Riemannian Transformations: Part 3 "Order, order, order!"

[@peterkelly8357]

Frans, your a genius

[@FransAbsil]

Well, I'd rather call it a passion for music, paired with curiosity and a structured approach to learning. But, thanks!

[@groovinhooves]

Aren't LRL/RLR transformations also valid? They all appear to result in functional harmonic succession, if not progression. For instance, C-major transforms LRL to B minor which equates to bII-i (Phrygian cadence). Conversely, C-major transform RLR to D-minor can be read as the deceptive cadence V-vi. Is it a requirement of L3 that the pathology always include "P"? I skipped lower division Schillinger Theory, so maybe I'm missing something in the 'rules' here, but I'd rather be found to be ignorant than remain so by failing to ask a question. Cheers!

[@FransAbsil]

@Mark Hoover. Thanks for the detailed observation of the two missing 3-step paths in the Tonnetz diagram in this video. Indeed they may be interpreted in a functional harmony as more or less special cadences. However a NRT characteristic is that the triad pairs may be used as floating and alternating harmonies. That yields the ambiguous pairs such as C-Fm and C-Gm that may act as Riemannian TF with a single common note, but also occur naturally in diatonic harmony. Their cadential feeling makes them less likely candidates for an NRT progression. C to {Abm,Ebm,F#m,C#m,G#m} are obvious 'remote' pairs (no overlap) with a characteristic 'sound' (chromatic); the {Bm,Dm} have proven not to have the same potential for moods and colouring, and therefore will not show up in the list of 3rd order triad connections. But you are right; when enumerating all options, they should be included.

[@groovinhooves]

@@FransAbsil Thank you for your response, Frans (NB: KZbin did not send me a notification, so I'm glad that I returned here). I think I see: the sense of ambiguity is paramount and any overlap with functional tonic-and-dominant harmony is merely happenstance? Ok, now that I've doubled back and paid better attention to the video on L2 transformations, noting that the motions to relative Dominant and Subdominant are not discussed despite falling within the L2 range, I think I am understanding the nuance.

[@FransAbsil]

@Mark Hoover Sorry for the fact that YT did not notify you of my response. Same thing happens to me occasionally. Others have reported this, but there is no clear information on how to solve this problem. And indeed, interpreting the horizontal-diatonic axis transformations as Riemannian is a matter of nuance and depending on the context.

[@TheSnowLeopard]

Hi, thanks for the clear and straightforward tutorials! I like your version of the Tonnetz diagram, do you have any printable versions?

[@FransAbsil]

I am planning to create a companion webpage on my site with the graphics and the full scores. How would you prefer the Tonnetz diagrams? Colour or black and white? PDF or JPG? I assume without Riemannian transformation paths and coloured triad triangles. Be aware that some authors depict major triads pointing upward. If you have any specific requirements, send me an email.

[@TheSnowLeopard]

@@FransAbsil Hi, the companion webpage sounds excellent. I suggest most people would prefer black and white, as many people use monochrome laser printers these days. Either PDF or JPG is fine, no other specific requirements.

[@FransAbsil]

Thanks for specifying preferences. I'll prepare a black-and-white version. However there is another design aspect. My Tonnetz diagrams are always centered around a 'main tonic' (e.g., C) and I ran into range problems for remote paths or other keys, forcing me to extend the graph or create new diagrams. How about a template Tonnetz that consists of blank triad triangles with vertices and empty pitch circles at the nodal points? The user could print the template, then write in triads an pitches as needed. What's your opinion on this?

[@TheSnowLeopard]

@@FransAbsil Yes, that option could work well.

[@FransAbsil]

@Snow Leopard. There now (Jun 30, 2020) is a free Tonnetz template PDF document in black-and-white on my website. Follow the link in the description.