Riemannian Transformations: Part 4 The Long and Winding Tonnetz Threads

[@sudsolar]

It would be great to see in your videos how you review the concepts of transformations, neutralization, regions and extended tonality of Schonberg's harmony structural functions. I really enjoy your videos.

[@FransAbsil]

@ionizacion Thanks for this suggestion. Maybe a coincidence, but the other day I was thinking about creating a video on how the Schoenberg and Schillinger diatonic harmony systems compare. However, the (strict) Schoenberg theory of harmony by itself already is covered in so many online sources, by knowledgeable authorities. There might be added value in demonstrating where the Schillinger generalisation approach starts to deviate from the strict rules, opening up many (undiscovered?) options. Will think about this a bit longer.

[@deRoland87]

this series was quite excellent and informative. It seems like a good and systematic approach towards categorizing triad motion with tons of helpful conceptual distinctions that also have a counterpart in the tonnetz diagram. I want to thank you for great work! I was sort of left with the following question at the end: There are exacty 24 (i.e. 12 + 12) triads, half major and half minor. So if I start from C major, there are 23 possible destinations, each yielding a unique transformation, with unique voice leading characteristics. If we start from a minor triad, there's another 23 unique possible destinations. Together these make 46, but each transformation has a unique inverse, so we're back at 23 * 2. Since the whole approach in your videos is quite encyclopedic and mathematical, I was surprised that you did not go a tiny bit further to give a mathematically exhaustive list of transformations. (Or at least, I think you did not. Or at least you do not bring it up.) Where there any transformations that are simply too rare? Or that proved resistant to description in terms of compound transformation?

[@FransAbsil]

@Roland Bolz Your observation is correct; this 4-episode series is not an exhaustive treatment of all possible {major,minor} triad combinations. In Part 4 the diagram shows 16 options surrounding the starting triad. Most of these are what are considered basic or compound Riemannian triad pairs, and some are traditional, diatonic (sub)mediant triad pairs. In these tutorials I wanted to discuss the difference between the 2 kinds, using the axis direction in the tonnetz, and then focus on the usage of the Riemannian Transformation potential, that has become such a strong (overused?) element in contemporary film music. That leaves out a number of triad motion options, whose characteristics (remoteness) have apparently made them less suitable for music composition. Thanks for the positive overall feedback on this series.

[@deRoland87]

@@FransAbsil yes thanks a lot for getting the point across so well about the film music vocabulary. That was very helpful!! I also agreed with your assessment that some of the more outlandish geometrical ideas are hard to apply as composers.