The Mathematics of Music: Constructive or Descriptive?
I claim that the subject is historically descriptive, PeriodicTable150.pdf (142.6 KB)
The way we hear music is determined by our physical ears and brains. By this I mean that our appreciation of music and our perception of music are correlated, that is, these two values are directly proportional.
I would say “Creative”, as my dear cuban master teached to me: “la música es el arte de bien combinar sonidos, silencios y ritmo” 
You can use mathematical constructs to describe music which is important for sharing and communicating intent. You can also use it to construct music and is the basis for the majority of music (imo) as many composers start with such constructs and build on them. Some even use algorithms to construct their compositions. Of course the art comes in with the variations. Good vibrations can change a good piece to a masterpiece.
@jofemodo. Great answer, I can hear Salsa!
My original intent was to present the Melakarta system of 72 distinct scales to Western musicians. But as I continued to study the math involved, I had to go back and learn something about Group Theory. The Intervals are strings of integers, and I have convinced myself that this is probably the most efficient format for storing these (and other even more exotic) scales. This format is independent of the tuning system; although it is based on Equal Temperament, it can be used for any 7 of 12 tone system, Pythagorean, Just, or whatever. You can pick any tonic and build the scale from the string of intervals. Or you can build other systems like 5 of 8, or 13 of 24.
I’m learning Python to implement some of these ideas. I haven’t done any real coding for 15-20 years, QBasic. Yeah, I’m a dinosaur. I wrote code for myself, just to prove that I understood the math. For instance, my trade was printing and I ran scanners when they took up a whole room and had oscilloscope readouts and core memories. So I wrote code to calculate and graph colors, converting from RGB to XYZ to Lab, etc. Started with arc lights and ended up with Photoshop on a G5 Mac.
So, in Python, I have the problem on converting from a string to a series of integers, to a series of floats, doing some calculations, converting back to integers, and then back to strings. Then the strings have to be identified by comparison to the original dictionary of strings. This is easy, although tedious, on paper. Combining 120 scales, 2 at a time is:
20 C 2 = 7140.
That’s a small number for a computer, but a large number to work out by hand. So I’m working on code for this calculation. But once it’s done, it’s done,; it will become a set of data. Of course there are duplications, sometimes 2 scales produce different scales, and other times the same 2 scales. Using the language of Group Theory, I claim that the catalog of the addition of these scales will converge to the kernel of the group.
Two extremes of approaches to Music Theory are represented by:
ARxIV:2010.01728v1
group actions, power mean orbit size, and musical scales
ARxIV:1202.4212v2
harmony explained: progress toward a scientific theory of music
The first paper presents a comprehensive analysis of scale groups, far beyond my naive classifications.
The second presents as the work of a crank, although mine might be also so described. The author’s criticism of John Mehegan’s seminal work seems excessive.
I read Mehegan in my early 20’s, and it was revelatory. 12 keys times 5 chords equals 60 chords. The stellated dodecahedron (5 x 12 = 60), or the stellated icosahedron (3 x 20 = 60).
Don’t bad-rap Mehegan! His books did more for music than any ‘medieval medical textbook’ ever did for medicine. (Paracelsus actually introduced Ayurvedic medicine to Europe).
Sorry, I’m ranting now. Reality check! More later.
I have their second lp…
And there he goes… bringing a different dimension to an otherwise facinating discussion 
.
Sounds really interesting though I dare not start to read as it will distract me from everything else I need to fit into this tiny brain!
It’s a proof I read it : -D
IT’s quite strongly related to the discussion on access to parameters that is going on elsewhere in that you are looking at exploring high numbers of dimensions and the mechanism for ordering.
The equally tempered construction of these paths through a multi dimension cloud of reference points has been something it took music several centuries to refine and to talk in terms of mixolydian et al, is probably as far as most musicians would take it.
They also tend to speak emotionally of differing modes and the emotional response they engender. That, perhaps, rather more interpretive.
But then in Little Red Riding Hood I tended to support the wolf. . .
I haven’t read the two papers are ARxIV, but the abstracts look problematic in the extreme, particularly the one about harmony. That particular paper, at least the abstract, starts with criticism of the history of music theory without substantive context. The other problem with that paper is just the word “harmony” itself evokes a centuries-old approach to music (Jean Rameau and Descartes, etc). Voice leading is a much more interesting term because it’s something that’s common to harmony AND counterpoint. It’s also a term, voice leading, that by and large is absent from Mehegan’s writings which are, well, interesting, useful, the first-of-their-kind and hugely flawed (his 5x12 matrix is troubled and extreme. Too extreme not to mention that the chord changes he provides as “jazz chord changes” are by and large, well, inaccurate. Still I learned a lot from those books which only goes to show that anything we read and study is better than reading and studying nothing!
Please don’t read my comments are coming from a wet-blanket thrower! It’s just that there’s a lot of great work out there in computational musicology, theory, composition, improvisation, etc…
Andd nwhere to begin even pointing is hard! There’s music21 (in python), The Geometry of Music (a book), and Barry Harris’ method of looking at jazz in particular. But those three are the tip of the tip of the tip of an iceberg or rather icebergs.
Again, I don’t mean to sound like a wet-blanket thrower (wet-blankets aren’t helpful!) because ANY excursion into computational “anything” related to music is a blessing! I mean, that’s how we all got to Zynthian in the first place! 
Hope this is helpful in some way …
Grrrr. And everything I’ve written is largely about pitch-centric approaches which is to say examinations the role of rhythm in music is largely undocumented in Western music theory (there are some exceptions). And “theory” itself is even a problematic term if only because theoretical knowledge doesn’t necessarily lead to practical knowledge and vice versa … I am very conflicted about all of this, as may be evident!
Aint that the truth !We have developed so many tools over the past forty or so years that have really allowed us to revisit much of what has been developed as ‘theory’.We attack these issues with our logic brain but in truth, I think we interact with music at a much lower chemical level, and I sure we felt rhythms before we grocked melody…
I wonder which group of creatures put melody together, initially, and roughly when they did it?
It becomes an exercise in post event notation, comparison and analysis.
I was reading about the differing requirements the brain makes with different fonts and how it’s used to attract more ‘awareness’ and attention from the viewer.
or to paraphrase…
I’ luv a good tune… Well your forced to . . . ’
Music21 seems like an incredible library, but mostly geared towards musical score analysis and manipulation and accessing a catalog of scores.
There is also project abjab (also in python) which is more geared towards musical composition (although outputting Midi needs Lillypond, so may not be practical)
These could even work on Zynthian providing the groundwork for a host of composition and output tools, generating melodies etc. possibly integrated with the UI/sequencer…
Hmm, the dream of a magical music box is coming closer…
… one could do a lot with music21 for composition but you’re right, it’s not geared for that specific purpose. Some years ago, David Cope produced some incredible results with his EMI (Experiments in Musical Intelligence). The first step in that was to analyse existing scores to find patterns and then … One could mix Mahler, Mozart, and whomever to create styles that existed, didn’t exist, or might exist only under x, y, and z conditions. He was totally excoriated by the majority of composers and performers who knew of his work! Because the outputs from EMI were often indistinguishable from the real thing …
A good story …
And +1 + 1 for abjab! Looks to be fascinating and useful!
After a disturbed night’s sleep (we had terrible storms last night), I find some new posts.
@printer1: I’ll upload the booklet explaining the Table. Mehegan wasn’t the be-all-end-all of music theory, he made mistakes. For instance, his chord progressions starting a blues with a major seventh. This never happens in classic blues.
Regarding ‘voice leading,’ I use the term Melodic Motion for the notes separated by a single Interval, along with the direction from the less harmonious note to the most harmonious.Raganomicon150.pdf (1.9 MB)
More later, life intervenes with art.
Thanks for the booklet Mickey … will check it out …
Major seven in blues … sure, we can find that all the time in bebop (Bud Powell, Charlie Parker), which is what Mehegan’s Vol I is about. It’s a common bebop idiom.
If you get a hold of the Charlie Parker Omnibook (a set of transcriptions) it’s easy to find examples ,in blues, of the I major 7 rather than the dominant chord built on I.
If we don’t get some 
out of this them I’m going to have to publish a public flattened fifth …
https://www.youtube.com/watch?v=dM-t3I5uFro
warning . . . which is what can happen if you let machines do it … 
Does anyone know about Scalia? Some years ago I downloaded a ‘List of mode names’ from an obsolete address of Manuel Op de Coul. When I just now goggled his name I got the site for Scalia, a scale building freeware. The list ranged from 7 tone Thai to 183 tone equal, 873 modes. I took the 12 tone modes and resorted from alphabetic to numeric order. I was, and remain, focused on the Melakarta system.
Interesting discussion. FYI I find this represantion much clearer and more computerphile.
https://ianring.com/musictheory/scales/
Basicaly describing every scale (12Tone equal temp) as a 12 Bit number and then sifting out copies (where group theory is involved). There are also hints to the “harmonicisity” (I know this isn’t wven a word, but hey…) of certain scales - you have to read it to the end though.
And remember: at the beginning there is one, and only one scale: the chromatic scale… 
I wonder if anyone has told the birds yet ? 
I would love to know how the human mechanism for tone recognition evolves. Rick Beato, whilst demonstrating his son’s considerable ability in perception, talks of pitched languages offering a route to ‘perfect pitch’ .
It would be interesting to know if someone who believes themselves to possess that ability would make of pitches that sit outside of a recognised power law based progression or similar categorisation structure.?
Stephen Kay’s Karma uses interrelated groups of parameters to procedurally generate MIDI arpeggiations. Some of the dynamics include;
The complexity of so many variables and without a clear mental map between the parameters and desired output is probably the main reason it is some what universally unloved.
Of course you knew I meant that constrained to our 12 tone equal temperament
But to your interessting musings: Although there are many many many (mostly counterintuitive) findings with actual perceptions (Zwicker Fastl 1999, Psychoacoutics 2nd Ed., Springer), my understanding is, it all boils down to the way our hearing mechanism is built (namely the Cochlea 
). And that little snail has the power law hardwired, so to say.
So I doubt, that anyone can overcome that physical fact, but then again people solemny swear they have beed abducted by aliens… 
The bird example is intriguing in another way. At least to me it sounds they never sing out of tune, which is certainly not true. Sometimes they even “use” certain simple intervals like an octave you instantly can pick out. Why is that? They can’t read music, can they? 
Trust me the rooks and ravens round here can’t even read the evening menu . . . .
Which is a very good thing from their perspective and mental well being … .