic system is the Augmented scale because it can only be transposed into four distict scales.Jazz musicians and academia have mistaken the interdependent nature of these scales and reversed the facts to a certain degree. This article was written to set the record straight.
The proverbial "take away" here is the interrelatedness of the two scale as an interdependent system. Part II will discuss the Diminished Augmented System as a Nine Tone Symmetrical Scale System.