Reddit reviews A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford Studies in Music Theory)

>So if it isn't a minor mode, isn't a diminished mode, and certainly isn't a major mode, than is it really even a usable scale?

You know, all that theorizing is well and good, but you're closing off your mind and your ears maaaan. I like how Dmitri Tymoczko talks about this in A Geometry of Music:

>>Broadly speaking, theorists have explained centricity in two ways. Internal explanations assert that the structure of a group of notes is sufficient to pick one out as a tonal center, without any effort on the composer’s part. External explanations focus on what composers do, asserting that composers make notes more prominent (or stable) by playing them more frequently, accenting them rhythmically or dynamically, placing them in registrally salient positions, and so on. Rather than being a property of collections considered abstractly, centricity is a property of collections as they are used in actual music. (177)

>>...

>>My own sympathies lie very much with the external view: in most practical cases, I believe the internal contributions to centricity are relatively weak and can easily be overridden. Consequently it is entirely possible to write diatonic music that is acentric, or chromatic music that emphasizes a particular note. I do not consider either kind of music to be "unnatural." Nor do I have any problem with any of the diatonic modes (or any other mode of any other scale): I am entirely convinced by the music of the Renaissance, of Debussy, Ravel, and Shostakovich, and of contemporary jazz and rock; I enjoy those numerous passages of twentieth-century music that make use of symmetrical scales while still asserting a tonal center; and as a composer I believe I can make virtually any note of virtually any collection sound like a tonic. (This last conviction has been reinforced by my experience with computers: by emphasizing particular notes through repetition, duration, loudness, and step- wise melodic motion, it is easy to create the effect of centricity in otherwise random sequences.) Any theorist who wants to argue against these convictions would have to fight an uphill battle: indeed, the very claim that the phrygian mode is deficient, or that centric music cannot use symmetrical scales, strikes me as evidence of a limited musical imagination. (180-181)

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Not that this is a masterpiece or anything, but I wrote this as a proof of concept to toss around Reddit threads about locrian mode. In my real compositions, I use locrian occasionally. And John Kirkpatrick has this folk tune, Dust to Dust, that uses some polyphony with a very ex tempore feel.

I get where you're coming from because I started in a very similar place, but what you want is really not compatible with the study of music theory. Music theory attempts to break down the structure of music to explain why it works by using other works as an example. It's not really about explaining cause and effect relationships or mathematics. (though there's certainly some of that)

Music theory has much more to do with pointing out similarities between pieces of music so you can say, "Well, these pieces of music are effective because they share these things in common, and you can use these techniques to similar effect."

It also has to do with auditory perception and psychology. "These notes are harmonically similar, so they will mask each other." "The change in harmonic relationships between these notes over time imply that something is going to change." Etc... It sounds like those are concepts you are already familiar with.

It's not that these things aren't science-adjacent, but it's not a formal science. It just integrates bits and pieces of science, history, convention, etc... Trying to reduce anything but specific subsections of music theory down to something adequately explainable by science or mathematics is not going to be helpful or satisfying.

The best recommendation I could give you is to find a good music history text that starts out somewhere around 570BCE or earlier and leads into modern day. I've found that for myself, the only satisfying way to understand modern music terminology and convention is to observe how it evolved. I think that's the most scientific approach you could take.

There are also a few extremely talented polymaths that have attempted to represent musical relationships in novel and useful ways using mathematics/geometry. Dmitri Tymoczko immediately comes to mind.

I also think you would enjoy reading this book and this book, as one explores some really fascinating and practical mathematical representations of musical ideas, and the other explores the tension/release mechanics that dictate/relate to much of the theory surrounding modern musical structure, rhythm, and harmonic progression.

Other than that, if you see a term that you don't understand, look it up. If you see a term in that term's definition that you don't understand, look that up. Follow that rabbit hole to the bottom. Draw a graph if you have to.

Diving down hierarchies of terms I don't understand in order to gradually pick apart texts is a skill I've had to develop as a software developer and DIYer, and training that muscle has been invaluable. It's the reason I don't kill plants anymore, how I was able to write a raycasting engine without prior 3D graphics experience, and how I taught myself music theory.

Probably the most accessible mathematical approach to the basics of music theory that is still really solid scholarship would be Dimitri Tymoczko's A Geometry of Music. About the only downside is that Tymoczko has several bones to pick and he makes sure to pick them! http://www.amazon.com/Geometry-Music-Counterpoint-Extended-Practice/dp/0195336674

Another classic is David Lewin's Generalized Musical Intervals and Transformations. http://www.amazon.com/Generalized-Musical-Intervals-Transformations-David/dp/0199759944

If I were you, I'd start with Tymoczko and then move to Lewin after.

They're absolutely anything but arbitrary.

Although we can't really be said to have created them ourselves, each one is intricately constructed to allow such things we take for granted like tonality. You change the system just a tiny bit and none of it works. Consider the five features of tonality:

1. Conjunct melodic motion (melodic motion by tiny steps)
   
2. Acoustic consonance (harmonies that sound good, however you want to define that)
   
3. Harmonic consistency - the use of chords that are similar to each other
   
4. Centricity - the emphasis of one note being more important than the others.
   
5. The use of macro-harmony; i.e., scales (into which all of the chords, i.e., micro-harmony, fall.)
   
   These 5 features mean that harmony and counterpoint (the movement of 2 voices against each other) constrain each other. And this means that compared with all the possible combinations of scale and harmony (a number beyond imagination), the amount of options that work ("work" meaning here that it displays the five features above), is only a small amount. And although you think it might be arbitrary then to write tonal music, it's not, as it corresponds very finely to how human's perceive sound (such as how humans group together sonic events that are close in pitch = conjunct melodic motion.)
   
   So no, they're not arbitrary. I recommend this book for more: Geometry of Music
   
   edit: Of course, this is only for tonality. However, it is much, much harder to write completely non-tonal music than one would think. And even non-tonality is often explained in its relationship to tonality.

I've been studying algorithmic composition for a while now, and AFAIK the best resources are books about modeling elements of music perception or composition.

Dmitri Tymoczko - A Geometry of Music
Fred Lerdahl

Also, watching brilliant live coders like Andrew Sorensen do their thing can be very enlightening.

IV. Voice-Leading Parsimony

("Parsimony" means "thriftiness, frugality; unwillingness to spend money.")

One interesting fact about P, L, and R: they leave 2 notes untouched, and the voice that does move only moves by a step. P and L only move one note by a half step, and R is a little more extravagant by moving a voice by a whole step. So these transformations are "parsimonious" (frugal) in the sense that they can get you new chords for very little effort (motion). It turns out that the triad is pretty cool for being able to do this: very few other chord types in the world can. (For example, you can't get from one French 6th chord or fully diminished 7th to another just by moving one voice a tiny amount.)

The next thing that Neo-Riemannian theory asks is "What happens if I chain a bunch of transformations together?" For example, what happens if I make a sequence by alternating P's and L's? Each step along the way changes only 1 half-step, but how many different notes does it use total? How long before I get back to my starting chord? (Will I go through all 12 major and all 12 minor triads? Or do I only use a fraction of the total?) Neo-Riemannian theory maps out the possibilities and describes them using a concept from modern algebra known as an algebraic "group." The transformations P, L, and R form a "group" of things that you can combine to make new things (e.g. imagine considering L-then-R to be a single transformation of its own). Group theory is used to explore the structure of the possibilities there.

V. Enharmonic Equivalence

(That is, the assumption that there are only 12 notes and that spelling doesn't matter, so G# = Ab.)

This doesn't sound very exciting, because we're pretty used to it by now. But it was a radical notion early in the Romantic period, and composers like Schubert got some cool effects out of exploiting it.

Earlier I asked "What happens if I make a sequence out of alternating P's and L's?" Well, it turns out that I go through 6 different chords, like this: CM - Cm - AbM - Abm - EM - Em (then back to CM). Every L takes me to a chord with a root a M3 lower, so that after 6 steps I've gone down by 3 major thirds and end up back where I started. This needs enharmonic equivalence to work, because without it I'd go C - Ab - Fb - Dbb... so that, in some weird conceptual world I'm actually not where I started. We're used to making that enharmonic shift, but it was relatively unfamiliar at the time. Partially that had to do with tuning, but also it had to do with the fundamental role of the diatonic scale at the time. Every interval had a meaning within a major or minor scale, and there were some combinations of intervals (like 3 M3's in a row) that couldn't be accomplished in any single scale. So shoving them all together like that, and forcing enharmonic equivalence on you, came very close to being a moment of atonality within tonal music!

This, again, is why the Neo-Riemannian approach of ignoring tonality and diatonic scales is useful: because there are pieces that do just that, in order to combine triads in weird ways (like the P-L sequence) that require enharmonic equivalence to make sense.

VI. The Tonnetz

In order to visualize the universe of possibilities that we've opened up with all this theorizing, Neo-Riemannian theory likes to create visual maps of the chord layouts that are possible. This kind of map is called a Tonnetz (German for "tone network"). Here's an example of a Tonnetz. Each letter represents a note (not a chord). Horizontal lines connect notes by perfect 5ths; diagonals that go up-right (or down-left) connect minor thirds; diagonals that go up-left (or down-right) connect major thirds.

The triangles that are formed in this picture represent triads: triangles pointing up are minor triads and triangles pointing down are major triads. So you can see the triangle framed by C, Eb, and G bolded in the picture, which of course is a C minor triad. Below it is the C,E,G of C major.

The nice thing about a Tonnetz like this is that it can also show our transformations. Consider the C major triad (just below the bolded triangle). Now look for the triangles that share a side with C major: they turn out to be exactly the 3 triangles that I can transform C major into via P, L, or R. So we can imagine those transformations as ways of flipping one triangle onto another inside the Tonnetz; we can make analyses of pieces by tracing out their chord progressions as if on a map.

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That's pretty much all I've got stamina for, tonight. I've left a bunch out, so I'd be happy to get corrections/additions (or questions!), but I hope this has been a plausible overview of the basics of Neo-Riemannian theory.

If this stuff piques your interest, here are two books that are very much worth taking a look at:

Audacious Euphony by Richard Cohn, who is one of the founders of the theory, and who explores its possibilities through many nice analyses in this book.

A Geometry of Music by Dmitri Tymoczko, who is critical of standard Neo-Riemannian theory in many ways. His book (which builds two articles he helped write for Science in, I think, 2006 and 2008) offers another perspective on some of the same issues, drawing on geometry rather than algebra for his underlying mathematics.

There 's [a beautiful book out there by Dmitri Tymoczko] (https://www.amazon.com/Geometry-Music-Counterpoint-Extended-Practice/dp/0195336674) about this

You're very welcome. I found the chart online via scaletrainer as I mentioned. I've also come across a reference to this printed book that sounds great in terms of a visual approach to chord theory using geometry.

Sure thing! This is going to be a bit of a doozy length wise because there's some background I should give first. You'll find some pictures in this link that will help you visualize some of the stuff I'm talking about here.

So let's start with the three basic transformations of Neo-Riemannian theory. We can use these transformations to turn some triads into others. A parallel transformation (P) will preserve a chord's root and fifth while swapping the quality of the third. So P applied to a C major triad will create a C minor triad and applied to an E minor triad will create an E major triad.

Then there's the leading tone transformation (L). When L is applied to a major chord, it moves the root down by half step to the leading tone (e.g. C major becomes E minor) and when applied to a minor chord, it moves the chordal fifth up a half step as if that chordal fifth was a leading tone (e.g. Bb minor becomes Gb major).

Finally, there's the relative transformation (R) which moves the chordal fifth of a major chord up by whole step (e.g. C major becomes A minor) or the root of a minor chord down by whole step (e.g. E minor becomes G major). This transformation relates relative major and minor keys.

Now notice, that the transformations all have something in common; no matter what triad we apply them to, we get a triad of opposing modality. If we use them on a major chord, we know a minor chord is the result and vice versa. Also notice that each transformation requires very minimal voice leading; the "biggest" transformation moves only one triad member by a whole step.

We can actually map all twelve chromatic pitches so that any equilateral triangle formed by immediately adjacent pitches is a triad. When we do this, we can arrange our map such that any two triangles that share an edge are related by one of our three transformations. Look at the first image here to see what I'm talking about. Technically, however, this pitch space is better thought of as a torus (Image 2) but I'm not trying to go too left field here.

Alternatively, we can just map all major and minor triads such that any two adjacent triads are related by one of our three transformations. Doing so gives us this hexagonal, chicken-wire fence shape that charts paths between chords via our transformations (Image 3).

Either/both of these representations help us visualize musical geometry, tonal relations, and voice leading in a very clear way. Before going on though, I should say that other maps are possible. For example, Allen Forte, in his Structure of Atonal Music, creates a neat map for trichordal set space. And tangentially, Klumpenhower and his networks operate like spiritual siblings to the same idea. But let's just worry about triadic and tonal spaces for now.

We can play around with these transformations and spaces a bit to see if we can't create a cycle. Cycles are any pattern of repeated transformations that (eventually) start and end with the same chord. Let's see what I mean. For this, you'll probably want to follow along on either the Neo-Riemannian pitch space or triad space maps.

Start with an E minor chord. Apply the P operation and get an E major chord. Apply the R transformation to get a C# minor chord. Apply P again, C# major. Apply R, Bb minor. Apply P, Bb major. Apply R, G minor. Apply P, G major. Apply, R, E minor.

So by just chaining P and R transformations back to back, we've managed to wind up back where we started. Hey, wait a minute! All of the pitches of these eight chords fit neatly into an octatonic scale!!! Because of this, we can call this P-R cycle an octatonic cycle because this chain of transformations produces an octatonic collection. You can see this more clearly in Images 4-6.

We can do the same thing to create the hexatonic collection, just by using a different set of operations. If we instead chain P and L operations together applied to any triad, we'll end up with a hexatonic cycle because again, we'll end on the same chord where we began. I'll leave it to you to map out all the changes for yourself but check out Images 7-9 to see what I mean.

I'm naturally skipping over a lot of juicy stuff in this discussion but I hope it at least sheds light on the basics of what I mean when I say crazy sh%t like "hexatonic cycles." There's this really nifty youtube video here that does a nice job of introducing plenty of the same concepts; please watch it! One of our Eternally Luminous Theory Monarchs has collected some resources for Neo-Riemannian theory that you can check out here and here.

There's also tons of lovely books and articles on the topic. Here are links to three; I would start with the Mason because it's designed to be a beginner's textbook in the field.

Cohn's Audacious Euphony

Tymoczko's A Geometry of Music

Mason's Essential Neo-Riemannian Theory for Today's Musician

Finally, there are some sweet videos on youtube that model chord progressions from real music on the tonnetz as the music plays. It shows just how audible this stuff is and it's also just cool to look at and listen to.

Adams: https://youtu.be/edyM_iH0jJc

Satie: https://youtu.be/nidHgLA2UB0

Chopin: https://youtu.be/c-HDDiWWWTU

I hope this helps and take care!

Do you know this book? https://www.amazon.com/Geometry-Music-Counterpoint-Extended-Practice/dp/0195336674

It is intended for musicians and uses topology to descrive music theory.

You could try Dmitri Tymoczko's book: https://www.amazon.com/Geometry-Music-Counterpoint-Extended-Practice/dp/0195336674/ref=sr_1_1?crid=1Y38JQAS2RAO8&keywords=tymoczko&qid=1574569471&sprefix=Tymoc%2Caps%2C147&sr=8-1

There are entire fields of study in this area. I've done a fair bit of work looking at harmonic theory, where the main focus point is in coming up with mathematical abstractions, i.e. structures, that capture various things we care about in harmonic theory.

For example, the set of integers is a mathematical structure, and the traditional thing we get taught is to put scale notes in correspondence with ordered integers. All this really does is capture our intuition that notes come in a particular order (low to high).

In practice, we don't just play notes in order of low to high, instead our melodies tend to jump around between notes, tending to prefer certain intervals. So a more elaborate example would be to use a graph structure that connects each note to other notes that are fundamentally related, by an octave, or a fifth, or a third, and so on.

Yet another example would be to connect chords to other chords that differ by only a single changed semi-tone. In this case, the act of moving a note to form a new chord could be described as a group operation. In fact, most mathematical approaches to music tend to rely on group theory, and other areas of abstract algebra.

Structures like these definitely can be used as tools for composition, or even can be used to build programmed composers. The core idea is to formalise our discovered or intuited knowledge of what makes good music sound good.

See:

• The Haskell School of Music - http://haskell.cs.yale.edu/euterpea/haskell-school-of-music/
  
  
• The Geometry of Music by Dmitri Tymoczko -http://www.amazon.com/Geometry-Music-Counterpoint-Extended-Practice/dp/0195336674

bad advice so far imo. You shouldn't try to learn something by randomly messing about until you eventual 'learn' it. Learn theory by reading books written on theory. Start with the basic conceptual stuff like what melody and harmony is and why it works the way it does. Learn your ABCs: major and minor scales, modes. How to build chords, Scale degrees and intervals. the cycle of fifths. The consonant < > Dissonant spectrum and how it relates to melody and harmony e.t.c.

THEN you can 'mess about', but in a structured way and explore the stuff you're learning as you learn it. Simply knowing scales is the equivalent of being able to say "hello" "yes" "no" "my name is" e.t.c. You've really got to get into the underlying relationships of intervals and harmony to begin getting a grasp of how to apply meaning (emotion/rhetoric/feeling) to your music.

the books by Michael Hewitt are a decent start as they apply this stuff in a computer music context. http://www.amazon.co.uk/Theory-Computer-Musicians-Michael-Hewitt/dp/1598635034

later down the line you can get into more complicated stuff like diatonic harmony, classical form, post tonal theory e.t.c.

http://www.amazon.co.uk/Classical-Form-Functions-Instrumental-Beethoven/dp/019514399X

http://www.amazon.com/gp/product/0195336674/ref=as_li_ss_tl?ie=UTF8&tag=masschairevio-20&linkCode=as2&camp=217145&creative=399369&creativeASIN=0195336674

It all depends on how far you want to go with it and ultimately how much control and scope you want to have. A lot of EDM producers are relatively theoretically mute. But it doesn't stop them from making decent music within the practice/genre they're versed in (but that's a different conversation a little outside the scope of your question ;) )

Also, study your favorite tracks, use what knowledge you have to deconstruct music you like, copy the chord progressions, arrangements, mimic timbre, vibe and theme e.t.c. Get familiar with the nuts and bolts of what makes the music you like sound so good to you, and then apply that general orientation in a creative manner to your own workflow.

Hope this helped!

I think theory as a whole has reached a very comfortable spot. Sure, we might still not have a tuning with perfect ratios of its harmonics on the octave, perfect fifth, mayor third, etc etc. But humanity knew how to adapt to what was already available and theory has gone beyond music to blend itself with non-functional sounds very useful for movies, video games or theater.

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I think the guinea pigs are the people themselves: we collectively decide what we like and the people who write for the big names take note.

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With that said there's a lot of experimentation with microtonality in both music (king gizzard & jacob collier are the first to come to my mind) and we have books that look to implement math into theory and expand whats possible:

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a geometry of music: a study in counterpoint: https://www.amazon.com/gp/product/0195336674/ref=ox_sc_act_title_1?smid=ATVPDKIKX0DER&psc=1

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The geometry of rhythm

https://www.amazon.com/Geometry-Musical-Rhythm-Godfried-Toussaint/dp/1466512024/ref=sr_1_1?__mk_es_US=%C3%85M%C3%85%C5%BD%C3%95%C3%91&keywords=geometry+of+rhythm&qid=1563542715&s=books&sr=1-1

Thanks for the reference. Although his ideas are interesting, from the amazon page it seems he is just presenting a theory he came up with. There doesn't seem to be any mention of experiments or testing his ideas.

I haven't read it so I can't really speak to it's contents but A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice seems like it would fit the mathematics + music theory bit nicely.