WAV files so you can listen to some homometric chords

/tuning-math/files/WarrenDSmith/

now includes 5 or so 1-second long sound files (WAV) so you
can listen to some of the homometric chord pairs from the "good sounding"
homometric pairs list I made. (If there is some way to play these files each on infinite repeat, that might help.)

So, if you believe in all the propaganda about Z-relations... and my "debugging theory"
of music... then good-sounding homometric chord pairs ought to provide a new kind of
debugging functionality never before heard/enjoyed by humans, thus instantly transporting you into new, never before experienced realms of sensory musical delight.

Well... really? So, you can listen to them yourself. It appears I succeeded in
making homometric chords that are pleasant to listen to -- as opposed to,
if you try just using the usual 12-tone scale, they'll always sound like crap (too many clashing note-dyads).

The optimization using the full freedom of the real numbers
to seek all-near-integer frequency ratios, was essential. I doubt that many (if any)
of your proposed 1-and-a-half-dimensional scales have enough freedom within them
to allow this -- probably only the full real numbers do.

You can judge subjectively for yourself if this is a new realm of musical sensory delight... who knows...

On Sat, Oct 8, 2011 at 7:26 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> Well... really? So, you can listen to them yourself. It appears I succeeded in
> making homometric chords that are pleasant to listen to -- as opposed to,
> if you try just using the usual 12-tone scale, they'll always sound like crap (too many clashing note-dyads).
>
> The optimization using the full freedom of the real numbers
> to seek all-near-integer frequency ratios, was essential.

Are they linearly homometric, or logarithmically homometric? They do
sound good, but I'm more interested in the latter than the former
case.

> I doubt that many (if any)
> of your proposed 1-and-a-half-dimensional scales have enough freedom within them
> to allow this -- probably only the full real numbers do.

The entire point of everything we're doing is to find generator chains
that DO approximate low integer ratios, up to some prime-limit. You
should read this:

http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space

And where did you get 1.5-dimensional from?

-Mike

On Sun, Oct 9, 2011 at 1:55 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> The entire point of everything we're doing is to find generator chains
> that DO approximate low integer ratios, up to some prime-limit. You
> should read this:
>
> http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space

Actually, if we're to start from the very beginning, it might be good
to start here:

http://xenharmonic.wikispaces.com/Monzos+and+Interval+Space

But then, the article on vals should be the next thing you read.

-Mike

> Are they linearly homometric, or logarithmically homometric? They do
> sound good, but I'm more interested in the latter than the former
> case.

--logarithmic.

> > I doubt that many (if any)
> > of your proposed 1-and-a-half-dimensional scales have enough freedom within them
> > to allow this -- probably only the full real numbers do.
>
> The entire point of everything we're doing is to find generator chains
> that DO approximate low integer ratios, up to some prime-limit. You
> should read this:
>
> http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space
>
> And where did you get 1.5-dimensional from?

--well, sure you are trying to approximate more integer ratios, BUT I doubt
it is possible to do that enough to get good sounding homometric chords.
See, even the simplest nontrivial homometric chords have 6 notes, and only 2 degrees
of freedom out of 6. The chance that one is going to have all 6*5/2=15
freq ratios all near an integer, is very low. With the full freedom of the real
numbers I can (& did) try millions of chords, completely covering the whole space of
possibilities, to find the very few chords which achieve this feat. Only about 30
such chords exist with <=7 notes.
If restricted to one of your discrete scales, we have FAR FAR less freedom -- it simply
is not possible to try millions, only 100 or at most about 1000 -- hence quite
likely we'll be unable to find even a single example.

"1.5 dimensional" well, you were speaking of 2-dimensional scales (generated by 2 freq ratios) but only allowing stacking the second generator boundedly high, so I loosely speaking called this "1.5 dimensional" scale. Nothing deep and strange intended :)

> read this:
>
> http://xenharmonic.wikispaces.com/Vals+and+Tuning+Space
>
> Actually, if we're to start from the very beginning, it might be good
> to start here:
>
> http://xenharmonic.wikispaces.com/Monzos+and+Interval+Space
>
> But then, the article on vals should be the next thing you read.

--What you call "p-limit" is called by the number-theorists "smooth"
(e.g. maybe "p-smooth," I'm not sure of their precise terminology).
Actually, both of these are poor names, and probably smooth is even dumber name
than yours, but "smooth numbers" has at least 30 years of use...
http://en.wikipedia.org/wiki/Smooth_number
http://mathworld.wolfram.com/SmoothNumber.html

You look like doing reasonable stuff on these 2 pages, but I could criticize you as follows:
it seems to me all the efforts there to make fancy theories are not necessary because
by brute force by computer, one could examine all possible scales (based on, say, 2 generators...) of your types with reasonably small parameters to find the best.

In short, I think brute force would be more effective than theorizing for you.
(But you've probably already tried it.)

So actually, if there is to be a musical application of the good-sounding homometric
chords, I suggest it should be the following.

Create artificial musical instruments (via computer) such that each note
that they play, is actually a chord from the good-sounding homosets.
(Scale all frequencies in the given set to move to a different note.)
(This is just as: each note played on a piano, is actually a chord in the
sense it has a lot of harmonic components at integer ratios.)

Now just play music like normal. BUT, you can vary between the two
homometric chords if desired.

Might this be better music and a better kind of musical instrument?

Well, I don't know. But let me say this. Musicians are into controlling individual notes,
such as adding vibrato, tremolo, distortion pedal, making each note be some sampled waveform got from someplace, etc. They think that matters. If it matters, presumably the reason is my debugging theory of music -- this adds more debugging functionality. OK, now if the proposed homoset-instruments were created, they would add more debugging functionality, of a new kind, to each note, hence might cause a more-pleasant musical experience. Specifically, each of their notes would force your brain to test whatever discrimination capabilities it has that are NOT based solely on the distance-set. (Normally, your brain is never forced to do that.)

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
>
> So actually, if there is to be a musical application of the good-sounding homometric
> chords, I suggest it should be the following.

I may be working with the wrong definition, but as far as I can tell generating homometric chord pairs with nice musical properties is pretty easy, and using them would require no special equipment.

On Mon, Oct 10, 2011 at 12:10 PM, WarrenS <warren.wds@gmail.com> wrote:
>
> Well, I don't know. But let me say this. Musicians are into controlling individual notes,
> such as adding vibrato, tremolo, distortion pedal, making each note be some sampled waveform got from someplace, etc. They think that matters. If it matters, presumably the reason is my debugging theory of music -- this adds more debugging functionality. OK, now if the proposed homoset-instruments were created, they would add more debugging functionality, of a new kind, to each note, hence might cause a more-pleasant musical experience. Specifically, each of their notes would force your brain to test whatever discrimination capabilities it has that are NOT based solely on the distance-set. (Normally, your brain is never forced to do that.)

This is part of the reason I think that the "debugging" concept makes
more sense if you get away from the paradigm where we're "debugging"
the auditory system specifically, and start considering more abstract
sorts of "debugging."

It can be a lot of fun to test your "discrimination capabilities." One
common example in the NYC jazz scene today is to play a 4/4 beat, then
accent every third sixteenth note, then turn that into a "faux"
backbeat (complete with bass on the faux 1 and 3 and snare on the faux
2 and 4), but then keep the original tempo in your head the whole
time, not getting "sucked" into the new time frame.

That kind of stuff is fun, and there was a whole group of us at UM who
are down here now who were obsessed with practicing that every chance
we could get; we'd hear the rhythmic clacking of the train along the
train tracks and immediately go into a 3/4 polyrhythm faux backbeat
over it. But sooner or later it started to feel like it was becoming a
chore, that music was becoming a test, and that our egoistic desire to
be in the 1% of musicians that have this skill was starting to take
away from other areas of music that required less thinking and more
feeling. Sooner or later we decided that it was more fun to not be so
OCD about the polyrhythms all the time, and allow yourself to get
swept away in the musical current at times, even if that left you
"vulnerable" to a predatory drummer coming in and bombing the
atmosphere with triplet groups of 5 all the time. So we hence cured
ourselves of this particular egotistical tendency, feeling we'd all
"matured" as musicians after the fact. In this case, we debugged the
OCD tendency to debug, you might say.

I've learned a lot of things from music as I've matured as a musician,
and while some of them make sense as increasing the powers of my
auditory system and cognization thereof, most of them seem to make
sense on a deeper level than that, sometimes involving predominantly
social concepts.

-Mike

On Mon, Oct 10, 2011 at 11:54 AM, WarrenS <warren.wds@gmail.com> wrote:
>
> --What you call "p-limit" is called by the number-theorists "smooth"
> (e.g. maybe "p-smooth," I'm not sure of their precise terminology).
> Actually, both of these are poor names, and probably smooth is even dumber name
> than yours, but "smooth numbers" has at least 30 years of use...
> http://en.wikipedia.org/wiki/Smooth_number
> http://mathworld.wolfram.com/SmoothNumber.html

Yes, smooth, same thing.

> You look like doing reasonable stuff on these 2 pages, but I could criticize you as follows:
> it seems to me all the efforts there to make fancy theories are not necessary because
> by brute force by computer, one could examine all possible scales (based on, say, 2 generators...) of your types with reasonably small parameters to find the best.
>
> In short, I think brute force would be more effective than theorizing for you.
> (But you've probably already tried it.)

There are more efficient ways of finding good temperaments than brute
forcing. The state of the art right now is Graham Breed's temperament
finder:

http://x31eq.com/temper/pregular.html

Just type in your desired limit, or "smoothness," and your desired
error, and off you go! The winner in the 11-limit for 5 cents is
Orwell temperament:

http://x31eq.com/cgi-bin/rt.cgi?ets=31_22&limit=11

Orwell's mapping matrix is

[< 1 0 3 1 3 ]
< 0 7 -3 8 2 ]>

Let's call this matrix "O". So if we want to see where 11/4 maps in
this chain, we first put it in monzo form:

[-2 0 0 0 1>

And then we multiply them, at which point we end up with [1 2>,
signifying that 11/4 maps to one step up on the first chain of
generators, and two steps up on the second chain. In this case, the
first generator is just 2/1. The second generator is a miracle of
mathematics that hits the (octave-equivalent) intervals 7/6, 11/8,
8/5, 15/8, 11/10, 9/7, and 3/2 in order.

More about the method here, although there might be an updated paper
that Graham can link you to:

http://x31eq.com/complete.pdf

-Mike

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> I may be working with the wrong definition, but as far as I can tell generating homometric chord pairs with nice musical properties is pretty easy

Nope. I was wrong about that.

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Oct 10, 2011 at 11:54 AM, WarrenS <warren.wds@...> wrote:
> >
> > --What you call "p-limit" is called by the number-theorists "smooth"
> > (e.g. maybe "p-smooth," I'm not sure of their precise terminology).
> > Actually, both of these are poor names, and probably smooth is even dumber name
> > than yours, but "smooth numbers" has at least 30 years of use...
> > http://en.wikipedia.org/wiki/Smooth_number
> > http://mathworld.wolfram.com/SmoothNumber.html
>
> Yes, smooth, same thing.

Not the same thing. P-limit refers to rational numbers, and are in general a ratio of two p-smooth numbers. Moreover, interest in smooth numbers is focused on "smooth" in comparison to size; "small" prime factors has in mind that they are small in comparison to the smooth number. One is not really concerned to call a prime number, or a product of two primes, "smooth", though it fits the definition. What number theorists really have in mind is numbers, especially very large numbers, with a lot of divisors, so 360 is the sort of thing one might think of as smooth, not so much 15 and certainly not 5.

Mike Battaglia <battaglia01@gmail.com> wrote:

> There are more efficient ways of finding good
> temperaments than brute forcing. The state of the art
> right now is Graham Breed's temperament finder:
>
> http://x31eq.com/temper/pregular.html

We are talking about regular temperaments, then, not
scales? I wasn't clear where the thread had gone.

Yes, you can brute force the search. I'm not sure what
that means, though. Maybe my searches are operating by
brute force. Brutality is in the eye of the beholder.
There are plenty of problems you need theory for, though.
The main thing is to know what it is you're searching for.
Then you need to know that pairing equal temperaments gives
you a rank 2 temperament, which you may call theory or
brute force, and it's quadratic in the number of equal
temperaments you seed with. It's useful to know which equal
temperaments are likely to lead to the results you're
looking for. You need an efficient badness function to
sort them by -- TOP-max is fine for equal temperaments, but
gets slower for higher ranks because you need to solve a
linear program instead of a linear least squares problem.
You need to have a way of searching equal temperaments that
doesn't show exponential behavior as the number of primes
increases. You need to be aware that if you want to search
beyond rank 2, that makes it more complex as well. After
everything's working, you may want to optimize it so that
it can run in an application (online or otherwise) without
a noticeable time lag. Once all that is sorted out, then
yes, it's pretty much brute force.

> More about the method here, although there might be an
> updated paper that Graham can link you to:
>
> http://x31eq.com/complete.pdf

That's one way of searching but it depends on having a
very clear idea what you're searching for. The method
behind the website is based on badness, and so the relevant
PDF is the one linked to from the web-app's launch page.
That being a level up from what you linked to:

http://x31eq.com/temper/

http://x31eq.com/badness.pdf

That explains how the equal temperament search is bounded,
and how good equal temperaments are likely to combine to
produce good temperaments of higher rank for the same
badness parameter. The exact algorithm for finding the
equal temperaments efficiently isn't in there, so you'll
have to read the code, in Python and Scheme and maybe
something else. The new algorithm for finding followers or
subsets of a higher rank temperament isn't documented at
all, and I only wrote it last week. It's running slower
than I'd like and I'm not sure it's correct. Brute force
works if you have a few minutes to spare.

Graham

Mike Battaglia <battaglia01@gmail.com> wrote:

> There are more efficient ways of finding good
> temperaments than brute forcing. The state of the art
> right now is Graham Breed's temperament finder:
>
> http://x31eq.com/temper/pregular.html

Oh, another thing . . .

The theory (that took years to work out) tells us that the
problem of finding good equal temperaments amounts to
finding short vectors in a lattice in an inner product space
(defined by a quadratic form). Pari has a function to do
that and I expect other packages do as well.

Graham

> Mike Battaglia:
> This is part of the reason I think that the "debugging" concept makes
> more sense if you get away from the paradigm where we're "debugging"
> the auditory system specifically, and start considering more abstract
> sorts of "debugging."
>
> It can be a lot of fun to test your "discrimination capabilities." One
> common example in the NYC jazz scene today is to play a 4/4 beat, then
> accent every third sixteenth note, then turn that into a "faux"
> backbeat (complete with bass on the faux 1 and 3 and snare on the faux
> 2 and 4), but then keep the original tempo in your head the whole
> time, not getting "sucked" into the new time frame.
>
> That kind of stuff is fun, and there was a whole group of us at UM who
> are down here now who were obsessed with practicing that every chance
> we could get; we'd hear the rhythmic clacking of the train along the
> train tracks and immediately go into a 3/4 polyrhythm faux backbeat
> over it. But sooner or later it started to feel like it was becoming a
> chore, that music was becoming a test, and that our egoistic desire to
> be in the 1% of musicians that have this skill was starting to take
> away from other areas of music that required less thinking and more
> feeling. Sooner or later we decided that it was more fun to not be so
> OCD about the polyrhythms all the time, and allow yourself to get
> swept away in the musical current at times, even if that left you
> "vulnerable" to a predatory drummer coming in and bombing the
> atmosphere with triplet groups of 5 all the time. So we hence cured
> ourselves of this particular egotistical tendency, feeling we'd all
> "matured" as musicians after the fact. In this case, we debugged the
> OCD tendency to debug, you might say.

-- WDS:
the debugging theory does not claim to be only about the auditory system, and speculates it also is about the whole brain (or anyhow a lot of it). So we are
in agreement. Now in the paper I do devote a lot of attention to auditory, because
that is easier to access more objectively. However, there is considerable evidence which I also discuss, that it affects plenty more brain that that. For example, brain scan evidence shows lot of brain affected/activated by music. This includes emotion areas and motor control areas that have nothing to do with plain auditory and which are not activated by plain speech. Various hormone levels (testosterone) are known to be affected by music.
Things like Tourette syndrome are known to be partially cured by music.

Now your story above in my view is yet another piece of confirmatory evidence.
Debug theory claims that you perpetually crave NEW musical experience i.e. new
kinds of debug+test functionality. After all, if you've tested
your brain on X and gained whatever neural benefits you can from it,
it is not much point to rerun that test over again
when you could now test Y. So listening to a piece of music you claim is "good" over and over, gets "boring" and even "annoying" after too many times. Like Xmas carols at malls.
How can you explain this phenomenon except via the debugging theory? Not too many rival ways. Certainly it is incompatible with some of the rival theories of music
like Pinker's cheesecake theory and the "rowing the boat" economic theory.

So you enjoyed this new musical experience but then too much of it got boring. If however you were to play that "boring" stuff to ME, I might find it quite nice.

(You're somewhat atypical though since I guess you are an exceptionally musical person.)

"WarrenS" <warren.wds@gmail.com> wrote:

> --What you call "p-limit" is called by the
> number-theorists "smooth" (e.g. maybe "p-smooth," I'm not
> sure of their precise terminology). Actually, both of
> these are poor names, and probably smooth is even dumber
> name than yours, but "smooth numbers" has at least 30
> years of use...
> http://en.wikipedia.org/wiki/Smooth_number
> http://mathworld.wolfram.com/SmoothNumber.html

I see Adleman gets mentioned there, so this may be
connected to our work. In "Modern Computer Algebra", by
von zur Gathen and Gerhard, there's a section on pages
554-555 called "Short vector cryptosystems". A key quote:
"Atjai showed that the problem of finding shortest vectors
in lattices . . . is ‘NP-hard’, and that it is as hard on
average as in the worst case."

Finding good equal temperaments amounts to finding short
vectors in a lattice. Sometimes lattice proofs only refer
to Euclidean lattices, and Cangwu badness isn't of that
kind, let alone whatever lattice you'd get from a minimax
approach. But I don't think this makes it easier. Atjai
was dealing with ". . . the type of lattice that
corresponds to simultaneous Diophantine approximations"
and they've already been tied to equal temperaments
(pp.481-2).

Then we have "Len Adleman had in 1995 reduced -- under some
reasonable but unproven assumptions -- the factorization of
integers to simultaneous approximation of sqrt(log q) for
the small primes p." That sounds relevant because it's
about small primes and logarithms of integers. Next, "Van
Emde Boas (1980) had shown that finding a shortest vector
in the max-norm is NP-hard. But it had remained an open
question whether this is true for the Euclidean norm . . ."

We don't need to worry about the NP-hardness because our
problems involve small enough numbers that they can be
solved by computers -- using brute force if you like. But
I think it's worth pointing out that this is ongoing
research. It's often asserted that music theory doesn't
lead to interesting mathematics. In this case, we aren't
very far behind.

Note, also, that the prime-limit terminology had been
around for a while. The usage can be traced back to
Genesis of a Music (1949). Although Partch was talking
about odd limits, many people assumed he meant prime
limits, and so the odd/prime distinction came about to
resolve this ambiguity.

Graham

> In "Modern Computer Algebra", by
> von zur Gathen and Gerhard, there's a section on pages
> 554-555 called "Short vector cryptosystems". A key quote:
> "Atjai showed that the problem of finding shortest vectors
> in lattices . . . is NP-hard, and that it is as hard on
> average as in the worst case."

--finding short vectors in lattices is hard for HIGH DIMENSIONAL lattices.

However in low dimensions, which is all you are interested in (like 1,2, or 3 dimensions),
it is not hard. In fact, Lovasz lattice reduction
techniques show that in any FIXED dimension,
these problems are in polynomial time.
There is something nasty that grows exponentially or even worse with
the number of dimensions, but if #dims fixed that does not matter.

"WarrenS" <warren.wds@gmail.com> wrote:
> > In "Modern Computer Algebra", by
> > von zur Gathen and Gerhard, there's a section on pages
> > 554-555 called "Short vector cryptosystems". A key
> > quote: "Atjai showed that the problem of finding
> > shortest vectors in lattices . . . is NP-hard, and that
> > it is as hard on average as in the worst case."
>
> --finding short vectors in lattices is hard for HIGH
> DIMENSIONAL lattices.
>
> However in low dimensions, which is all you are
> interested in (like 1,2, or 3 dimensions), it is not
> hard. In fact, Lovasz lattice reduction techniques show
> that in any FIXED dimension, these problems are in
> polynomial time. There is something nasty that grows
> exponentially or even worse with the number of
> dimensions, but if #dims fixed that does not matter.

It's not 1, 2, or 3 dimensions. My website goes up to the
31-limit, and that's 11 dimensions. If you were brute
forcing rank 2 temperaments in a really brutal way, you'd
have 20 dimensions to play with. Yes, the difference
between polynomial and exponential time matters as the
dimensions grow. (How could it even be polynomial in fixed
dimension? Isn't everything constant time if nothing
changes?) An exponential time algorithm can already be
unusable by this point. Fortunately I've improved on
that. Yes, the calculations we're doing can be done the
way we're doing them. Yes, it took a lot of theory to get
to that point.

But there's a thread in the archives where we looked
at much higher limits, and my code wasn't really up to it.

I don't know what "Lovasz lattice reduction techniques"
would apply. Maybe you could suggest some instead of
giving the impression that this is all so easy we should
know exactly what to do. I have plain LLL working but not
for this case. It requires a Euclidean lattice. (Yes, it's
a polynomial time algorithm, but it isn't guaranteed to
return the shortest vector. No magic pixie dust will give
solve an NP hard problem with a polynomial time
algorithm. If equal temperaments are a special case,
let's see the proof.)

From the Pari source code, I see that the algorithm it uses
for this is called Fincke-Pohst. It has something to do
with LLL but I can't find a friendly specification of it.
It isn't in The Concise Oxford Dictionary of Mathematics.

Graham

> It's not 1, 2, or 3 dimensions. My website goes up to the
> 31-limit, and that's 11 dimensions. If you were brute
> forcing rank 2 temperaments in a really brutal way, you'd
> have 20 dimensions to play with. Yes, the difference
> between polynomial and exponential time matters as the
> dimensions grow. (How could it even be polynomial in fixed
> dimension? Isn't everything constant time if nothing
> changes?)

--polynomial in the input length, integer input as bits.

> An exponential time algorithm can already be
> unusable by this point. Fortunately I've improved on
> that. Yes, the calculations we're doing can be done the
> way we're doing them. Yes, it took a lot of theory to get
> to that point.
>
> But there's a thread in the archives where we looked
> at much higher limits, and my code wasn't really up to it.
>
> I don't know what "Lovasz lattice reduction techniques"
> would apply. Maybe you could suggest some instead of
> giving the impression that this is all so easy we should
> know exactly what to do.

--not so easy. There is a literature on this where various techniques related to LLL lattice
reduction are used to get the lattice into a not-so-bad form, then various post-processing techniques are used to get useful output (such as shortest vector) from that. I'm not sure
how many of the proposed algorithms have actually been programmed.

I'm not sure how far you can go before it gets infeasible, as a guess maybe 30 dimensional lattices are feasible today? Quite likely you are not going to do it better than PARI does it, in which case you might as well just use PARI or whatever software you can get.

--- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:
> --What you call "p-limit" is called by the number-theorists "smooth"
> (e.g. maybe "p-smooth," I'm not sure of their precise terminology).
> Actually, both of these are poor names, and probably smooth is even dumber name
> than yours, but "smooth numbers" has at least 30 years of use...
> http://en.wikipedia.org/wiki/Smooth_number
> http://mathworld.wolfram.com/SmoothNumber.html

http://www.mersenneforum.org/showthread.php?t=5630
asks:
"...are there any more?
than:

1 2 octave
3 4 4th
8 9 tone
24 25 chroma
80 81 syntonic comma
125 126 septimal-semicomma
224 225 septimal-kleisma
2400 2401 Breedsma
3024 3025 Lehmerisma
4224 4225
4374 4375 Ragisma
6655 6656
9800 9801 Gauss-Comma or 'Kalisma'
10647 10648
123200 123201
194480 194481
336140 336141
601425 601426
633555 633556
709631 709632
5142500 5142501
5909760 5909761
11859210 11859211
......
"

see also
http://mathforum.org/kb/message.jspa?messageID=270203&tstart=0
"
123200 = 2^6 * 5^2 * 7 * 11 123201 = 3^6 * 13^2
10647 = 3^2 * 7 * 13^2 10648 = 2^3 * 11^3
9800 = 2^3 * 5^2 * 7^2 9801 = 3^4 * 11^2
6655 = 5 * 11^3 6656 = 2^9 * 13
4374 = 2 * 3^7 4375 = 7 * 5^4
4095 = 3^2 * 5 * 7 * 13 4096 = 2^12
3024 = 2^4 * 3^3 * 7 3025 = 5^2 * 11^2
2400 = 2^5 * 3 * 5^2 2401 = 7^4
"

out from the sequence:
http://oeis.org/A002072

as used in:
http://xenharmonic.wikispaces.com/Ragismic+microtemperaments

bye
Andy

See http://xenharmonic.wikispaces.com/List+of+Superparticular+Intervals , which has all the epimoric ratios up through 13-limit.

Note that some of those are not on this list you found, because they don't satisfy the (arbitrary) rule that the limit must be ceil(log2(N)).

--- In tuning-math@yahoogroups.com, "Andy" <a_sparschuh@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "WarrenS" <warren.wds@> wrote:
> > --What you call "p-limit" is called by the number-theorists "smooth"
> > (e.g. maybe "p-smooth," I'm not sure of their precise terminology).
> > Actually, both of these are poor names, and probably smooth is even dumber name
> > than yours, but "smooth numbers" has at least 30 years of use...
> > http://en.wikipedia.org/wiki/Smooth_number
> > http://mathworld.wolfram.com/SmoothNumber.html
>
> http://www.mersenneforum.org/showthread.php?t=5630
> asks:
> "...are there any more?
> than:
>
> 1 2 octave
> 3 4 4th
> 8 9 tone
> 24 25 chroma
> 80 81 syntonic comma
> 125 126 septimal-semicomma
> 224 225 septimal-kleisma
> 2400 2401 Breedsma
> 3024 3025 Lehmerisma
> 4224 4225
> 4374 4375 Ragisma
> 6655 6656
> 9800 9801 Gauss-Comma or 'Kalisma'
> 10647 10648
> 123200 123201
> 194480 194481
> 336140 336141
> 601425 601426
> 633555 633556
> 709631 709632
> 5142500 5142501
> 5909760 5909761
> 11859210 11859211
> ......
> "
>
> see also
> http://mathforum.org/kb/message.jspa?messageID=270203&tstart=0
> "
> 123200 = 2^6 * 5^2 * 7 * 11 123201 = 3^6 * 13^2
> 10647 = 3^2 * 7 * 13^2 10648 = 2^3 * 11^3
> 9800 = 2^3 * 5^2 * 7^2 9801 = 3^4 * 11^2
> 6655 = 5 * 11^3 6656 = 2^9 * 13
> 4374 = 2 * 3^7 4375 = 7 * 5^4
> 4095 = 3^2 * 5 * 7 * 13 4096 = 2^12
> 3024 = 2^4 * 3^3 * 7 3025 = 5^2 * 11^2
> 2400 = 2^5 * 3 * 5^2 2401 = 7^4
> "
>
> out from the sequence:
> http://oeis.org/A002072
>
> as used in:
> http://xenharmonic.wikispaces.com/Ragismic+microtemperaments
>
> bye
> Andy

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@> wrote:
>
> See http://xenharmonic.wikispaces.com/List+of+Superparticular+Intervals ,

> which has all the epimoric ratios up through 13-limit.
> Note that some of those are not on this list you found,
> because they don't satisfy the (arbitrary)
> rule that the limit must be ceil(log2(N)).

Unfortunately that list above mentioned is also incomplete.
Especially there are lacking all the most possible tiniest epimoric-ratios above 17-limit and higher.
That missing ratios can be found in the Lehmner's [1964] original paper:

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.ijm/1256067456&view=body&content-type=pdf_1

or too

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ijm/1256067456

They got overtaken into

http://oeis.org/A117581
..."Stormer came to this problem from music theory."...

let's continue with the absent

17-limit
336141/336140
= |-2 2,-1 -5 0 3,1>
=~0.0051503286544407261790149930928240154889362971324646...cents

19-limit
11859211/11859210
= |-1 -4,-1 1 0,1 0 4>
=~0.0001459822345734361587251184422247917738158580844868...cents

23-limit = same as alredy 19-limit [sic!]

29-limit: 177182721/ 177182720
= |-11 6,-1 0 -3, -1 2 0,2>
=~0.0001459822345734361587251184422247917738158580844868...cents

31-limit:
1611308700/1611308699
= |2 6,2 -4 -1 2,-2 0 -2,1 0 2>
=~1.0744272960258761942391391079690316340654680117... × 10^-6 cents

37-limit:
3463200000/3463199999
= |8 2,5 -5 0,1 -2 0,-1 0 1,1>
=~4.9989433163452969931312173314975841249417149176... × 10^-7 cents

41-limit:
63927525376/63927525375
= |13 -3,-3 -7 4,1 0 0,-1 0 0,0 1>
=~2.7081199200153076302804852330098376829882909767... × 10^-8 cents

43-limit
421138799640/421138799639
...

47-limit
1109496723126/1109496723126

53-limit
1453579866025/1453579866025
...

59-limit
20628591204481/20628591204481
...

61-limit
31887350832897/31887350832896

67-limit = same as already 61-limit

71-limit
119089041053697/119089041053696
...
=~1.453730783075290310632265781335655012641478824... × 10^-11 cents

&ct.

Also there lack for instance some intermediate epimoric-ratios alike
even in ancient Baroque 17- & 19-limit,

http://groenewald-berlin.de/text/text_T126.html
Quote from:
"Andreas Reinhard, 1604 - Abraham Bartulus, 1614

built from the bases:

... 81/80 (sK), 136/135, 153/152, 171/170 und 256/255 ...."

Look them again closer en-detail
for labeling them with adequate names
of the people, that had applied them once in the past:

Proposals of denominating theirs epimoric terms:

136/135 = |3 -3,-1 0 0,0 1> =~12.37...cents
'upper 17-limit "Reinhard"-isma [1604]

256/255 = |8 -1,-1 0 0, 0 -1> =~6.78...cents
'lower 17-limit "Reinhard"-isma [1604] or
maybe Fermat-isma_F1*F2*F3 = Mersenn-isma too?
because 255 = 3*5*17 = F1*F2*F3 = M_8 = 2^8-1
http://en.wikipedia.org/wiki/Fermat_number
http://en.wikipedia.org/wiki/Mersenne_prime

153/152 = |-3 2,0 0 0,0 1 -1> =~11.35.cents
'upper 19-limit "Bartulus"-isma [1614]
or even 'Ganassi-isma [1543]
hitherto as up to now Macolm-isma[1721]
http://groenewald-berlin.de/text/text_T029.html
Quote:
"Bisher war man geneigt, diese Entdeckung Alexander Malcolm, ca. 1721, zuzuschreiben. Der Ruhm hierfür gebührt aber eindeutig Ganassi."
translation
'Previously there had been a tendency,
to attribute this discovery to Alexander Malcolm,
ca. 1721. But the fame of founding
this must clearly by awarded to the prior Ganassi. "
or even to
http://en.wikipedia.org/wiki/Aristides_Quintilianus
in his ouevre
http://www.britannica.com/EBchecked/topic/451614/Peri-musike
in order to mention my earliest knowledge of
the diverse 17&19-limit combinations,
as far as I'm aware about antique greek music-theory at the moment.

171/170 = |1 -2,-1 0 0,0 -1 1> =~10.15...cents
'lower 17-limit "Bartulus"-isma [1614]
if not some kind of 'Quntilianus'-isma
as suggest
http://www.jstor.org/stable/40265828
http://theses.gla.ac.uk/1262/01/1996hughesphd-1.pdf
Quest:
Does anybody in that group here know more
details about the ancient-greek usage of 17&19-limits?

alike
http://www.huygens-fokker.org/docs/intervals.html
mentions
513/512 = |-9 3,0 0 0,0 0 1> =~3.38...cents
as undevicesimal comma, "Boethius"-isma comma

1216/1215 = |6 -5,-1 0 0,0 0 1> "Erasthostenes-Comma
"Eratosthenes"-isma comma

for that both ones see
http://en.wikipedia.org/wiki/Tetrachord
"Eratosthenes chromatic tetrachord" tri-section
obtained from subdivision 4:3 = (20:19) * (19:18) * (6:5)
into two small semitones and the 5-limit minor-3rd.

Historical remark:
Back in 1704 Andreas Werckmeister donated his pupil
http://en.wikipedia.org/wiki/Johann_Gottfried_Walther
the books of Reinhard-&-Bartholus as gift for reading
gererally about intervals.
Probably he studied the above 17&19-limits epimoric-ratios
together in collabortion with his favourite cousin J.S.Bach.

Hence there exist good reasons, that J.S.Bach
assembeld just from that coeval practice
his own kind of 'double'-emimoric tempered tuning
Probably first alike
Michael Praetorius's common usual
epimoric-'meantonics' subdivision of the SC=80:81
into four parts, as Werckmeister deeply excoriated:

C 323:324 G 322:323 D 321:322 A 320:321 E

But later Kirnberger reverted back to that old-fashion in his #3:
http://groenewald-berlin.de/text/text_T093.html
Quote:
"Kirnberger gibt für die vier strittigen Quinten Werte an ... C-G = 323/216, G-D = 322/215,333, D-A = 320/214 und A-E = 321/ 214,666." Kirnberger nennt es einen "mathematischen Kunstgriff", dass 321/320 x 322/321 x 323/322 x 324/323 = 81/80 ist."

See for partial translation of the original, the both documents:
http://harpsichords.pbworks.com/f/K_III.html
http://harpsichords.pbworks.com/f/Kirn_1871.html

Excursus:
And appearenty -from-time-to-time- some people revinvent that
anew again-and-again once more:
http://www.paulgreenhaw.com/Writings/objet_petit.pdf
"With 289:288 being absorbed into 18:17 â€" thus giving rise to a 17:16. Notice every interval in the resultant scale is superparticular."
{end-of-excursus}

Finally my own contribution:
/tuning/topicId_103187.html#104314
Consists also in such an epimoric graduation of 5ths:

Ab
Eb
Bb
F
C
512/513 'Boethius'-isma =
G
323/324 from 'Michael-Pretorius'-meantonics there @ C 324/323 G
D
288/289 differecne (17/16):(18/17) =~-6.0008cents
A
272/273 in list as : "273/272 =~6.3532...cents = (3*7*13)/(2^4*17)
E
728/729 in list as: "729/728 =~2.3764...cents = |
B
F#
C#
G#

Then turn the deviations of 3rds from 5/4 also out as epimoric.
That yields for the amounts of graduation in the 3rds sharpening.
It achieves the graduation:

Ab-C: 81/80 = |-4 4,-1> Syntonic-Comma, hence ditone Ab-C = 81/64
Eb-G: 96/95 = |5 1, -1 0 0, 0 0 1> the "Biome"-comma
Bb-F: 136/135 = |3 -3, -1 0 0, 0 1> upper "Reinhard"-isma
F-A : 256/255 = |8 -1, -1 0 0, 0 1> lower "Reinhard"-isma
C-E : 4096/4095 =|12 -2, -1 -1 0, -1>tridecimal or Sagittal schismina
G-B : 1216/1215 =|6 -5, -1 0 0, 0 0 1> "Erasthostenes"-comma
D-F#: 256/255 = |8 -1, -1 0 0, 0 1> the same amount as already F-A
A-C#: 136/135 = |3 -3, -1 0 0, 0 1> same as Bb-F
E-G#: 91/90 = |-1 -2,-1 1 0,1> again the "Biome"-comma too as Eb-G
B-D#: 81/80 like Ab-C and all the other remote-keys
F#A#: 81/80
C#E#: 81/80
G#B#: 81/80 back again, cycle complete.

bye
Andy

--- In tuning-math@yahoogroups.com, "Andy" <a_sparschuh@...> wrote:
> Unfortunately that list above mentioned is also incomplete.
> Especially there are lacking all the most possible tiniest epimoric-ratios above 17-limit and higher.

Okay, well if you're looking for a complete list of epimoric ratios of arbitrary limit, it is here: http://oeis.org/A000027 =)

For 13-limit (which is what I said), that list *is* complete.

Keenan