New stuff on Transformations, and more.

Hi patrons! I've been plugging away at the book and have a pretty solid scaffolding for all the chapters, as well as some sections of them written and completed. The diligence of researching this book means I'm having to track down references from the past 80 years (or sometimes more) of music theory publications, so that every concept in the Exciting Universe is properly attributed to the person who either invented it, or first published it. It's been a difficult but enlightening experience. Whereas I used to merely learn & regurgitate stuff into the website, now I'm following trails of citations and scouring old books that I might never have bothered with before, and I'm uncovering some interesting smells along the way.

Among my discoveries was the first person to ever represent a pitch class set as a binary word: Daniel Starr, in a 1978 article for Journal Of Music Theory, titled "Sets, Invariance, and Partitions". I also discovered

I also ended up in personal correspondence with the man who invented the "Deep Scale" property, Terry Winograd! He sent me a copy of an unpublished 1966 paper he wrote for a class with Norman Carey, in which all the arcane details of Deep Scales are explored and given robust mathematical proofs. Priceless stuff.

I also learned about a thing called "Chopin's Theorem", which describes how the complement of a generated scale will have the same generator. Which makes obvious sense if you consider it for a moment, but that theorem never had a name until Emmanuel Amiot gave it one. Chopin himself had nothing to do with it, but Amiot named it after the "poet of the piano" because the generator symmetry between complements is illustrated by the major/pentatonic pairing in Opus 10, N. 5 etude in G flat major.

The real thing I came online to tell you about was that last night I added some new stuff to the publicly-facing Scale Detail pages. In the "Transformations" section, there used to be just T0. T1, T2... and T1I, T2I, T3I... but now I've added the special transformations for "M" so the number of transformations is doubled to include T0M, T1M, T2M... and T0MI, T1MI, T2MI... . 

The "M" Transformation is a cool one, not always mentioned in the 12-tone literature; rather than a transposition (T) or inversion (I), it does a multiplication (M). As is often the case in duodecimal stuff, we find that multiplications by factors of 12 are degenerate, ie you *could* do multiplying by 2, 3, 4 or 6, but you'd end up with a lot of repeated pitches - a poor mapping. Multiplying by 1 leaves a sonority unchanged. Multiplying by 11 (mod 12) is the same transformation as inversion, aka M1I. M10 is the same as M2I. M9 is the same as M3I, and so on.

So, the only non-degenerate M-factors are 1, 5, 7, and 11. Of those, M1 is a no-op (it accomplishes nothing), M7 is the inverse of M5, and M11 is the same as I. So what's left? The only interesting M-factor is 5.

So, when you see "M" in the abbreviation of a transformation, it means M5 has been applied. Every pitch is multiplied by 5 (mod 12) to become a new pitch. Effectively what this does is swap a bunch of even-numbered tones around. For more detail than that there's an excellent chapter about it in "Simple Composition" by Charles Wuorinen.

The last thing I'll mention is that I'm following a scent (a citation in a paper by Jeremiah Goyette) that I think leads to the use of binary representation as a method of finding the Prime form of a scale. This is humbling because I thought I invented that method myself, but now I'm finding that it might have existed a few years before I was born - in the early 70s - it's just that subsequent writers on the subject have buried it. Goyette also mentions the six PCS where Forte and Rahn diverge in their definitions of prime, a truth that I calculated by brute force with my software three years ago... again I thought that I had discovered a previously unknown sixth Divergent Prime, well it turns out that someone already did that.

I could go on, and on... but I'm going to cut myself off there. Big thanks to a few new patrons who have signed on! I'm so deeply grateful for your patronage, you have no idea. This is a spare-time pursuit, I'm not a full-time academic, and this pursuit isn't supported by anyone except you. Your patronage means that once in a while I can buy a book, or upgrade my software, pay some of the server/hosting fees and such. I hope that over time this Patreon thing grows to the point where I can at least sort of break even :D

In Ontario a lot of public events and gatherings have been cancelled, and we are encouraged to practice "social distancing" or "social isolation" for *at least* the next two weeks. This has been (and will be) financially devastating for musicians who depend in whole or part on live performances for their living. Our local orchestra has cancelled their entire season, and many musicians I know are in a panic, not sure if they'll become homeless. If you have tickets for an upcoming show that has been cancelled (especially chamber, small-venue, community orchestra, indie), consider donating the refund or saved cost directly to a musician who is going to have trouble paying their rent this month, and next. May you and yours stay safe and healthy.
xo Ian