Yahoo Tuning Groups Ultimate Backup makemicromusic Mad Calliopist

Jim,

{you wrote...}
>Thanks for fixing my URL - sorry for my sloppiness!

It happens.

>And thanks for daring to subject your ears to the experiment!

I know where the 'stop' button is.

>I'm aware of course that this attempt at music-making is very limited. >There is very little rhythmic variety. The waveform generation is very >basic, essentially boring sine waves.

True enough. Please take any comments in stride...

>Mostly what I am trying to do is to explore some of the harmonic or >melodic possibilities in 53 note equal temperament, to show that music can >be made that sounds pretty much like regular 12 note equal temperament

Um, OK, but many people want to use new tunings so they *don't* sound "pretty much like regular 12 note equal temperament"!

>... but then the details don't quite work out the same way, the actual >musical structure that works with 53 is a different beast after all.

Aha, and it is the *structure* that is missing from your experiment.

One of the difficulties in bridging a gap between explorations like yours and out-and-out compositions is where do you start? You've prefaced your entry to the group by explaining you have no musical training, so I'm hesitant to try to give much in the way of direction, as I am imagining a lot would be over your head.

However, when one programs tones like these in what ends up being a fairly random (or completely random) manner, it doesn't take very long for the listener to think "there is nothing coming up I'm interested in listening to".

I've never taken a poll, but I also think that people that would like to just sit and listen to a tuning, and its various combinations, are in a distinct minority compared to people who would like to listen to a tuning in a musical context. Me, for instance.

One thing I might suggest would be to use the freeware program Scala to experiment with 53 (or other tunings) - you can play the scales and intervals, and try out chords as well. The other thing that might really be fun for you would be to work with Robert Walker's Fractal Tune Smithy, where you can set a tuning and then use varying parameters to generate 'fractal' music. The bonus is that the program is the development of a person who is not only and scientist and programmer, but also a musician, and while it is algorhythmic in nature it *does* generate material that is much more musical than pure random settings.

Both Scala and FTS work hand in hand together, and may open up many other tuning 'doors' to you. Links (I believe they are still current) can be found at:

http://www.microtonal.org/resources.html

>One thing that I would really love is to find out about other work that is >headed in a similar direction.

Did you check out a posting about a piece in 19tet that came up about a week or so ago, "the Juggler"?:

http://www.aaronandlorna.com/audio/juggler.mp3

While you are at it, check out a couple of other xentonal works on the same site from this page:
http://www.aaronandlorna.com/aaron_works.html

Gene Smith has a whole bunch of stuff at his site:
http://www.xenharmony.org/

...but he doesn't show how large the files are, and some are quite large. Also, maybe Joe Pehrson can chime in with a link to his files.

>I've heard Easley Blackwood's music in 13 through 24 note equal temperament.

Sorry, not a fan. In fact, I'm not really an ET kind of person, but others around here are.

>I'm trying to do something similar, except of course this is algorithmic >music where his is composed.

Bear in mind that even algorithmic compositions vary from total randomness to controlled freedom; you'll probably find that for many listeners - and probably yourself, after time - you tire of complete randomness. There is a lot of literature, and a fair amount of widgets, available for algo comp stuff. I'm not the most up-to-date on it, but slowly there is a synergy towards (at least on a computer platform) getting all this to work together.

>And the 53 note scale hits the basic conventional ratios 3/2, 4/3, 5/4, >5/3, etc very nicely, so the starting point is far less strange.

Oh, Jim, you're not going to tell me you're afraid of "strange" things, are you?

Cheers,
Jon

🔗Joseph Pehrson <jpehrson@...>

🔗kukulaj <kukulaj@...>

Thanks for the comment and pointers, Jon. You're right that almost all
in-depth commentary would be over my head! I've looked at a few music
theory books, and understand essentially zero!

Randomness versus structure is a curious business. I was rather
worried that the piece ends up with too MUCH structure! The way I
generate the tones is to set up a lattice - in this case I think it's
a four dimensional torus. The sequence of sounds played is then
generated by a sort of helical walk that covers the torus. At each
point on the toroidal mesh is a sound. There's a cost function that
wants the sound at a point to be harmonically close to the sounds at
the neighboring points, of which there are eight on the four
dimensional torus. I simulate a Bolzmann distribution on this system,
starting at a high temperature and gradually reducing the temperature.
This sort of system is pretty much guaranteed to have a second order
phase transition at some temperature. Right at that temperature the
fluctuations will have a fractal character. I didn't implement any
fancy mechanisms to detect the right temperature. But if the
temperature goes any lower, the crystal freezes, i.e. the sound
becomes just a steady unvarying single sound. So I just listen in at
various temperatures, and just at the border between under-correlation
and over-correlation, that's where I stop. But that's my ear judging
the boundary - not a terribly reliable judge!

I'm imposing structure in several ways: I define the shape of the
torus and the path of the walk that generates the sound sequence; I
define the harmonic cost function; and I pick the sequence of
temperatures through which the system evolves. It's funny. When I
listen to the piece, all this structure is very apparent to me. I
rather worry that so much structure will be too boring. Yet here you
are, telling me - if I am getting the message straight - that for all
you can tell, the sound sequence sounds pretty much purely random!
That's a big surprise for me! For me, such a suprise is the mark of a
successful experiment. It means I can learn something! So thanks for
the education!

Here's one feature of this piece: if you repeat it, the last note
leads right back to the first. The piece is a loop. There is really no
variation at all that I imposed, i.e. the space is totally
homogeneous. I can imagine that the piece really needs a story line to
create a focus of interest. Most music sort of builds up some tension
and then resolves it, i.e. has a beginning and an end. My piece, being
homogeneous and circular, has no beginning or end.

One way I could create a story line in the context of my current
approach would be to vary the cost function from point to point in the
mesh. From a software perspective it's easy to do in whatever
arbitrary way. The challenge is to do it in a way that makes sense
musically! No doubt all those books that I don't understand would give
me some clues! On the other hand, they probably leave out the Bolzmann
distributions! Well, I hope all of my explanation here in
physics/computer terms was not so un-musical to be overly bothersome!

Thanks again,
Jim

🔗Jonathan M. Szanto <JSZANTO@...>

Jim,

I hope you can sense that the spirit in which I am share this is one of assistance and not discouragement. I can't emphasize this enough! And, with that said, I now go to:

{you wrote...}
>Thanks for the comment and pointers, Jon. You're right that almost all >in-depth commentary would be over my head! I've looked at a few music >theory books, and understand essentially zero!

OK. And to make you feel fairly dang comfortable, if we were to switch places and you put math books in front of me, the effect would be quite similar (though I made it as far as first year calculus, so I guess that's more than your music education.

>Randomness versus structure is a curious business. I was rather worried >that the piece ends up with too MUCH structure! The way I generate the >tones is to set up a lattice ...

And at this point I'm going to skip most of your description. I should try to go back and listen, which maybe I'll do as background to other tasks this evening, but even without that, what does it say (again) to you that I thought it was at least somewhat random?

For many musicians, one of the failings of contemporary classical music of the avant-garde variety (which really flourished from the late 40's and has only lost some of it's grip as a progressive voice in the last couple of decades) is primarily that: all of the structures that were being place on music, in an effort to 'liberate' it's emphasis on tonality and common rhythmic structures, fail to yield any aurally detectable results; they can only be discerned upon analyzing the score.

This isn't the forum to debate it, but it is entirely up to you whether you want the structure you have imposed on your creation to be heard.

>It's funny. When I listen to the piece, all this structure is very >apparent to me.

I don't think it is funny, I think it is pretty remarkable. Have you, by any chance, taken a random sampling of friends (your enemies wouldn't listen to this willingly, would they? :) who *aren't* in your field of expertise and had them listen, and found that the majority could *hear* the structure? I have my doubts.

>I rather worry that so much structure will be too boring. Yet here you >are, telling me - if I am getting the message straight - that for all you >can tell, the sound sequence sounds pretty much purely random!

I don't know, Jim, what can we infer? I've been actively playing, composing, and listening to music for 40 of my 50 years - I simply think this means that musical structures and mathematical/physical structures are not completely analogous.

>It means I can learn something! So thanks for the education!

I love it when my completely missing something helps others... :)

>Here's one feature of this piece: if you repeat it, the last note leads >right back to the first. The piece is a loop.

OK, I gotta stop you: if you assume that after eleven minutes no one is going to immediately start the piece over again, and taking into account a complete lack of timbral variation, no purposeful dynamic (loudness) variation, and what 'appears' to be a fairly random rhythmic structure - did you think I could notice that it ended on the same one of 53 discrete pitches that it started on? Not this puppy!

>My piece, being homogeneous and circular, has no beginning or end.

I should note that I very much like circular and related structures, and have used that very format myself, from very small musical works to one long song cycle that in seven movements covers the events of a week-long cycle, and completely dove-tails back to the beginning. You might also want to take a listen to (and study of) important musical pieces that are similar - there are movements in some of Bartok's string quartets which fit this bill quite nicely.

Damn, I'm/we're wandering from microtonal music, and I try to keep focussed there. Rats.

>One way I could create a story line in the context of my current approach >would be to vary the cost function from point to point in the mesh.

Jim, it is only my perspective, but I think the piece is way too non-obvious for this approach. Most ears will simply not discern a structure unless it is evinced in much more that the pitch structure.

>Well, I hope all of my explanation here in physics/computer terms was not >so un-musical to be overly bothersome!

How would you know if you didn't attempt that? But I *do* have a suggestion for you: this list, along with another, sort of splintered off from the original tuning list that began back in the 90's. I started this one to focus directly on the act of making microtonal *music*; there is another group that focuses specifically on the mathematical nature of tuning. Not only would your perspective be of interest there, but the cross-section of your audience would be enormously more suited to understanding the structural points you are working with. I think you should take a look at:

/tuning-math/

Paul, Gene, Dave and the others over there will definitely know more about this stuff than I. If you want to change course a bit and have your tuning framed in a somewhat more conventional 'musical' template (though I hate like hell to align myself with convention!), feel free to either post here or write me off-list.

BTW, if you go over there, ask about Dave Keenan's very cool 3 or 4 dimensional tuning construct, as built within an Excel spreadsheet and done with 3D glasses, allowing you to fly around a tuning (if I remember even somewhat correctly).

Wow, I'm tired now.

Cheers,
Jon

🔗Paul Erlich <perlich@...>

--- In MakeMicroMusic@yahoogroups.com, "kukulaj" <kukulaj@y...> wrote:
> Thanks for the comment and pointers, Jon. You're right that almost
all
> in-depth commentary would be over my head! I've looked at a few
music
> theory books, and understand essentially zero!
>
> Randomness versus structure is a curious business. I was rather
> worried that the piece ends up with too MUCH structure! The way I
> generate the tones is to set up a lattice - in this case I think
it's
> a four dimensional torus. The sequence of sounds played is then
> generated by a sort of helical walk that covers the torus. At each
> point on the toroidal mesh is a sound. There's a cost function that
> wants the sound at a point to be harmonically close to the sounds at
> the neighboring points, of which there are eight on the four
> dimensional torus.

what ratios would these eight points approximate relative to the
starting point?

one problem i see with this is that the ear has no way to judge, and
thus no reason to prefer, successive sine waves in simple ratios as
opposed to any old intervals. instead, it's largely for non-sine
repeating waveforms (i.e., timbres with harmonics), sounding
*simultaneously* rather than *successively*, that simple ratios have
a psychoacoustical "attraction".

honestly, i couldn't even tell that you were preferring simple ratios
for the melodic intervals, despite my normally very good ear!

> I'm imposing structure in several ways: I define the shape of the
> torus and the path of the walk that generates the sound sequence; I
> define the harmonic cost function; and I pick the sequence of
> temperatures through which the system evolves. It's funny. When I
> listen to the piece, all this structure is very apparent to me. I
> rather worry that so much structure will be too boring. Yet here you
> are, telling me - if I am getting the message straight - that for
all
> you can tell, the sound sequence sounds pretty much purely random!
> That's a big surprise for me! For me, such a suprise is the mark of
a
> successful experiment. It means I can learn something! So thanks for
> the education!

i don't hear any structure either, in that it's impossible to tell
the difference between any section of the piece and any other. where
are we? at the beginning? at the end? it's all one big amorphous swim
around the torus.

> Here's one feature of this piece: if you repeat it, the last note
> leads right back to the first. The piece is a loop. There is really
no
> variation at all that I imposed, i.e. the space is totally
> homogeneous. I can imagine that the piece really needs a story line
to
> create a focus of interest. Most music sort of builds up some
tension
> and then resolves it, i.e. has a beginning and an end. My piece,
being
> homogeneous and circular, has no beginning or end.

well then, ignore my last remark, since you already knew that!

> On the other hand, they probably leave out the Bolzmann
> distributions!

you might want to read _formalized music_ by iannis xenakis, probably
the only music textbook that *does* incorporate boltzmann
distributions!

> Well, I hope all of my explanation here in
> physics/computer terms was not so un-musical to be overly
>bothersome!

hi jim, i remember your name from a while back, i have a degree in
physics too . . . . it might be better to direct future discussions
on the more math/physics/theory aspects of this to the tuning list or
especially the tuning-math list (for example, please tell me more
about your "helical walk" and how it differs from what i thought it
was, a "random walk"), while continuing to post sound examples and
work out issues such as *composition* here on this list.

originally there was only the tuning list (which is probably where i
remember you from), but the diversity of mindsets there
caused "splinter groups" such as this list, the tuning-math list, and
others to appear. some lament this and refuse the added effort of
keeping up with the various lists; i think most are happy and keep up
with whatever lists interest us. just thought i'd fill you in since i
think you were away while all this happened.

🔗Rick McGowan <rick@...>

"kukulaj" wrote...

> One thing that I would really love is to find out about other work
> that is headed in a similar direction.

What do you mean exactly? Exploring other tunings? There's plenty of it.
Exploring algorithmic melody in non-12 equal tunings? Something else?

> I've heard Easley Blackwood's music in 13 through 24 note
> equal temperament.

Just FYI, we generally distinguish temperament from tuning. So these are
13-24 equal tunings. They're not properly temperaments of anything; they
just consist of a number of equal sized intervals per octave.

I've also written at least one non-trivial piece of music in all the equal
tunings from 7 through 19. Some (such as 15 and 17) I've used extensively.
Of course, some tunings are more interesting/different/useful than others.
It all depends on what you're trying to do.

> ... Most music sort of builds up some tension
> and then resolves it, i.e. has a beginning and an end. My piece, being
> homogeneous and circular, has no beginning or end.

It's "wallpaper music". Eric Satie started doing this (well, high grade
wallpaper) in the 1920s, and others have followed. Probably any serious
composer born after 1910 has done one or more experiments in this vein.
Some years ago I even did a few of these on the Commodore 64 (with
animations of a psychedelic buzzing fly). There are definite uses for such
music, mostly as an environmental element of one type or another. Or for
meditative effect. But it doesn't tend to fascinate the intellect on its
own.

Rick

🔗Robert Walker <robertwalker@...>

Hi Jim,

Just a few ideas - how about trying repeating sines
to get more variety than a pure
sine wave, i.e. sin(sin(sin(x))) say,
gives a kind of rounded square wave.

Triangle waves are also very rich in harmonics,
also sawtooth, but need to be rounded
to get away from the harsh sounding sharp edges.
Rich harmonics are good for just intonation chords.

As for structuring, music usually comes in sections
with cadences that act like the punctuation in a
sentence, and if you have no such device to break
it up then the feeling is like a story that
is written all one sentence with no paragraphs
or sentences or commas - the listener gets
tired easily because there are no cues to say
when you can kind of pause a bit and gather
momentum to start up again.

if you want a cadence that a musician will instantly
recognise as such, then you can try motion
by fifths. Even - playing a section then
playing it again transposed up a fifth,
then down again will give quite a feeling of
structure to it even if random.

More commonly one would use triads
and alternate up / down by fifths for
a while, i.e.

1/1 4/3 3/2 15/8 9/4
to
1/1 5.4 3/2

using any octave positions for the notes.

That establishes the tonic as the key of the
piece, then you follow with a longer
cadence for the "full stop" as it were
at the end of the sentence, which
can be the same but descend maybe by
four or five fifths or more.

To make it a more decisive cadence, then you
often add an extra note to the triad to make
a dominant seventh - to any of the triads except
the tonic. Use e.g. the 8/9 below the root of the
triad, or else the 7/8 or 9/10. The 8/9 sounds most
decisive:
1/1 4/3 3/2 15/8 9/4
to
1/1 5.4 3/2

while the 7/8 is a more restful harmonic series chord
and is suitable if you want it more leisurely
and less decisive (well this is just personal
view on it expect opinions will differ).

Melody lines can follow notes of the
triads, though often they contain
scale passages with many "passing tones" - notes
usually on off beats that don't belong to the
triad of the harmony.

E.g. if you just play an ascending scale
c d e f g
with c e and g on the beat then that by itself
will already give a feeling of the tonic based triad
even if you have no accompaniment.

This is only one of many systems of harmonizaation
now, or from the past, and you can invent your
own ones too, e.g. as a result of exploring a tuning.

But really doesn't have to be harmony, and you don't
have to use that partiular harmony system,
any kind of epsiodes can help structure it.

A change of timbre, or of speed can help.
Or a variation in the volume of the tune so that
you have a louder phrase, then maybe repeat it
exctly again, maybe quieter, maybe transposed
down a bit or something, or whatever.
Long slow increases and decreases of volume or
speed over a matter of several seconds or minutes
can be very effective too.

Rhythm also helps to give it a structure, which means
to have a repeating pattern that can get broken up
in various ways but is always the same length - the
bar. Usually divided into a number of smaller sections
- the beats, e.g. three or four beats or maybe six or eight.

Those usually then get divided into halfs or quarters or sometimes
thirds. More rarely you get structures with five beats or seven,
sometimes more.

Not to be afraid of repetition there. A repeating
rhythm, with a varying pitch can be really
effective.

...............

Maybe some of that info could be useful to help add structure
to the piece that a musician will find easy to recognize.

Even perhaps episodic, some of it as it is, with an occasional incursion
of a recognisable cadence, could be kind of intriguing.
With the cadences there to ground it and group it into
sections, then it may be that the melodic structure that you can hear yourself
will become more noticeable. Maybe you can find some way
to use the cadences to bring out what you hear yourself
for others to be able to hear it more plainly.

I think that as well as using a richer timbre or maybe
several of them, you could also do with more triads and diads and
less of just the single melody line too to really
show off the harmonic resources of 53-et.

You should drop in at cnfractalmusic and say hello there.

/cnfractal_music/

Robert

I have just added chords and cadences to Fractal Tune Smithy.
incidentally. Until now, it was based on a system of passing
tones which gave a kind of harmonic strucutre to the tunes
but it wasn't based on conventional harmony really at all
even if it sounded kind of a bit as if it did. I really
don't understand in fact why the passing tone tunes
in FTS sound as much like conventional harmonies as they do.

http://www.tunesmithy.co.uk/fts_download.htm

🔗Jonathan M. Szanto <JSZANTO@...>

🔗Paul Erlich <perlich@...>

🔗Rick McGowan <rick@...>

🔗Paul Erlich <perlich@...>

🔗Robert Walker <robertwalker@...>

🔗kukulaj <kukulaj@...>

Thanks, folks, for all the helpful comments! My little experiment has
two characteristics: it's algorithmically, pseudo-randomly generated;
and it uses pitches from the 2**(k/53) series. Really the algorithmic
character overwhelms the microtonal character. Thanks, Robert Walker,
for the pointer to the fractal music group. Probably what I am doing
will make more sense there.

I am amazed, Paul Erlich, that you remember my name from the tuning
list! That was probably 1997 or so! For better or worse, I have so
many little projects clamoring to be bigger projects, none of them get
enough time to really blossom. It's only about every five years or so
that I get a chance to make any progress with music!

Perhaps it's worthwhile for me to try to clarify here a bit of what I
was doing, before I run off to other groups.

The torus and helix business actually has nothing at all to do with
the tuning! What I am trying to do with the torus and helix is to
create the kind of temporal structure that Robert Walker was suggesting.

An analogy might help. Think of an image on a TV/CRT screen, and the
way that image gets scanned by the electron beam. Each horizontal line
scanned is just a little different than the last line. This is very
similar to what I am doing. There's a little sequence of notes, and
then the next sequence is a slight variation on that, and then the
next is a slight variation on that. Actually, if I just made the
sequence that way, the torus would be two dimensional. In fact, it was
four dimensional! So a short series of variations on a little phrase
then itself gets repeated with a slight variation! And then that whole
series gets repeated with variation! At each level, the whole piece
gets multiplied in length by some small integer factor. That's why the
whole thing ends up being 11 minutes long!

Looking at a specific step in the temporal sequence, it will have 8
neighboring steps: the note played immediately before it in the
temporal sequence, the note played immediately after it, the note
played 4 time steps before, the note played 4 time steps later, the
note played 16 steps before, the note played 16 steps later, etc.
Actually I think this piece didn't use a size 4 for every dimension of
the torus, but maybe this is enough to get the idea across. The
combination of notes played at one time step is pushed by the
algorithm to be harmonically close to the notes played at each of
these neighboring steps.

The tonal structure is embedded in the cost function. I forget the
precise details, because I kept changing them as I experimented! But
roughly: at each time step, any subset of the notes can be played, out
of a range of maybe three octaves. The cost function makes it very
expensive to play too many notes at the same time. The cost is
something like the sum of the "harmonic distances" between every pair
of notes being played simultaneously. So e.g. if five notes were to be
played simultaneously, the cost would add up the harmonic distances of
the ten different pairs of notes involved. The harmonic distance is
something I just hard-coded in a table, where octaves are really
close, perfect fifths (31 microsteps) are slightly less close, major
thirds (17 microsteps) a bit further apart, etc.

The cost function is also applied, of course, to notes played at
neighboring points. This way the little phrase played on the next turn
of the helix could be very similar but perhaps with one note of the
phrase moved by a perfect fifth.

I completely agree with Rick McGowan, that what I am doing is making
wallpaper music. I think of it like creating the aural equivalent of
watching puffy clouds roll by on a summer day, looking for faces and
animals etc. Not likely something that would pull people into a
concert hall! But maybe a sort of spare parts box of curious doodles.
The hope is that novel possibilities buried in the structure of
2**(k/53) music might appear amidst all the boring shapes,
possibilities that we would have a hard time discovering without the
doodling because of our powerful perceptual/conceptual habituation to
the 12 note system.

My thinking on this was powerfully reinforced by reading Blackwood's
_Structure of Recognizable Diatonic Tunings_ book. As I understand it,
conventional Bach to Mahler music is based on puns inherent in the
twelve note system. There are lots of these puns, but I suppose the
most fundamental is that four perfect fifths makes a major third. Of
course there's also three major thirds makes a unison, or twelve
perfect fifths makes a unison, etc. 2**(k/53) has a totally different
set of puns. The old puns are not available. For example, in
2**(k/53), four perfect fifths comes around to 18 steps, whereas a
major third is 17 steps. So the basic step is pretty darn close to
81/80 - what gets swept under the rug in 12 note music is the basic
building block of 53 note music!

Anyway, all this thinking doesn't justify the music if it doesn't work
when people listen! I really appreciate folks taking the time to
listen and the effort to help with comments! When I get some time to
dive back in, this will help give me good direction to improve the
algorithms to make something that better engages the ear!

Thanks again!
Jim

🔗Rick McGowan <rick@...>

🔗kraig grady <kraiggrady@...>

🔗Paul Erlich <perlich@...>

--- In MakeMicroMusic@yahoogroups.com, "kukulaj" <kukulaj@y...> wrote:

> My thinking on this was powerfully reinforced by reading Blackwood's
> _Structure of Recognizable Diatonic Tunings_ book. As I understand
it,
> conventional Bach to Mahler music is based on puns inherent in the
> twelve note system. There are lots of these puns, but I suppose the
> most fundamental is that four perfect fifths makes a major third. Of
> course there's also three major thirds makes a unison, or twelve
> perfect fifths makes a unison, etc. 2**(k/53) has a totally
different
> set of puns. The old puns are not available.

only one of them is -- see below.

> For example, in
> 2**(k/53), four perfect fifths comes around to 18 steps, whereas a
> major third is 17 steps. So the basic step is pretty darn close to
> 81/80 - what gets swept under the rug in 12 note music is the basic
> building block of 53 note music!

if you took a look at my post yesterday to the tuning list that i
posted a link to here on this list (since it was a response to rick's
post here), you'll see that this is exactly what i was talking about,
with regard to 12-equal and also the blackwood (coincidentally) 13-
though-24-equals that rick brought up. if you follow all the links in
that post, you'll be able to see that just as 12-equal has
these "puns" that you mention including 81/80 and others, 53-equal
has 15625/15552, 32805/32768 (this is the only one in common between
12 and 53, unless you count *powers* of 32805/32768),
1600000/1594323, 2109375/2097152, and others -- and of course many
more if you allow prime factor 7 into all this. scalewise, the 81/80
pun implies the diatonic/pentatonic family of scales, while, for
example, the 15625/15552 implies a family of scales generated by the
minor third and having, as an example, 11 notes per octave:

http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm

that's already way too much theory for this list, so i'd love it if
you'd follow up this conversation with me on the tuning list or
tuning math list! great to see you thinking about music again after
your long hiatus!!

Mad Calliopist

🔗James Kukula <kukulaj@...>

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