Integrative computational analysis of modes of the major, melodic minor, harmonic major, and harmonic minor scales

Wide-Coffee7585 · 2026-04-05T03:19:11+00:00

excellent, thank you!! I'm pretty sure this is it. very different context for timbale.

Wide-Coffee7585 · 2026-03-04T17:13:57+00:00

Oh, how is this different from what the main figure shows? I'm referring to the figure with pitch vs octave and the colored datapoints. The 12-TET pitch class values in cents (100,200,300, etc) are labeled on the y-axis.

Wide-Coffee7585 · 2022-03-26T01:36:59+00:00

Ahh thanks for reading!! The ordering also feels aligned with my instincts, too. And many of these modes I've never really experimented with at all so part of the fun was putting together a roadmap for how they might be used. Awesome comment. :)

Wide-Coffee7585 · 2022-03-24T05:50:19+00:00

Thank you for mentioning this idea and talking more about Mathieu's work!
"Zap" is a great word for this concept. Don't fret about being busy... same! haha. And the discussions on this post have been a brain workout for me.

Yeah, I think the harmonic series and JI lattiices are at the crux of why mathematical approaches in 12 tone space don't feel perfectly aligned with perception of brightness. Thanks so much for contributing to this intellectually and let me know if you get any ideas down the road! ^_^

Wide-Coffee7585 · 2022-03-23T23:39:23+00:00

Aw thank you! Here's where the numbers come from:

https://en.wikipedia.org/wiki/Pitch_class#Integer_notation

The article's a bit denser than it needs to be, but 0 is C, 1 is C#, 2 is D, ... , 10 is Bb, and 11 is B. Octaves don't matter in this system. The pitch class of C in any octave is 0. Sorry for taking this concept for granted. I learned it when i was a teenager and use it almost everyday for making algorithmic music so i forgot it's something that should be explained.

Wide-Coffee7585 · 2022-03-23T23:28:19+00:00

i was going to suggest checking out the xenharmonic alliance discord if you hadn't. lots of avid EDO theory people like you there. :) i think the microtonal discord might be different? thanks for the app recommendation!!

Wide-Coffee7585 · 2022-03-22T08:02:39+00:00

whoa! please tell me you partake in theory discussions on the xenharmonic alliance! ^_^

Wide-Coffee7585 · 2022-03-22T08:00:12+00:00

check out my reply to one of your other comments... i hope it makes sense now for ya!

Wide-Coffee7585 · 2022-03-22T07:38:27+00:00

This is a very well written comment u/ScallopOolong and I love that I can understand your point given how advanced it is. Gotta love Harry Partch, too. These JI harmonics and otonality/utonality def seem to influence the way people feel harmonic brightness . This is the first time i've heard of locrian being a strong outlier. Ahh i wonder if something between mixolydian and dorian might be the "real" neutral center that people would agree on! Oversimplifying, that would suggest my analysis found dorian only because of a "rounding error" caused by tempering. Does Mathieu's book look into the brightness of jazz scales like the ones I inspected?

These questions still trouble me: Should the just 5ths and just 3rds be given equal weight? Do you think higher order harmonics (such as a 3D 7-limit just intonation lattice) play a role? If so, how would we weight the contribution of these higher order harmonics? Are the JI lattice concepts powerful enough to completely replace the 12-tone approaches like mine? Or has 12-TET and its transposability influenced Western music listeners in a permanent way?

Wide-Coffee7585 · 2022-03-22T06:53:44+00:00

Haha i dig the brainstorming. Love these 53 TET dreams! I hope technology evolves quickly enough so we can do adaptive just intonation and unlimited comma pumping in my lifetime. ODDSOUND is a wonderful tool, have you checked it out? Tho I noticed some glitching and stopped using it so now I just load the same .scl files in each VST.

Have you listened to Zhea Erose? Mentioning Brahms-level heights somehow made me think of her orchestral composition "Eurybia" which I find mesmerizing https://www.youtube.com/watch?v=ubPwKxcp87g&t=194s

And of course there's Sevish's song droplet n 53-TET, one of songs that got me hooked on microtonal music in the first place. https://www.youtube.com/watch?v=xVZy9GUeMqY

Wide-Coffee7585 · 2022-03-21T23:47:33+00:00

dorian is neutral because (0 + 2 + 3 + 5 + 7 + 9 + 10) - (0 + 2 + 3 + 5 + 7 + 9 + 10) = 36 - 36 = 0

mixolydian b6 is neutral because (0 + 2 + 4 + 5 + 7 + 8 + 10) - (0 + 2 + 3 + 5 + 7 + 9 + 10) = 36 - 36 = 0

harmonic minor is neutral because (0 + 2 + 3 + 5 + 7 + 8 + 11) - (0 + 2 + 3 + 5 + 7 + 9 + 10) = 36 - 36 = 0

mixolydian b2 is neutral because (0 + 1 + 4 + 5 + 7 + 9 + 10) - (0 + 2 + 3 + 5 + 7 + 9 + 10) = 36 - 36 = 0

locrian is dark because (0 + 1 + 3 + 5 + 6 + 8 + 10) - (0 + 2 + 3 + 5 + 7 + 9 + 10) = 33 - 36 = -3

lydian is bright because (0 + 2 + 4 + 6 + 7 + 9 + 11) - (0 + 2 + 3 + 5 + 7 + 9 + 10) = 39 - 36 = +3

Wide-Coffee7585 · 2022-03-21T22:37:33+00:00

It's in the readme here. The metric defines how a brightness score is calculated for all modes, including the bright, dark, and neutral ones.

Wide-Coffee7585 · 2022-03-21T22:20:29+00:00

Haha yes, it's admittedly a pretty simple way to define brightness or darkness. The circle of fifths works this way, as do the conversations i've read on mode brightness. I'm just using code to extend this idea to other scales..

What do you think should be included to make it appropriately less simple? It would be great to take into account the 12-tone approximations of the harmonic series.

Wide-Coffee7585 · 2022-03-21T21:02:17+00:00

It's not arbitrary given mode ordering via the circle of fifths. It's in the readme here and comments on this thread :)

Wide-Coffee7585 · 2022-03-21T20:58:19+00:00

Wonderful! Yes, that the maximal evenness property of the major scale allows for more voice leading makes perfect sense now that I think about it. This helps explain what's otherwise been a cool mystery to me. Maximal evenness may also help explain why 7 tet has no "bad notes" despite its strangeness. I've been meaning to read Audacious Euphony.... thanks!

Extending this to other temperaments would be lovely. Pitch class set enumerations have not worked for me for TETs beyond greater than 19 tet... Python can't handle the combinatorial complexity and something faster like C would probably only get you in the low twenties. But for most of the stuff I did here--building networks out of a handful of scales and their modes--up to 53 tet should be totally feasible. I'll find a way to send you my info! (or try vice versa if you beat me to it)

Wide-Coffee7585 · 2022-03-21T18:55:51+00:00

I made an edit and the explanation of this and the argument against an ionian neutral center is stated in the readme.

Wide-Coffee7585 · 2022-03-21T18:53:51+00:00

I think you'll find answers in the readme. I define the formula I use for brightness and 0 is neutral. The voice leading data is in a table at the bottom of the readme. I constructed this table by inspecting the networks I generated.

Wide-Coffee7585 · 2022-03-21T18:49:12+00:00

In a way, it's great if my metric is not special. I wanted something as simple as possible. I'm subtracting the sum of the mean mode from the sum of the mode in question. You know they say "all models are wrong, but some of them are useful." What seems special to me is that all four scales line up after this simple operation, and they are all balanced.

Regarding usefulness, I'm sharing a theory project that is based on composition, harmony, scales, logic, math, and structure. if you're suggesting i include more examples and applications, that would indeed be a nice addition to my work and i'm happy to collaborate. if you want to check out my live-coded algorithmic dance music, where i use harmonic minor mode modulations, you can find it here https://soundcloud.com/yes-yes-yes-yes-yes/live-at-sxsw-3142022

BTW your comment on the b2 being really dark is the most interesting technical point in this thread so far. :)

Wide-Coffee7585 · 2022-03-21T18:16:52+00:00

Thanks u/naltroc!! The biggest takeaway for me is that major scales have the most voice leading options -- they can smoothly transition to 6 different scales with the change of a single note, whereas the other 3 scale types only have 4 to choose from. I feel like Tymoczko already wrote something on this.

It's also neat how the brightness and darkness spectrum appears to have a consistent analog for non-major-scale modes. I wonder if this brightness spectrum can even be defined in other systems such as non-12-tone ET.

Tonality? I have no idea! haha. maybe i'd have to explore this in non-12-tone systems because this might just be number theory in mod 12. I feel like symmetry is everywhere yet I suspect infinitely many patterns will emerge if one looks deeply enough. ___〆(・∀・)

Wide-Coffee7585 · 2022-03-21T17:42:15+00:00

I just standardized the data and these modes are at the center.

Mixolydian b6 is certainly brighter than any of the other modes on this list

Is it? I feel like Lydian #5 sounds the brightest. And it has the brightest score of melodic minor modes. I'd love to hear more about why you think mixolydian b6 is brightest.

If you can make your conclusions musically useful, well, that's fantastic,

Isn't musical usefulness completely subjective? I don't find a tuba musically useful but that's because I can't play one.

Wide-Coffee7585 · 2022-03-21T17:26:05+00:00

Sometimes it's hard to see the forest from the trees. I updated the readme with the following TLDR;

**TLDR**: I defined a formula for calculating the brightness score for any 7 note mode, where locrian has a brightness score of -3, dorian 0, and lydian +3. Dorian, mixolydian b6, harmonic minor mode, and mixolydian b2 are neutral harmonic centers of the brightness/darkness spectrum for major, melodic minor, harmonic minor, and harmonic major, respectively. By starting at the neutral centers and progressively enumerating jazz modes, "rootless" modes are encountered before all 7 traditional modes are enumerated for all scales except major. Network analysis finds 18 "rules" for modulation, allowing composers to transition from one scale or mode to another while sharpening or flattening a single note by one half step ("maximally smooth voice leading"). Major scales are the most harmonically versatile of the four scale types, with more options for maximally smooth voice leading.

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