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Message: 7550 Date: Sun, 05 Oct 2003 13:47:13 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "dkeenanuqnetau" <d.keenan@u...> Yahoo groups: /tuning-math/message/3756 * [with cont.] wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > I followed that conversation and, although I have strong
convictions
> > about what was discussed, I just didn't have the time to get
> involved
> > in it. My thoughts on this are: > > > > 1) Johnny is already very familiar with cents, so that is what
works
> > for him. For the rest of us it would take a bit of training to
be
> > able to do the same, and then might we need a calculator to
> determine
> > the intervals? When you are writing chords, where do all the
cents
> > numbers go, and how can you read something like that with any > > fluency? (But that is for instruments of fixed pitch, which do
not
> > require cents, which brings us to the next point.) > > > > 2) Tablatures were mentioned in connection with instruments of
fixed
> > pitch, where cents would be inappropriate. I hate tablatures
with a
> > vengeance! Each instrument might have a different notation, and
> this
> > makes analysis of a score very difficult. We need a notation
that
> > enables us to understand the pitches and intervals, regardless of > > what sort of instrument is used.
> > These were proposed as notations for performers, not composers or > analysers. As such I see no great problem with the above, or > scordatura. >
***A notation for performers, as opposed to analysers? I thought Carl Lumma very clearly stated there should't be two different kinds of music notation for different purposes... ?? J. Pehrson
Message: 7551 Date: Sun, 05 Oct 2003 13:51:32 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor" <gdsecor@y...> wrote: Yahoo groups: /tuning-math/message/3768 * [with cont.]
> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > [Secor]
> > >> That's something that I don't like about the Sims notation --
> down
> > >> arrows used in conjunction with sharps, and up arrows with
flats.
> > > > [Keenan]
> > >I think Manuel exempts sharps and flats from this criticism.
> > > > Yes indeed, for example, Eb/ is always the nearest tone to 6/5 > > as E\ is always nearest to 5/4. > > > > Manuel
> > My objection is to alterations used in conjunction with sharps and > flats that alter in the opposite direction of the sharp or flat by > something approaching half of a sharp or flat. For example, 3/7 or > 4/7 is much clearer than 1 minus 4/7 or 1 minus 3/7, but I would
not
> object to 1 minus 2/7 (instead of 5/7). > > I have been dealing with this issue in evaluating ways to notate > ratios such as 11/9, 16/13, 39/32, and 27/22, and I am persuaded
that
> the alterations for these should not be in the opposite direction > from an associated sharp or flat. In other words, relative to C, I > would prefer to see these as varieties of E-semiflat rather than E- > flat with varieties of semisharps. But (for other intervals) > something no larger than 2 Didymus commas (~43 cents or ~3/8
apotome)
> altering in the opposite direction would be okay with me. > > --George
***Johnny Reinhard is frequently talking about the follow of mixing the direction of symbols in accidentals. It seems like a mistake that should be avoided (if possible), even for the smaller ones... J. Pehrson
Message: 7552 Date: Sun, 05 Oct 2003 13:54:28 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "dkeenanuqnetau" <d.keenan@u...> Yahoo groups: /tuning-math/message/3773 * [with cont.] wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > I spent some time wrestling with 27-ET last night, and it proved
to
> > be a formidable opponent that severely limited my options. There
is
> > one approach that allows me to do it justice (using 13 -- what
else
> > is there?)
> ... > > The only other option I could see was to notate it as every fourth > note of 108-ET (= 9*12-ET), using a trinary notation where the 5-
comma
> is one step, the 7-comma is 3 steps, and the apotome is 9 steps,
but
> that would be to deny that it has a (just barely) usable fifth of
its
> own. >
> > With this it looks as if I am going to be stopping at the 17
limit,
> > This might be made to work for ETs, but not JI. The 16:19:24 minor > triad has a following. >
> > with intervals measurable in degrees of 183-ET.
> > I don't understand how this can work. >
> > Once I have made a > > final decision regarding the symbols, I hope to have something to > > show you in about a week or so.
> > I'm more interested in the sematics than the symbols at this stage.
I
> wouldn't spend too much time on the symbols yet. I expect serious > problems with the semantics.
***It seems there has been more emphasis on the *semantics* and not enought emphasis on the *symbols* in the entire Sagittal project! (Some are not well-enough differentiated...) J. Pehrson
Message: 7553 Date: Sun, 05 Oct 2003 14:14:10 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "dkeenanuqnetau" <d.keenan@u...> Yahoo groups: /tuning-math/message/3797 * [with cont.]
> > OK. So you have gone outside of one-comma-per-prime and > one-(sub)symbol-per-prime. But you have given fair reasons for
doing
> so in the case of 11 and 13. >
> > I am also outlining a 23-limit approach; I went for
> the
> > 19 limit and got 23 as a bonus when I found that I could
approximate
> > it using a very small comma. The two approaches could be
combined,
> > in which case you could have the 11-13 semiflat varieties along
with
> > the 19 or 23 limit, but the symbols may get a bit complicated --
> more
> > about that below.
> > You don't actually give more below about combining these
approaches.
> But I had fun working it out for myself. I'll give my solution
later.
>
> > I thought more about this and now realize that the problem with
> 27-ET
> > is not as formidable as it seemed. If we use the 80:81 comma for
a
> > single degree and the 1024:1053 comma for two degrees of
alteration,
> > we will do just fine, even if the *symbol* for 1024:1053 happens
to
> > be a combination of the 80:81 and 63:64 symbols (by conflating > > 4095:4096). For the 27-ET notation we can simply define the > > combination of the two symbols as the 13 comma alteration, and
there
> > would be no inconsistency in usage, since the 63:64 symbol is
*never
> > used by itself* in the 27-ET notation. The same could be said
about
> > 50-ET.
> > You're absolutely right. >
> > Are there any troublesome divisions above 100 that we should > > be concerned about in this regard?
> > Not that I can find on a cursory examination. >
> > I anticipate that you believe that the JI purists would still
want
> to
> > have this distinction, so we should go for 19.
> > Correct. I'll skip the 183-ET based one. >
> > I
> > > wouldn't spend too much time on the symbols yet. I expect
serious
> > > problems with the semantics.
> > > > I don't know what problems you are anticipating, ...
> > Well none have materialised yet. :-) >
> > I have found that the semantics and symbols are so closely
connected
> > that I could not address one without the other,
> > Yes. I see that now. >
> > Both the 17-limit and 23-limit approaches use 6 sizes of > > alterations. In the sagittal notation these are paired into left
> and
> > right flags that are affixed to a vertical stem, to the top for > > upward alteration and to the bottom for downward alteration.
These
> > pairs of flags consist of straight lines, convex curved lines,
and
> > concave curved lines. With this arrangement there is a
limitation
> > that two left or two right flags cannot be used simultaneously.
> > Given these constraints I think your solution is brilliant. >
> > *23-LIMIT APPROACH* > > > > And here is the 23-limit arrangement, which correlates well with
> 217-
> > ET (apotome of 21 degrees):
> > I don't think you can call this a 23-limit notation, since 217-ET
is
> not 23-limit consistent. But it is certainly 19-prime-limit. >
> > Straight left flag (sL): 80:81 (the 5 comma), ~21.5 cents (4
degrees
> > of 217) > > Straight right flag (sR): 54:55, ~31.8 cents (6deg217) > > Convex left flag (vL): 4131:4096 (3^5*17:2^12, the 17-as-flat
> comma),
> > ~14.7 cents (3deg217) > > Convex right flag (xR): 63:64 (the 7 comma), ~27.3 cents (5deg217) > > Concave left flag (vL): 2187:2176 (3^7:2^7*17, the 17-as-sharp > > comma), ~8.7 cents (2deg217) > > Concave right flag (vR): 512:513 (the 19-as-flat comma), ~3.4
cents
> > (1deg217) > > > > The difference between this and the 17-limit approach is that I
have
> > removed the 715:729 alteration and added the 512:513 alteration, > > while reassigning the 17-commas to different flags. No
combination
> > of flags will now exceed half of an apotome. > > > > With the above used in combination, the following useful
intervals
> > are available: > > > > sL+sR: 32:33 (the 11-as-semisharp comma), ~53.3 cents (10deg217) > > sL+xR: 35:36, ~48.8 cents, which approximates > > ~sL+xR: 1024:1053 (the 13-as-semisharp comma), ~48.3 cents
> (9deg217)
> > vL+sR: 4352:4455 (2^8*17:3^4*5*11), ~40.5 cents, which
approximates
> > ~vL+sR: 16384:16767 (2^14:3^6*23, the 23 comma), ~40.0 cents > > (8deg217) > > xL+xR: 448:459 (2^6*7:3^3*17), ~42.0 cents, which approximates > > ~xL+xR: 6400:6561 (2^8*25:3^8, or two Didymus commas), ~43.0
cents
> > (8deg217) > > > > The vL+sR approximation of the 23 comma deviates by 3519:3520
> (~0.492
> > cents). > > > > All of the above provide a continuous range of intervals in 217-
ET,
> > which I selected because it is consistent to the 21-limit and > > represents the building blocks of the notation as approximate > > multiples of 5.5 cents.
> > Once I understood your constraints, I spent hours looking at the > problem. I see that you can push it as far as 29-limit in 282-ET if > you want both sets of 11 and 13 commas, and 31-limit in 311-ET if
you
> can live with only the smaller 11 and 13 commas. But to make these > work you have to violate what is probably an implicit constraint,
that
> the 5 and 7 commas must correspond to single flags. Neither of them > can map to a single flag in either 282-ET or 311-ET and so the
mapping
> of commas to arrows is just way too obscure. > > 217-ET is definitely the highest ET you can use with the above > additional constraint. > > I notice that left-right confusability has gone out the window.
***This is exactly the problem that I have been discussing on the main list... the lef-right confusability between the 5-comma single flag and the 7-comma single flag. But
> maybe that's ok, if we accept that this is not a notation for > sight-reading by performers.
***WHAT?? This is obviously very much *not* ok! However, it is possible to improve the
> situation by making the left-right confusable pairs of symbols
either
> map to the same number of steps of 217-ET or only differ by one
step,
> so a mistake will not be so disastrous. At the same time as we do
this
> we can reinstate your larger 11 and 13 commas, so you have both
sizes
> of these available. The 13 commas will have similar flags on left
and
> right, while the 11 commas will have dissimilar flags. It seems
better
> that the 11 commas should be confused with each other than the 13 > commas, since the 11 commas are closer together in size. >
***I can't understand why everybody is so concerned about the 11 and 13 commas when even the 5 and 7 commas are confusable! The trees are getting in the way of the forest yet again! J. Pehrson
Message: 7554 Date: Sun, 05 Oct 2003 15:04:25 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor" <gdsecor@y...> wrote: Yahoo groups: /tuning-math/message/3817 * [with cont.]
> > It didn't take me very long to reach a definite conclusion. I
recall
> that it was the issue of lateral confusibility that first led to
the
> adoption of a curved right-hand flag for the 7-comma alteration in > the 72-ET notation. Before that all of the flags were straight. > Making the xR-sR symbol exchange would once again give the 7-comma > alteration a straight flag, which would negate the original reason > for the curved flag. The 72-ET notation could still use curved
right
> flags, but they would no longer symbolize the 7-comma alteration,
but
> the 54:55 alteration instead, which tends to obscure rather than > clarify the harmonic relationships. Also, since the JI notation > would use straight flags for both the 5 and 7-comma alterations,
then
> lateral confusibility would make it more difficult to distinguish > between two of the most important prime factors, and we would be > giving this up without receiving anything of comparable benefit in > return. > > Notating the 7-comma with the xR (curved) flag, on the other hand, > makes a clear distinction between ratios of 5 and 7 in JI, 72-ET,
and
> anywhere else that 80:81 and 63:64 are a different number of > degrees.
***I don't believe this is correct. The curved flag, as I have repeatedly mentioned, is *still* not a distinct enough symbol to avoid the lateral confusibility situation. I'm glad, at least, that all of this was being thought about, but still there is no solution. It also minimizes the use of curved flags in the ET
> notations, introducing them only as it is necessary or helpful: 1)
to
> avoid lateral confusibility (in 72-ET); 2) to distinguish 32:33
from
> 1024:1053 (in 46 and 53-ET, *without* lateral confusibility!); and
3)
> to notate increments smaller than 80:81 (in 94-ET). Lateral > confusibility enters the picture only when one goes above the 11 > limit: In one instance one must learn to distinguish between > 1024:1053 and 26:27 by observing which way the straight flag points > (leftward for the smaller ratio and rightward for the larger).
***This was one of the questions I had *immediately* when I saw the "complete" matrix of Sagittal symbols. This lack of differentiation is bothersome.l
> Another instance does not come up until the 19 limit, which
involves
> distinguishing the 17-as-sharp flag from the 19 flag. > > So I think we have enough reasons to stick with the convex curved > flag for the 7 comma. (I will also give one more reason below.) > > By the way, something else I figured out over the weekend is how to > notate 13 through 20 degrees of 217 with single symbols,
***You know, I don't want to shine the "harsh light of reality" on this project (if I could) but it seems an *awful* lot of though has gone into notating higher level ETs. This is an interesting intellectual exercise but, practically speaking how many people (besides, I guess, Marc Jones :) actually *use* higher-level ETs. Names, please?? And yet, the simplest of systems, like 72-tET has not yet been clearly differentiated. Maybe the whole *premise* of Sagittal is wrong. (Dare I say it?) Maybe there should be *different* notational systems for different purposes and a *unified field theory* shouldn't even be attempted. But, I believe this sentiment has been echoed before (was it Carl Lumma??) J. Pehrson
Message: 7555 Date: Sun, 05 Oct 2003 15:12:22 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "dkeenanuqnetau" <d.keenan@u...> Yahoo groups: /tuning-math/message/3819 * [with cont.] wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > Something we'll have to keep in mind is how much primal
uniqueness
> > should be traded off against human comprehension of the symbols.
I
> > think that the deciding factor should be in favor of the human,
not
> > the machine -- software can be written to handle all sorts of > > complicated situations;
> > I agree. But it is also possible to disambiguate dual purpose flags
by
> say adding a blob to the end of the stroke for one use and not the > other. >
***I think not. That's not enough differentiation in any way...
> > I had never given much thought to notating divisions above 100,
but
> I
> > would like to see how well the JI notation will work with these. > > Which ones between 94 and 217 would you consider the most
important
> > to be covered by this notation (listed in order of importance)?
> > I don't know order of importance. 96, 105, 108, 111, 113, 121, 130, > 144, 149, 152, 159, 166, 171, 183, 190, 198, 212. >
> > And if 217 seems suitable, then we should stick with > > it. (Over the weekend I happened to notice that it's 7 times 31 -
-
> > in effect a division built on meantone quarter-commas!)
>
***So here we're up to 217 ET (a real popular one...) and yet, the 72- tET symbols aren't sufficiently differentiated.
> > I spent some time this past weekend figuring out how all of this
was
> > going to translate into various ET's under 100, and every
division I
> > tried could be notated without any lateral mirroring whatsoever. > > (Even 58-ET, which had given me problems before, now looks very
> good.) >
***Huh? Must have missed 72 then.
> > It didn't take me very long to reach a definite conclusion. I
> recall
> > that it was the issue of lateral confusibility that first led to
the
> > adoption of a curved right-hand flag for the 7-comma alteration
in
> > the 72-ET notation. Before that all of the flags were straight.
****Ugggh. How awful.
> > Making the xR-sR symbol exchange would once again give the 7-
comma
> > alteration a straight flag, which would negate the original
reason
> > for the curved flag.
> > Yes. I was considering putting a blob on the end of the straight 7 > flag, but no. I agree with you now. Keep the curved flag for the > 7-comma. It is most important to get the 11-limit right. The rest
is
> just icing on the cake, and a little lateral confusability there
can
> be tolerated. >
***I think not. J. Pehrson
Message: 7556 Date: Sun, 05 Oct 2003 15:17:11 Subject: Re: Pitch Class and Generators From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "paulerlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/3829 * [with cont.]
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., graham@m... wrote:
> > > In-Reply-To: <B8C6611F.38DC%mark.gould@a...> > > > Mark Gould wrote:
> >
> > > > >From my recollection, I think the C12 group is isomorphic
with
> > > > relation to > > > > two of its sub groups C3 and C4
> >
> > > I don't know what that means.
> > > > C12 is isomorphic to the direct product C3 x C4. It can be
> expressed
> > in terms of a single generator of order 12, but its subgroups can
> be
> > expressed in terms of generators of degree 3 and 4 respectively,
> and
> > these also generate C12. In fact, since 3 and 4 are relatively
> prime,
> > any integer can be expressed as a linear combination of 3s and
4s,
> so
> > any 2^(k/12) can be expressed as a product of major and minor 12-
et
> > thirds, without the use of octaves.
> > yes, but to claim (as balzano did) that the fundamental importance
of
> the diatonic scale hinges on this fact is to pull the wool over the > eyes of the numerically inclined reader. the fact is that around
the
> time of the emergence of tonality in diatonic music, many musicians > advocated a 19- or 31-tone system in which to embed the diatonic > scale, and 12 won out only because of convenience. it is only with > the work of late 19th century russian composers that the cycle-3
and
> cycle-4 aspects of C12 became musically important. > > in fact, the diatonic scale emerged over and over again around the > world without any 'chromatic universe' whatsoever, let alone an
equal-
> tempered one. the important properties of the diatonic scale must,
i
> feel, be found in the scale itself, in whatever tuning it may be > found (with reasonable allowances for the ear's ability to accept > small errors) -- any 'chromatic totality' considerations should
wait
> until, and be completely dependent upon, the establishment of the > fundamental 'diatonic' entity upon which the music is to be based. > this was my approach in my paper, and more recently, in my
adaptation
> of fokker's periodicity block theory.
***This is extremely interesting and echos Easley Blackwood's work, I believe... J. Pehrson
Message: 7557 Date: Sun, 05 Oct 2003 04:37:48 Subject: Re: hey gene From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >
> > Just one more question. Can you prove that the 7-limit is the > > only one that works? What a strange thing.
> > Yes, but it hardly matters, since higher limit versions make > progressively less sense.
The algebraic proof is simple, so I'm going to give it despite the fact that it is both algebra and a proof. Ignoring the metric issue, we can represent octave classes of p-limit intervals by reduced monzos--tuples of prime exponents with the 2 ignored. If n is the number of odd primes in the p-limit, then these are n-1-tuples. We can represent (otonal or utonal) simplex chords by taking the sum of the vertices, which gives as an n-tuple such that the sum of the coefficients is +-1 mod n. If we add [-1, ..., -1] to all these tuples we move [1, ..., 1] to the origin, and now the sum of coefficients is 0 or 2 mod n. Only if n is 2 or 4 does this reduce to a group (0 or 2 mod 4 is just the even numbers, and of course 0 mod 2 means even), and since summing the coordinates is a homomorphism, only the 3-limit and the 7-limit can possibly have chords which form a group, and it is easily checked they both do.
Message: 7558 Date: Sun, 05 Oct 2003 15:32:04 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor" <gdsecor@y...> wrote: Yahoo groups: /tuning-math/message/3835 * [with cont.]
>> Here's something to keep in mind as we raise the prime limit. I
am
> sure that there are quite a few people who would think that making
a
> notation as versatile as this one promises to get is overkill.
****!!!! My last post. I
> think that such a criticism is valid only if its complexity makes
it
> more difficult to do the simpler things.
***And it *has*, in *my* opinion. Let's try to keep it simple
> for the ET's under 100 (as I believe we have been able to do so
far),
> keeping the advanced features in reserve for the power-JI composer > who wants a lot of prime numbers.
***I agree. How many *active* composers are actually using higher level ETs though. Names? Past 100?? Names? That should be the *last* accomodation on the list, despite it's intellectual fascination, in *my* opinion... Nobody *plays* music like that. At least not *living* musicians... If we build everything in from the
> start and do it right, then there will be no need to revise it
later
> and upset a few people in the process. >
***I'm already very upset... :)
> > > By the way, something else I figured out over the weekend is
how
> to
> > > notate 13 through 20 degrees of 217 with single symbols, i.e.,
> how to
> > > subtract the 1 through 8-degree symbols from the sagittal
apotome
> > > (/||\). The symbol subtraction for notation of apotome
> complements
> > > works like this: > > > > > > For a symbol consisting of: > > > 1) a left flag (or blank) > > > 2) a single (or triple) stem, and > > > 3) a right flag (or blank): > > > 4) convert the single stem to a double (or triple to an X); > > > 5) replace the left and right flags with their opposites
> according to
> > > the following: > > > a) a straight flag is the opposite of a blank (and vice versa); > > > b) a convex flag is the opposite of a concave flag (and vice
> versa).
> > > > You gotta admit this isn't exactly intuitive (particularly 5a).
I'm
> > more interested in the single-stem saggitals used with the
standard
> > sharp-flat symbols, but it's nice that you can do that.
> > Believe it or not, the logic behind 5a) is pretty solid, while it
is
> 5b) that is a bit contrived. The above is an expansion of what I > originally did for the 72-ET notation before any curved flags were > introduced. Allow me to elaborate on this. Consider the following: > > 81:80 upward is a left flag: /| > 33:32 upward is both flags: /|\ > so 55:54 upward is 33:32 *less* a left flag: |\ > Since an apotome upward is two stems with both flags: /||\ > then an apotome *minus 81:80* is the apotome symbol *less a left > flag*: ||\ > which illustrates how we arrive at a symbol for the apotome's > complement of 81/80 by changing /| to ||\ according to 4) and 5a) > above. > > Using curved flags in the 72-ET native notation to alleviate
lateral
> confusibility complicates this a little when we wish to notate the > apotome's complement (4deg72) of 64/63 (2deg72), a single *convex > right* flag. I was doing it with two stems plus a *convex left* > flag, but the above rules dictate two stems with *straight left*
and
> *concave right* flags. As it turns out, the symbol having a single > stem with *concave left* and *straight right* flags is also 2deg72, > and its apotome complement is two stems plus a *convex left* flag > (4deg72), which gives me what I was using before for 4 degrees. So > with a little bit of creativity I can still get what I had (and > really want) in 72; the same thing can be done in 43-ET. This is
the
> only bit of trickery that I have found any need for in divisions > below 100. >
***OK, this is starting to get a little beyond my expertise, I confess... But, still, the part that I *do* understand, the left- right flag business for 81/80 and 64/63... has not yet been satisfied...
> As you noted, it is nice that, given the way that we are developing > the symbols, this notation will allow the composer to make the > decision whether to use a single-symbol approach or a single-
symbols-
> with-sharp-and-flat approach. And the musical marketplace could > eventually make a final decision between the two.
***Heh... Does anyone *seriously* wonder about this outcome. The sharps and flats will "win" every time. With the tradition behind it, I can't see how this could even be brought forward as a proposition... So while we can
> continue to debate this point, we are under no pressure or
obligation
> to come to an agreement on it. >
***I disagree.
> > > I will prepare a diagram illustrating the progression of
symbols
> for
> > > JI and for various ET's so we can see how all of this is going
to
> > > look. > > > > > > Stay tuned!
> > > > Sure. This is fun.
> > More fun (if more complicated) than I had ever expected! > > --George
***I'm the only person not having so much fun :( I want something USABLE for my music!!! J. Pehrson
Message: 7559 Date: Sun, 05 Oct 2003 18:49:58 Subject: Re: Ekmelische Musik From: monz hi Joe, --- In tuning-math@xxxxxxxxxxx.xxxx "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx pitchcolor@a... wrote: > > Yahoo groups: /tuning-math/message/3708 * [with cont.] > >
> > > Ekmelic is a generic German term used to describe prime > > > harmonics greater than 5.
> > > > You mean ekmelisch; I don't know if that's exactly true. If I > > remember correctly it comes from the Greek words ek=out and > > melos=series so it means "out of the normal range". So in that > > sense it can be seen as the equivalent of Ivor Darreg's term > > "xenharmonic". The opposite term is emmelisch. > > > > Manuel>> > > > > Hi and thanks for the info; I meant to point out that the > > term is widely used in reference to just intonation rather > > than equal tunings, as your note verifies. "Ekmelisch" > > may be roughly equivalent to "xenharmonic" but I think > > it is important to point out that "ekmelisch" does > > actually refer to "harmonics;" hence "series," rather > > than to some arbitrary "unusual tuning." > > More specifically, it refers to harmonics which are above > > those associated with traditional Western music - those > > of 7 and beyond. Martin Vogel used the term to describe > > prime harmonics 7 and beyond in his books "the future of > > Music", "the number 7 in music", and "on the relations of > > tone" (all in German of course) The use of the term > > "ekmelisch" in other texts (Ernst Bindel et al) is > > consistent with this, however, the international > > conferences which were hosted by the late Herf-Richter > > were titled "Musik miot Mikrotönen, Ekmelische Musik," > > which would suggest that it was being used as a broader > > umbrella, especially since Ezra Sims was there and 72-equal > > played a major role in those Salzburg conferences. Anyone > > know what's going on over there nowadays? > > > > Aaron
> > > ***I'm assuming, then, that the Sims notation was used in > these Salzburg conferences??
no, Joe. Richter-Herf used his own form of 72edo notation, which is closer to my 72edo-HEWM. (read my HEWM page). Sims contributed to those conferences, so i imagine that both notations were encountered. i've posted about Herf here in the past: Yahoo groups: /tuning/messages/23086?expand=1 * [with cont.] (second half of the post) modern use of the terms "ekmelische" and "ekmelic" (German and English, respectively) is due to Herf. Herf's interesting theory did indeed have a JI/harmonic basis, but he used 72edo for notational purposes. -monz
Message: 7560 Date: Sun, 05 Oct 2003 21:20:51 Subject: base 12 From: Gene Ward Smith Here is a list of 11-limit intervals in 1728-et, base 12. Given the importance of 12 and 72, base 12, which succesively uses 12, 144, 1728, might conceivably be of some use notationally. 12/11 [1, 6, 1] 11/10 [1, 7, 10] 10/9 [1, 9, 11] 9/8 [2, 0, 6] 8/7 [2, 3, 9] 7/6 [2, 8, 0] 6/5 [3, 1, 11] 11/9 [3, 5, 8] 5/4 [3, 10, 4] 14/11 [4, 2, 1] 9/7 [4, 4, 3] 4/3 [4, 11, 9] 11/8 [5, 6, 2] 7/5 [5, 9, 11] 10/7 [6, 2, 1] 16/11 [6, 5, 10] 3/2 [7, 0, 3] 14/9 [7, 7, 9] 11/7 [7, 9, 11] 8/5 [8, 1, 8] 18/11 [8, 6, 4] 5/3 [8, 10, 1] 12/7 [9, 4, 0] 7/4 [9, 8, 3] 16/9 [9, 11, 6] 9/5 [10, 2, 1] 20/11 [10, 4, 2] 11/6 [10, 5, 11]
Message: 7561 Date: Sun, 05 Oct 2003 21:20:51 Subject: base 12 From: Gene Ward Smith Here is a list of 11-limit intervals in 1728-et, base 12. Given the importance of 12 and 72, base 12, which succesively uses 12, 144, 1728, might conceivably be of some use notationally. 12/11 [1, 6, 1] 11/10 [1, 7, 10] 10/9 [1, 9, 11] 9/8 [2, 0, 6] 8/7 [2, 3, 9] 7/6 [2, 8, 0] 6/5 [3, 1, 11] 11/9 [3, 5, 8] 5/4 [3, 10, 4] 14/11 [4, 2, 1] 9/7 [4, 4, 3] 4/3 [4, 11, 9] 11/8 [5, 6, 2] 7/5 [5, 9, 11] 10/7 [6, 2, 1] 16/11 [6, 5, 10] 3/2 [7, 0, 3] 14/9 [7, 7, 9] 11/7 [7, 9, 11] 8/5 [8, 1, 8] 18/11 [8, 6, 4] 5/3 [8, 10, 1] 12/7 [9, 4, 0] 7/4 [9, 8, 3] 16/9 [9, 11, 6] 9/5 [10, 2, 1] 20/11 [10, 4, 2] 11/6 [10, 5, 11]
Message: 7563 Date: Mon, 06 Oct 2003 18:58:44 Subject: Re: A common notation for JI and ETs From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor" <gdsecor@y...> wrote: > > Yahoo groups: /tuning-math/message/3835 * [with cont.] >
> >> Here's something to keep in mind as we raise the prime limit. I
am
> > sure that there are quite a few people who would think that
making a
> > notation as versatile as this one promises to get is overkill.
> > ****!!!! My last post. >
> > I > > think that such a criticism is valid only if its complexity
makes it
> > more difficult to do the simpler things.
> > ***And it *has*, in *my* opinion. > > Let's try to keep it simple
> > for the ET's under 100 (as I believe we have been able to do so
far),
> > keeping the advanced features in reserve for the power-JI
composer
> > who wants a lot of prime numbers.
> > ***I agree. How many *active* composers are actually using higher > level ETs though. Names? Past 100?? Names? That should be the > *last* accomodation on the list, despite it's intellectual > fascination, in *my* opinion... > > Nobody *plays* music like that. At least not *living* musicians... > > > If we build everything in from the
> > start and do it right, then there will be no need to revise it
later
> > and upset a few people in the process.
> > ***I'm already very upset... :) >
> > > > By the way, something else I figured out over the weekend is
how to
> > > > notate 13 through 20 degrees of 217 with single symbols,
i.e., how to
> > > > subtract the 1 through 8-degree symbols from the sagittal
apotome
> > > > (/||\). The symbol subtraction for notation of apotome
complements
> > > > works like this …
> > ***OK, this is starting to get a little beyond my expertise, I > confess... But, still, the part that I *do* understand, the left- > right flag business for 81/80 and 64/63... has not yet been > satisfied...
Joseph, I think you're getting a little carried away with yourself. This is the 7th posting you've made in rapid succession, and you seem to be under some sort of compulsion to make up for lost time by "replying" to things that we said over a year ago (and for which some of the specific details may not even apply anymore). Even when we start talking about something about which you have little or no interest, instead of snipping it out, you act as if we've gotten off the subject, and you start repeating something that you've already told us over and over about on the main list. You're coming off like a cross between a broken record and Mystery Science Theatre 2003, except that ...
> > > > I will prepare a diagram illustrating the progression of
symbols for
> > > > JI and for various ET's so we can see how all of this is
going to
> > > > look. > > > > > > > > Stay tuned!
> > > > > > Sure. This is fun.
> > > > More fun (if more complicated) than I had ever expected! > > > > --George
> > ***I'm the only person not having so much fun :(
... in Mystery Science Theatre at least the characters making the comments about the dialogue in the movie were having fun. If reading all of this is making you even more upset, then stop here. The point in having mentioned our development of the notation on tuning-math is so that you could *see* that we *have been* discussing a lot of the things that you're concerned about. But we've also been discussing things that *others* are concerned about (such as notating JI meaningfully without sacrificing precision, or about notating large- number ETs, or about notating groups of ETs in a reasonably consistent manner). If you don't think those other things are important, then you're entitled to your opinion ...
> I want something USABLE for my music!!! > > J. Pehrson
…, but please have a little consideration for others by not demanding that we sacrifice other things that may be important to them so that you can have the *optimal* 72-ET notation (which is what you've *really* been demanding), as opposed to something *usable* (as you claim above). If you can calm down and restrain yourself from firing off shotgun postings for at least a couple of days, then Dave and I may be ready to work with you to give you something that I believe will be *useable* for 72-ET. I believe that I understand what you need, but Dave and I need a little time to discuss how best to address the issues you raised. --George Love / joy / peace / patience ...
Message: 7565 Date: Mon, 06 Oct 2003 21:32:32 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
> Thanks. It looks like your web page should give me the basic
concept.
> (I've studied exterior algebra (Grassman) too.)
You might see what you can glean from my site (www.xenharmony.org) also.
Message: 7566 Date: Mon, 06 Oct 2003 21:37:52 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
> I really hate to ask, but what do wedgies have to do with mapping > generators to primes?
I take the wedgie, and from it generate what I call the subgroup vals. Then I hermite-reduce these, and apply a further reduction to make the generators of the generator/period pair as small, greater than one, as possible. This gives a standarized period/generator for the temperament in question.
Message: 7569 Date: Tue, 07 Oct 2003 18:52:51 Subject: Re: Please remind me From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
> So it's not exactly like falling off a log? (haha)
Since Maple has a hermite reduction algorithm, it wasn't hard to code. I take it that you
> run minimax or RMS after that, to get an exact generator.
Among other possibilities. For minimax I set up a linear programming problem and solve it using Maple's simplex implementation. What is hard or easy depends a lot on what functions you have already available.
Message: 7570 Date: Tue, 07 Oct 2003 01:16:33 Subject: [tuning] Re: Polyphonic notation From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> If the pretenses don't matter, we can let history and/or
sensicality
> >> prevail. > >> > >> -Carl
> > > >meaning?
> > Whatever we think it means. I tend to see history as in, the > last few years on this list -- our contribution is substantial. > Given the term MOS, I don't see much of a role for sensicality
what does sensicality mean?
> but it seems the term applies just as well to the fractional- > octave case.
after kraig told us it doesn't, daniel and other chimed in in agreement. these are people who know the originator of the term, erv wilson, far better than i do. so i'm inclined to revert to my previous usage as it'll make communicating with the erv-ites easier, and these are pretty much the main people who use the term anyway.
Message: 7571 Date: Tue, 07 Oct 2003 01:27:02 Subject: Re: Pitch Class and Generators From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "paulerlich" <paul@s...> wrote: > > Yahoo groups: /tuning-math/message/3829 * [with cont.] > >
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> > > --- In tuning-math@y..., graham@m... wrote:
> > > > In-Reply-To: <B8C6611F.38DC%mark.gould@a...> > > > > Mark Gould wrote:
> > >
> > > > > >From my recollection, I think the C12 group is isomorphic
> with
> > > > > relation to > > > > > two of its sub groups C3 and C4
> > >
> > > > I don't know what that means.
> > > > > > C12 is isomorphic to the direct product C3 x C4. It can be
> > expressed
> > > in terms of a single generator of order 12, but its subgroups
can
> > be
> > > expressed in terms of generators of degree 3 and 4
respectively,
> > and
> > > these also generate C12. In fact, since 3 and 4 are relatively
> > prime,
> > > any integer can be expressed as a linear combination of 3s and
> 4s,
> > so
> > > any 2^(k/12) can be expressed as a product of major and minor
12-
> et
> > > thirds, without the use of octaves.
> > > > yes, but to claim (as balzano did) that the fundamental
importance
> of
> > the diatonic scale hinges on this fact is to pull the wool over
the
> > eyes of the numerically inclined reader. the fact is that around
> the
> > time of the emergence of tonality in diatonic music, many
musicians
> > advocated a 19- or 31-tone system in which to embed the diatonic > > scale, and 12 won out only because of convenience. it is only
with
> > the work of late 19th century russian composers that the cycle-3
> and
> > cycle-4 aspects of C12 became musically important. > > > > in fact, the diatonic scale emerged over and over again around
the
> > world without any 'chromatic universe' whatsoever, let alone an
> equal-
> > tempered one. the important properties of the diatonic scale
must,
> i
> > feel, be found in the scale itself, in whatever tuning it may be > > found (with reasonable allowances for the ear's ability to accept > > small errors) -- any 'chromatic totality' considerations should
> wait
> > until, and be completely dependent upon, the establishment of the > > fundamental 'diatonic' entity upon which the music is to be
based.
> > this was my approach in my paper, and more recently, in my
> adaptation
> > of fokker's periodicity block theory.
> > > ***This is extremely interesting and echos Easley Blackwood's work,
I
> believe... > > J. Pehrson
glad you got to read my comments, even if only a year and a half late . . .
Message: 7572 Date: Tue, 07 Oct 2003 04:36:38 Subject: Re: A common notation for JI and ETs From: Joseph Pehrson --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> Yahoo groups: /tuning-math/message/7004 * [with cont.]
> Joseph, > > I think you're getting a little carried away with yourself.
***Indeed... Well, thanks anyway for answering, George. My guess is that I'm not going to be able to find the time to continue this in any case, so we're a bit "saved by the bell..." as it were :) Anyway, good luck to you and to Dave with it all!!!! Joseph
Message: 7574 Date: Wed, 08 Oct 2003 20:10:51 Subject: sorry gene -- couldn't reach you offlist From: Paul Erlich for some reason e-mail to your address bounces -- in case you're reading this, i just wanted to say: mathematician needed! Yahoo groups: /Polytopia/message/194 * [with cont.]
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