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Message: 7250 Date: Mon, 11 Aug 2003 13:26:42 Subject: Re: Bosanquet keyboards and linear temperamentss From: Graham Breed Carl Lumma wrote:
> "12-based"? Bosanquet issued layouts for negative, positive, and > doubly-positive temperaments, IIRC.
"Negative" means the fifth is smaller then that of 12-equal. "Positive"
means the fifth is greater than that of 12-equal. "Doubly-positive"
means the fifth is larger than that of 12-equal, and the Pythagorean
comma divides in 2. This is all based on 2.
I had a brief look at Bosanquet's book, and didn't see a doubly-positive
mapping. But he did class all equal temperaments using this scheme. To
get an ET's positivity, take the number of notes modulo n. 17 is singly
positive, and 17 = 5 mod 12. An n note scale is mthly positive if n =
5*m mod 12. For singly negative scales, m=-1 and -1*f = 7 modulo n. So
the singly negative scales are 7, 19, 31, 43, ...
Wilson in one of the Xenharmonikon 3 papers extended this scheme to
singly positive and negative scales with respect to 5 and 7, which
allows more ETs to be played on a Bosanquet keyboard. Other files in
the Wilson Archive show he has considered more general cases as well.
Graham
Message: 7251 Date: Mon, 11 Aug 2003 22:38:13 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> is the inner product defined over a triangular-taxicab metric?
That's a normed vector space, but for a (positive definite) inner product, what you get is a standard Euclidean vector space.
Message: 7252 Date: Mon, 11 Aug 2003 13:42:00 Subject: Re: tctmo! From: Graham Breed Carl Lumma wrote:
> Everything we do starts from the fundamental assumption of JI. > Or so it seems to me.
The CGI won't do it yet, but my Python and OCaml modules are quite capable of working with an arbitrary set of prime intervals. The test script finds linear temperaments for Rayleigh's tubulong formula, as given in one of Brian McLaren's Xenharmonikon articles. The "prime intervals" are simply a minimal set of intervals required to construct every interval in the scale. For prime-limit JI, these are the logarithms of the relevant prime numbers (modulo the log of 2 for octave equivalence). In terms of group theory, they're the generators of a finitely generated group and provide a homomorphism into the reals. The only special treatement given to JI is that the prime intervals are supplied for you, there's a formula for expressing any ratio in terms of them and you can get a list of consonances that correspond to an odd limit.
> Za?
Is that supposed to mean something?
>>"Straightness" is another thing I never understood.
> > It has to do with the angle between the commas. If A and B are > commas that vanish, and a and b are their lattice points, then the > interval C between a and b also vanishes. The thing is, A and B > could both be simple, but if the angle between them is wide, C could > still be complex. So you have to account for this in a heuristic.
So it's an angle on the lattice? It's something that would be nice to
have (and the heuristic certainly won't work without) but I've never
been able to calculate it.
Graham
Message: 7253 Date: Mon, 11 Aug 2003 20:52:20 Subject: Re: tctmo! From: Carl Lumma
>> It strikes me as quite possible that group theory is a better >> basis from which to explain this stuff, but can a group theory >> crash course be fit into a short document?
> >The kind of groups we are most interested in are free abelian groups >of finite rank, and they can be equated with row vectors of integers.
I don't see how this answers my question, but I understand it, at least on some level. But, once again, isn't this stuff assumed by the basic ratio math everybody already uses? Wouldn't anything but an abelian group be catastrophic? If so, I don't see why it's so important. -Carl
Message: 7254 Date: Mon, 11 Aug 2003 23:44:39 Subject: Re: tctmo! From: Graham Breed Paul Erlich wrote:
> the pythagorean diatonic is improper but would seem to have the > property you're trying to describe. so would blackjack . . .
I don't agree that the Pythagorean diatonic is improper. The two sizes
of tritone are too close for most listeners to order them correctly when
disjoint. And I don't agree that Blackjack has the property of diatonic
modulation.
Now Carl, I don't think Part III of Rothenberg's series should be in the
references. All you consider is propriety and efficiency.
The result of tempering out all commas can be thought of geometrically
as collapsing the (hyper)plane into a (hyper)torus. Periodicity blocks
are different because no commas are tempered out, so one block is
different to another.
When you talk about lattice distance, you should say how the distance is
measured.
Graham
Message: 7255 Date: Mon, 11 Aug 2003 22:45:39 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >
> > is the inner product defined over a triangular-taxicab metric?
> > That's a normed vector space, but for a (positive definite) inner > product, what you get is a standard Euclidean vector space.
i must have phrase my question incorrectly.
Message: 7256 Date: Mon, 11 Aug 2003 22:47:18 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote: >
> > the pythagorean diatonic is improper but would seem to have the > > property you're trying to describe. so would blackjack . . .
> > I don't agree that the Pythagorean diatonic is improper. The two
sizes
> of tritone are too close for most listeners to order them correctly
when
> disjoint.
how about the 17-equal diatonic?
> And I don't agree that Blackjack has the property of diatonic > modulation.
where does it fail?
> The result of tempering out all commas can be thought of
geometrically
> as collapsing the (hyper)plane into a (hyper)torus.
that's what i told carl in my initial reply.
Message: 7257 Date: Mon, 11 Aug 2003 23:11:16 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > > wrote: > >
> > > is the inner product defined over a triangular-taxicab metric?
> > > > That's a normed vector space, but for a (positive definite) inner > > product, what you get is a standard Euclidean vector space.
> > i must have phrase my question incorrectly.
The short answer would be "no".
Message: 7258 Date: Mon, 11 Aug 2003 23:25:08 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote: >
> > how about the 17-equal diatonic?
> > I don't know. But Pythagorean's too marginal. >
> >>And I don't agree that Blackjack has the property of diatonic > >>modulation.
> > > > where does it fail?
> > The large and small scale steps are so different that if the
pattern
> changes a melody is qualitatively different.
that seems subjective. how about the 17-equal diatonic case? you don't know?
>
> > that's what i told carl in my initial reply.
> > If that was to "tcmo!" I don't have it.
no, it was "review . . ."
Message: 7259 Date: Mon, 11 Aug 2003 23:31:54 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> I think > both are valid metrics, but complexity isn't an inner product.
who said it was?
> What's special about kees's lattice? Triangular-taxicab distances
on an
> fcc lattice won't give the right results for 9-limit intervals.
that's where "wormholes" come in.
Message: 7260 Date: Tue, 12 Aug 2003 09:50:36 Subject: Re: tctmo! From: Carl Lumma
>>>and complexity is the smallest >>>number of intervals in the relevant odd limit that make up the >>>comma.
>> >> This sounds like taxicab distance. Paul uses d, but d is >> supposed to be like taxicab distance (at least, on the right >> lattice)...
> >It'd be a taxicab distance on the right lattice, assuming the roads form >an n-dimensional lattice triangular in cross section with wormholes. >What's d?
Denominator. -Carl
Message: 7261 Date: Tue, 12 Aug 2003 09:51:47 Subject: Re: tctmo! From: Carl Lumma
>> What's an interval vector?
> >A pitch difference expressed as a list of integers, which is much the >same as a free albelian group.
So you need some kind of space. How do you 'factor' irrational intervals to get this space? -Carl
Message: 7262 Date: Tue, 12 Aug 2003 01:18:41 Subject: Re: tctmo! From: Graham Breed Gene Ward Smith wrote:
> What pitch difference? A comma has a size, namely itself; we can take > the log of that also, of course.
The size of a comma depends on the metric you apply. An interval is the ratio of two frequencies, or the difference between two pitches, where pitch is the logarithm of frequency. To get the right error heuristic, the error has to be as a pitch difference.
> A comma is made up of intervals? I thought it *was* an interval.
Yes, an interval is an element of an abelian group, remember? So intervals can be produced from other intervals. The interval 81:80, for example, can be made up of 3:2 * 3:2 * 3:2 * 3:2 * 1:5. But it's simpler to break it down into 3:5 * 3:4 * 3:2 * 3:2. That involves 4 intervals in the 5-limit, so 81:80 has a 5-limit complexity of 4.
> If you want an inner product, what about the one I use to define > geometric complexity?
Yes, that'll do. But so would a standard dot product of octave-specific
vectors, scaled by the size of the prime intervals. That would be
simpler, and give much the same result for sufficiently small (as a
pitch difference) commas.
Graham
Message: 7263 Date: Tue, 12 Aug 2003 09:53:52 Subject: Re: tctmo! From: Carl Lumma
>Homomorphisms are important, and we may as well call them such. But >we'll need to explain them in musical terms. Something like "each >just interval has a counterpart in the temperament, and adding >intervals should give the same result whether the mapping occurs >before or after the addition".
Hey, that's good. Mind if I use it? -Carl
Message: 7264 Date: Tue, 12 Aug 2003 00:25:24 Subject: Re: Bosanquet keyboards and linear temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
> > We can now generalize this for any linear temperament...
> > Erv did that a long time ago.
I'm sure he would have, however he apparently hasn't explained it very well. I didn't try to explain it either, but instead just _implemented_ it -- in a spreadsheet that draws the layout, given the period, the generator, the number of notes and an aspect-ratio parameter. See Yahoo groups: /tuning/message/25742 * [with cont.] The URL for the spreadsheet given in that message will soon be obsolete. The current location is The page cannot be found * [with cont.] (Wayb.) The spreadsheet doesn't do the colouring, but that's still somewhat a matter of taste. To generalise that, I'd colour white the most central mode of that proper MOS whose notes per octave is between 5 and 9 and is closest to the magic number 7 (I'd use an improper MOS if there's no proper MOS with 5 to 9 notes and failing that I'd use the smallest MOS with 10 or more notes per octave) and then expand that symmetrically to the next proper MOS with black, then the next with red, then blue, yellow, green, orange, purple as required.
Message: 7265 Date: Tue, 12 Aug 2003 19:54:21 Subject: Re: tctmo! From: Graham Breed
> Hey, that's good. Mind if I use it?
Yes, of course.
Message: 7266 Date: Tue, 12 Aug 2003 01:07:21 Subject: Re: Bosanquet keyboards and linear temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <D.KEENAN@U...> wrote:
> The page cannot be found * [with cont.] (Wayb.)
I'm planning to put up a page on Bosanquet lattices on xenharmony, and would like to link to this. I presume that is OK? We really need to set up a system of links, and I still like the idea of a web ring.
Message: 7267 Date: Tue, 12 Aug 2003 20:43:47 Subject: Re: tctmo! From: Graham Breed Carl Lumma wrote:
> So you need some kind of space. How do you 'factor' irrational > intervals to get this space?
You can take the most prominent partials, relative to the fundamental, as a set of prime intervals, to take the place of prime numbers for JI of harmonic timbres. Sharp minima in the dissonance curve will usually be simple combinations of these intervals. If you don't have dissonance curve, taking a cross set will do. For example, on p.170 of Sethares' magnum opus, a typical spectrum for Swstigitha sarons is given as f, 2.76f, 4.72f and 5.92f. I'll take 2.76 as the equivalence interval and use a standard two-dimensional cross set (the equivalent of the 5-limit)
>>> import temper, math >>> swastigitha = temper.PrimeDiamond(2) >>> swastigitha.primes = [math.log(x)/math.log(2.76) for x in
(4.72, 5.92)]
So that's it, a set of consonances is defined according to the empirical
spectrum. Temperaments can be generated from it like any other set of
consonances
>>> [et.basis[0] for et in temper.getLimitedETs(swastigitha)]
[1, 4, 5, 9, 11, 13, 15, 17, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32]
>>> temper.getLinearTemperaments(temper.getLimitedETs(swastigitha))[0]
11/36, 305.7 milli-equivalence generator basis: (1.0, 0.30573972381117814) mapping by period and generator: [(1, 0), (0, 5), (-1, 9)] mapping by steps: [(23, 13), (35, 20), (40, 23)] highest interval width: 9 complexity measure: 9 (10 for smallest MOS) highest error: 0.000170 (0.170 milli-equivalences) unique The equivalence interval here is 1757.6 cents, so the generator of that linear temperament is 537.4 cents, and the worst error 0.3 cents. Alternatively, using a regular octave as the equivalence interval, to fit better with other instruments
>>> swas_oct = temper.PrimeDiamond(3) >>> swas_oct.primes = [temper.log2(x) for x in (2.76, 4.72, 5.92)] >>> [et.basis[0] for et in temper.getLimitedETs(swas_oct)]
[4, 5, 8, 9, 12, 13, 17, 19, 21, 22, 28, 30, 32, 34, 37, 39, 40, 41, 43, 46]
>>> temper.getLinearTemperaments(temper.getLimitedETs(swas_oct))[0]
2/7, 101.0 milli-equivalence generator
basis:
(0.33333333333333331, 0.10099919983161132)
mapping by period and generator:
[(3, 0), (5, -2), (7, -1), (8, -1)]
mapping by steps:
[(12, 9), (18, 13), (27, 20), (31, 23)]
highest interval width: 2
complexity measure: 6 (9 for smallest MOS)
highest error: 0.006523 (6.523 milli-equivalences)
101 millioctaves is 121.2 cents.
Graham
Message: 7268 Date: Tue, 12 Aug 2003 01:09:15 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote: >
> > What pitch difference? A comma has a size, namely itself; we can
take
> > the log of that also, of course.
> > The size of a comma depends on the metric you apply.
Ah. We're talking norms, then.
Message: 7270 Date: Tue, 12 Aug 2003 22:59:07 Subject: Re: Comments about Fokker's misfit metric From: Graham Breed Carlos wrote:
> I have been reading Fokker's papers "Les mathematiques et la musique" > (1947) and the more recent "On the expansion of the musician's realm of > harmony" (1967).
Those are online at
Adriaan Fokker: Les tempéraments égaux * [with cont.] (Wayb.)
Stichting Huygens-Fokker: On the Expansion of ... * [with cont.] (Wayb.)
Sorry for the digression, but Paul! look!
The first paper clearly associates a periodicity block with 31-equal.
Here's a quote:
"""
Ces trois vecteurs nous définissent dans le réseau un parallélépipède,
une base de périodicité. Seuls les notes intérieures à ce
parallélépipède seront indépendantes.
Leur nombre est défini par son volume, qui, par les méthodes de la
géométrie analytique, se trouve à l'aide du déterminant formé avec les
coordonnées des arêtes:
| 4 -1 0 |
| 2 2 -1 | = 31
| 1 0 3 |
Cette méthode nous fournit le tempérament égal de trente-et-un
cinquièmes de ton, tel qu'il a été calculé par Christiaan Huygens.
"""
The last sentence translates as "This method provides us with the equal
temperament of thirty-one fifth tones, as was calculated by Christiaan
Huygens."
> Now it occurs to me that the metric used weights too much the
significance
> of the upper intervals, like the seventh, the eleventh and so for. I
would
> say that it could make more sense, specially if one is thinking of doing > tonal music, to really create a metric that will weigth more the most > basic intervals. Something like
If you're thinking of tonal music, it makes sense to restrict the search to meantones. That is, temperaments where 81:80 is tempered out. These include 12, 19, 31, 43, 50 and 55 note equal temperaments.
> If you do use this metric just looking to the three intervals indicated, > the result of Fokker still holds and the 31 equally tempered scale is > still the first that produces a considerable reduction in the metric or > distance value. But my guess is that if the same procedure is used with > more harmonics then the result could vary significantly depending on the > weight factors selected.
I'm all for treating intervals equally. If you care about it, it should
be as much in tune as possible. I also find that more complex intervals
have to be better tuned to be comprehensible. But that's all a matter
of personal preference.
However, I also optimize on complete odd limits, whereas Fokker is here
only using the prime numbers. When you hit the 9-limit, 3 is
automatically weighted higher than 5 and 7.
Although 31-equal doesn't have such a good fifth, it still compares well
with 41 in the 11-limit. I haven't checked the detail of Fokker's
method, but probably it's the prime bias showing through here.
31-equal's approximation to 11:8 is 9.4 cents out, but 41-equal is only
4.8 cents out. In both cases, these are worse than the approximations
to 3, 5 and 7. It's easy to see how 31 comes out looking a lot better
than 41.
But for the 11-limit as a whole, the worst error in 31-equal is 11.1
cents, for 9:5. The worst error for 41-equal is 10.6 cents for 11:10
which is only slightly better. So including these second-order
intervals, even when 9 is included as well, 31-equal is roughly as good
as 41-equal, when you take the difference in complexity into account.
Perhaps somebody could repeat Fokkers calculations with all 11-limit
intervals to see what the result would be.
Graham
Message: 7271 Date: Tue, 12 Aug 2003 23:05:21 Subject: Bosanquet lattices up on xenharmony From: Gene Ward Smith This is among the new pages now up. Check it out if interested, or if you failed to get Yahoo's Photos area to work for you.
Message: 7272 Date: Tue, 12 Aug 2003 23:58:33 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Here's the doc... > >> > >> * [with cont.] (Wayb.)
> > > >what happened to the corrections/reactions i already posted??
> > They were added. > > -C.
not very well.
Message: 7273 Date: Tue, 12 Aug 2003 03:27:44 Subject: Re: Bosanquet keyboards and linear temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <D.KEENAN@U...> > wrote: >
> > The page cannot be found * [with cont.] (Wayb.)
> > I'm planning to put up a page on Bosanquet lattices on xenharmony, > and would like to link to this. I presume that is OK?
Sure, no problem.
> We really need to set up a system of links, and I still like the idea > of a web ring.
I'm afraid I prefer trees to rings (you don't end up where you've already been without knowing it). I think of John Starret's site as the root. Microtonalists</A> Those marked with an ... * [with cont.] (Wayb.) Ask John to list you there and link to your site. I've not described the algorithm in words anywhere before now, although I hope I made it easy for anyone to reverse engineer the spreadsheet. Here it is in a nutshell: I calculate the pitches in cents (not octave reduced) for sufficiently many chains of generators (36 in the spreadsheet), with the centers of the chains spaced out by periods, and each chain having N/2 notes on either side of its center. So these pitches are calculated in an array indexed by period-number (+-big_number) in one dimension and generator-number (+-N/2) in the other dimension. To plot the pitches onto the keyboard, the left/right coordinate (from the players point of view) is simply the pitch in cents, and the front back coordinate is simply the pitch's generator-number multiplied by the aspect ratio parameter. The lines joining the dots on the plot show the chains of generators. Given that one usually has to tweak not only the aspect ratio parameter but also the generator size in order to obtain an equilateral hexagonal layout, it strikes me that there are only a finite number of distinct hexagonal layouts corresponding to linear temperaments having a generator between say 1/12 and 1/2 of the period. I'd be interested to know how many, and how far apart their generators typically are.
Message: 7274 Date: Tue, 12 Aug 2003 17:16:19 Subject: Re: tctmo! From: Carl Lumma
>> So you need some kind of space. How do you 'factor' irrational >> intervals to get this space?
> >You can take the most prominent partials, relative to the fundamental, >as a set of prime intervals, to take the place of prime numbers for JI >of harmonic timbres.
Ok, sure. I'll take out "just intonation" and replace it with "consonant intervals". -Carl
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