The Pythagoras Project

The Pythagoras Project is a systematic accumulation and online storage of simple harmonic exercises.

These exercises form the basis for much of the jazz language of the 1950's and 1960's, and provide an excellent resource for pianists and guitarists, as well as composers and arrangers.

This material is free and intended for wide divulgation, with the intent of it becoming a common idiom for all serious jazz musicians

Below is the collection of exercise. More are being added on a continuing basis.

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These are all 3 or 4 voice harmonic exercises, and are fairly simple to execute,
but conceptually sometimes quite difficult.

The exercise concept is described in the text that precedes each exercise.


This is a way to enrich a major chord situation lasting 2 bars or more.
Play the major 6 chord (drop 2), which is a 1 - 5 - 6 - 3 voicing, in any inversion.
Now shift all 4 notes to an adjacent dim.7 chord. This is equivalent to playing a dominant b9 chord.
Alternate between these two chords, moving up and down the inversions.
Notice that a single scale is being used: the maj scale with an added #5.


This is a way to enrich a melodic line that follows a major (add #5) scale (e.g. bridge of My Funny Valentine)
Play the major 7 chord (drop 2), which is a 1 - 5 - 7 - 3 voicing, in root position.
Now shift all 4 notes to the adjacent notes in the major (add #5) scale.
The chords will sound almost like alternating major and 7(b9) chords, but each with an altered note.


This is a good way to enrich a major chord situation inside a slow moving tune (ballad).
This is a variation of 002, introducing some 8th note motion.
In this exercise, the altered note is resolved to a note that is either in the major 6 or the dim.7 chord, depending which the starting chord is.


This is essentially a way to enrich a major chord situation lasting 2 bars or more.
The top and bottom voices move at a 10th apart from each other.
The progression has three segments.
This first progresses up the major scale from I to ii to iii with a 1 - 5 - 7 - 3 voicing.
The second progresses from ii to iii, but with the ii in first inversion.
The third progresses from ii to iii in second inversion.
The 2nd inversion has the 9th of the may chord on top, so a LONG - SHORT movement can be performed between soprano and bass.



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This is an option for playing V7 to I
The V7 is played with the shell (1 - b7) in the L.H., and m7b5 on the 3rd in the R.H.
The I is played as 1 - 3 in the L.H., and Maj6 on the 5th in R.H.
The extensions are made to move down chromatically to their resolved chord tones.
The R.H. chords can be played in all inversions, to change the top note.



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Inserting diminished between m7 chord
Note that the dominant chord does not have a 7th! Yet, due to context, it sounds like a dominant. The b9 contributes to its dominant sound.
The chords can also be played with 4 voices.


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