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Message: 7275 Date: Tue, 12 Aug 2003 03:47:34 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >It depends on whether you are talking to a mathematician or not.
> > Gene, > > 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.
Message: 7276 Date: Tue, 12 Aug 2003 05:22:35 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> 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.
One important reason is that you want the mechanics of mappings (homomorphisms) to be available. There's an adage in mathematics that the morphisms are more important that the objects, and that holds here. It's not very exciting to know that an equal temperament considered as a group is isomorphic to the integers; it really requires a mapping (telling us, at the very least, where the octave is) to make any sense. The mappings also have kernels, which leads us to commas.
Message: 7277 Date: Tue, 12 Aug 2003 10:48:18 Subject: Re: tctmo! From: Graham Breed Carl Lumma wrote:
> 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.
But that's why it's important! Groups are intended to generalize
arithmetic operations. The positive rationals under multiplication do
constitute a group. Unless you stay with JI of harmonic timbres, you
need to consider a more general case. Tempered intervals can be added
as well, but they don't work with ratio math. Abelian groups constitute
a good level of abstraction that wasn't invented for us.
There are some other properties of rational numbers we still need. The
number itself represents the frequency ratio between two notes, so we
need a mapping from notes to real numbers representing frequency or
pitch. For JI and regular temperaments, this mapping is a homomorphism
of intervals. The complexity of a ratio gives some idea of dissonance,
but this doesn't really generalize outside of JI, and is one of the
things we're currently discussing. The difference between rationals
gives the beat ratio. In the general case, that's a function of
frequencies.
The thing is, although group theory is applicable, I don't know if it's
useful enough in this context for people who don't already know it. The
same things can be expressed using matrix algebra, which also allows us
to solve a set of commas to get a temperament. You have to know about
either matrix adjoints or wedge products to do this, but there's nothing
so far that requires group theory. It only provides a language for
expressing certain generalizations.
Graham
Message: 7278 Date: Tue, 12 Aug 2003 00:11:26 Subject: Re: tctmo! From: Graham Breed 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's what i told carl in my initial reply.
If that was to "tcmo!" I don't have it.
Graham
Message: 7279 Date: Tue, 12 Aug 2003 11:02:49 Subject: Re: tctmo! From: Graham Breed Me:
>>The error heuristic I use is size/complexity where size is the >>magnitude of the pitch difference
Carl:
> That sounds the same as Paul's heuristic.
I think Dave used it first. Paul's heuristic is an approximation assuming the comma to be a ratio.
>>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?
> You seem to like worst error more than average error. Any reason?
In this case, worst error is what the heuristic works for. I'm using
RMS error in software because it's easier to optimize for.
Graham
Message: 7280 Date: Tue, 12 Aug 2003 10:04:21 Subject: Re: tctmo! From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> The thing is, although group theory is applicable, I don't know if
it's
> useful enough in this context for people who don't already know it.
Academic theorists have taken up groups, so it will not be utterly unfamiliar to people coming from that quarter.
Message: 7281 Date: Tue, 12 Aug 2003 00:13:23 Subject: Re: tctmo! From: Graham Breed
> If that was to "tcmo!" I don't have it.
Or even "tctmo!"
Message: 7282 Date: Tue, 12 Aug 2003 11:19:27 Subject: Re: tctmo! From: Graham Breed Carl Lumma wrote:
> This is about music, right?
It's about tuning systems, which may be used to make music. There are already tuning systems derived from inharmonic timbres. A general method of finding temperaments will be most useful when considering timbres that haven't been considered before.
> What's an interval vector?
A pitch difference expressed as a list of integers, which is much the
same as a free albelian group.
Graham
Message: 7283 Date: Tue, 12 Aug 2003 12:05:10 Subject: Re: tctmo! From: Graham Breed Gene Ward Smith wrote:
> Academic theorists have taken up groups, so it will not be utterly > unfamiliar to people coming from that quarter.
So we are intending this "for dummies"? ;-)
It wouldn't do any harm to point out that directed intervals (remember
the "directed"; I usually don't, and it is important) form an abelian
group and the specific representation we use is of a free abelian group.
If people don't understand that, they can ignore it -- possibly it can
be in a footnote.
Beyond that, we're already using the term "generator". 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".
The link with unison vectors and the kernel of a homomorphism can be in
a footnote.
Equal temperaments as cyclic groups is in Balzano (1980) so we'll have
to reference it!
Otherwise, what group theory do you want?
Graham
Message: 7284 Date: Tue, 12 Aug 2003 00:26:57 Subject: Re: tctmo! From: Graham Breed Carl Lumma wrote:
> Ok, ok. But this is a caveat. We've got JI in our brains and one > can only go so far from it.
The theory shouldn't have to rely on empirical details about the human brain.
> It means "?".
The heuristics involves numerators and denominators, and interval vectors in general don't have numerators and denominators. But, as I saw one of them derived, it's only an approximation to something that you can calculate directly for all interval vectors as I use them. So there's an avoidable loss of generality.
> The heuristic works fine without it for linear temperaments. But > for more commas, one needs some way of taking into acount the > 'difference vector(s)'. Straightness is Paul's idea, but I'm not > sure he ever suggested a way to calculate it. Maybe he would like > to say something here.
For 5-limit linear temperaments.
Graham
Message: 7285 Date: Tue, 12 Aug 2003 12:35:54 Subject: Re: tctmo! From: Graham Breed Gene Ward Smith wrote:
> Ah. We're talking norms, then.
The size of an interval as a pitch difference is a norm but not a
p-norm. In general terms, we're working with directed intervals so we
need the sign as well. That isn't a norm. It is really the physics
definition of a vector -- a magnitude and a direction. The norm gives
us the magnitude. If the direction is the one-dimensional equivalent of
an angle, what we need is a metric, the way I was taught:
"... the notions of /length/ and /angle/ are represented by a /metric/."
(Ian D. Lawrie, "A Unified Grand Tour of Theoretical Physics", pp. 29-30)
For a regular tuning, we need a homomorphism into R, which can be
interpreted as a pitch difference.
For a well temperament, the mapping has to be from notes to pitches.
The size of an interval depends on what note it starts on, so it isn't a
homomorphism.
Graham
Message: 7286 Date: Tue, 12 Aug 2003 00:29:37 Subject: Re: tctmo! From: Graham Breed Paul Erlich wrote:
> we haven't yet been able to pin down the tempering rules on which the > error heuristic works, though it seems we must use the triangular- > taxicab metric on kees van prooijen's lattice to get the complexity > heuristic to work.
The error heuristic I use is size/complexity where size is the magnitude
of the pitch difference and complexity is the smallest number of
intervals in the relevant odd limit that make up the comma. I think
both are valid metrics, but complexity isn't an inner product. You
could approximate it by straight line distance on a triangular lattice
or rectangular octave-specific lattice.
The error measured is exactly the worst case of the worst tuning of any
interval in the relevant limit for an optimized temperament with the
given comma.
What's special about kees's lattice? Triangular-taxicab distances on an
fcc lattice won't give the right results for 9-limit intervals.
Weighting the 3 direction as half the size of the others will
approximate them (probably as well as your heuristic). Straight line
distances with scaled axes won't be much different, and are probably the
best bet as they are inner products, so we can get angles from them.
Graham
Message: 7287 Date: Tue, 12 Aug 2003 00:30:13 Subject: Re: tctmo! From: Graham Breed Paul Erlich wrote:
> that seems subjective. how about the 17-equal diatonic case? you > don't know?
Yes, it's subjective.
> no, it was "review . . ."
Right!
Graham
Message: 7291 Date: Wed, 13 Aug 2003 09:51:24 Subject: Re: Comments about Fokker's misfit metric From: monz@xxxxxxxxx.xxx hi Carlos, i haven't really followed this thread, but when i saw this i thought i could help illustrate something ...
> From: Carlos [mailto:garciasuarez@xx.xxxx > Sent: Wednesday, August 13, 2003 8:38 AM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: Re: [tuning-math] Re: Comments about Fokker's misfit metric > > > Graham, thanks for your comments. > > Ok, if I understand you and Paul allright you suggest > to include many more intervals in the metric no[t] just > the fundamental prime intervals of Fokker. I am not > sure what the physical sense of that is. As Paul points > out, in the case of the minor third, you certainly would > like to keep it in the metric as it is a nice consonant > ratio (for our ears today).
the point of using the minor-3rd as well as the major-3rd
and 5th (in the 5-limit) is that *all* other intervals
can be calculated as various combinations of those three.
in the 7-limit, in addition to the primary 3:2, 5:4, and
7:4, you would also need 6:5, 7:6, and 7:5.
it can be demonstrated on a lattice as follows:
(use "Expand Messages" mode if viewing on the stupid
Yahoo web interface)
5-limit
-------
primary:
5:4
/
/
/
/
1:1 ------- 3:2
primary and secondary:
5:4
/ \
/ \
/ \
/ \
1:1 ------- 3:2
7-limit
-------
primary:
5:4
/
/
/ 7:4
/.'
1:1 ------- 3:2
primary and secondary:
5:4
/| \
/ | \
/ 7:4 \
/.' '. \
1:1 ------- 3:2
it can be seen easily on the lattices that
adding lines to illustrate the secondary intervals
"completes" or "closes" the structure.
-monz
Message: 7292 Date: Wed, 13 Aug 2003 18:48:54 Subject: Re: Comments about Fokker's misfit metric From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:
> Graham, thanks for your comments. > > Ok, if I understand you and Paul allright you suggest to include
many more
> intervals in the metric no just the fundamental prime intervals of
Fokker.
> I am not sure what the physical sense of that is. As Paul points
out, in
> the case of the minor third, you certainly would like to keep it in
the
> metric as it is a nice consonant ratio (for our ears today). > > At any rate, following your suggestion I have computed the misfit > (unweigheted misfit that is, based on a simple cuadratic mean) for
all non
> repetitive ratios up to the 11th limit. This means that when an
interval
> has a complementary to the octave, I have only retained one of them
(like
> the fifth and not the fourth and so on). > > Also I have enforced that the octave should be absolutely just.
Fokker
> actually discusses this aspects and he indicate that he sees no
reason to
> treat the octave any different. This is not my case I enforce the
octave
> to be exact and all the other intervals to be approximate. > > The intervals I have considered then are > > Just fifth 3/2 > Just mayor third 5/4 > Just minor third 6/5 > Harmonic seventh 7/4 > Subminor fifth 7/5 > Subminor third 7/6 > Supersecond 8/7 > Major tone 9/8 > Super major third 9/7 > Acute minor seventh 9/5 > Trumpet interval 11/10 > The 11th harmonic 11/8 > Meshaqah quartertones 11/6 > Unamed_1 11/9 > Unamed_2 11/7 > > The names are those of Ellis. Interestingly I could not find a name
for the
> two last ones. > > I have then computed the misfit in four cases: > > - looking at the 5th, the major 3rd and the harmonic 7th > - looking at all the above + the minor 3rd > - looking at all the above + 7 and 9 limit ratios > - looking at all the above plus all the 11 limit ratios > > The results still show that the 31 equally tempered scales is a
very good
> choice in all cases. However, when one looks to more and more
ratios
> (like in the fourth case above) the option of 41 and 53 become more > attractive.
actually, 53-equal suffers from an additional problem with this type of calculation, whether fokker's original or modified as you have above. the problem is "inconsistency". if you consider each 11-limit interval's best approximation in 53, you will think that you can approximate 4:7 with 43 steps, 4:11 with 77 steps, and 7:11 with 35 steps. unfortunately, if you try to approximate a 4:7:11 triad in 53- equal, you'll run into the problem that 43 + 35 does not equal 77 steps. thus you'll have to use a worse approximation for at least one of these intervals than your original analysis suggested you'd need.
> My conclusion out of this is that 31 ET is a very nice compromise
of
> accuracy and compexity.
certainly. it's also a meantone which is hugely important for the performance of western music. you'll find that 72-equal is much more accurate than 31 for most of the intervals you've looked at, and indeed 72 has gotten a huge amount of discussion on the tuning list for this very reason. the advantage of 72 is that it's a multiple of 12 so can be taught as an extension of existing technique to modern instrumentalists, allowing one to incorporate these higher ratios into performable contemporary compositions (as, for example, joseph pehrson has done). the disadvantage is that it's not a meantone, so one can't use it to render the western repertoire c. 1480-1780 in a more authentic and pleasing tuning. for this, a true revival of 31 would be a welcome development.
Message: 7293 Date: Wed, 13 Aug 2003 18:49:48 Subject: Re: Comments about Fokker's misfit metric From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carlos <garciasuarez@y...> wrote:
> I undertstand that you like to include in the metric function then > intervals like 9/8 an others. Is that right?.
i do, yes.
Message: 7294 Date: Wed, 13 Aug 2003 18:51:49 Subject: Re: Comments about Fokker's misfit metric From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx <monz@a...> wrote:
> hi Carlos, > > > i haven't really followed this thread, but when i saw > this i thought i could help illustrate something ... > >
> > From: Carlos [mailto:garciasuarez@y...] > > Sent: Wednesday, August 13, 2003 8:38 AM > > To: tuning-math@xxxxxxxxxxx.xxx > > Subject: Re: [tuning-math] Re: Comments about Fokker's misfit
metric
> > > > > > Graham, thanks for your comments. > > > > Ok, if I understand you and Paul allright you suggest > > to include many more intervals in the metric no[t] just > > the fundamental prime intervals of Fokker. I am not > > sure what the physical sense of that is. As Paul points > > out, in the case of the minor third, you certainly would > > like to keep it in the metric as it is a nice consonant > > ratio (for our ears today).
> > > > the point of using the minor-3rd as well as the major-3rd > and 5th (in the 5-limit) is that *all* other intervals > can be calculated as various combinations of those three.
i don't follow.
Message: 7296 Date: Wed, 13 Aug 2003 16:40:59 Subject: Re: Comments about Fokker's misfit metric From: monz@xxxxxxxxx.xxx hi paul,
> From: Paul Erlich [mailto:perlich@xxx.xxxx.xxxx > Sent: Wednesday, August 13, 2003 11:52 AM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] Re: Comments about Fokker's misfit metric > > > --- In tuning-math@xxxxxxxxxxx.xxxx <monz@a...> wrote:
> > hi Carlos, > > > > > > i haven't really followed this thread, but when i saw > > this i thought i could help illustrate something ... > > > >
> > > From: Carlos [mailto:garciasuarez@y...] > > > Sent: Wednesday, August 13, 2003 8:38 AM > > > To: tuning-math@xxxxxxxxxxx.xxx > > > Subject: Re: [tuning-math] Re: Comments about Fokker's misfit
> metric
> > > > > > > > > Graham, thanks for your comments. > > > > > > Ok, if I understand you and Paul allright you suggest > > > to include many more intervals in the metric no[t] just > > > the fundamental prime intervals of Fokker. I am not > > > sure what the physical sense of that is. As Paul points > > > out, in the case of the minor third, you certainly would > > > like to keep it in the metric as it is a nice consonant > > > ratio (for our ears today).
> > > > > > > > the point of using the minor-3rd as well as the major-3rd > > and 5th (in the 5-limit) is that *all* other intervals > > can be calculated as various combinations of those three.
> > i don't follow.
OK, well, as i said, i wasn't really following the thread, so maybe my comment has nothing to do with anything. i was just pointing out that whatever the metric is measuring, if it's meant to be used over the whole tuning system, it only needs those 3 intervals in the 5-limit, or those 6 in the 7-limit, etc., to cover the metric for any interval in the system. -monz
Message: 7297 Date: Wed, 13 Aug 2003 00:02:17 Subject: Re: tctmo! From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >When you talk about lattice distance, you should say how the
distance
> >is measured.
> > An earlier version mentioned "taxicab", but I pulled it, because > Gene's metric isn't taxicab. > > Paul, I just noticed you said the complexity heuristic doesn't work > but with taxicab on a kees lattice. What are the failing cases?
all the others! i mean, it's always a good heuristic, but it's exact for kees' lattice . . .
Message: 7298 Date: Wed, 13 Aug 2003 16:44:11 Subject: Re: interval vector From: monz@xxxxxxxxx.xxx
> From: pitchcolor@xxx.xxx [mailto:pitchcolor@xxx.xxxx > Sent: Wednesday, August 13, 2003 2:19 PM > To: tuning-math@xxxxxxxxxxx.xxx > Subject: [tuning-math] interval vector > >
> > > What's an interval vector?
> > > > > > A pitch difference expressed as a list of > > integers, which is much the same as a free albelian > > group. > > > >
> > In case this has not been clarified, the above use > of the term 'interval vector' is non-standard. > According to Forte / Rahn pitch-set theory, which > is still standard in academia, an interval vector > is an ordered enumerated list of six (twelve-tone) > interval classes (ic) which are present in a pitch set.
Aaron is right about that, guys. since Gene honored me by using my name to describe what you mean here by "interval vector", it's best to just call it a monzo. :) -monz
Message: 7299 Date: Wed, 13 Aug 2003 01:05:35 Subject: Inharmonic temperaments (Was: tctmo!) From: Graham Breed I wrote:
> 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
... These timbres are used with a pelog tuning. The second best linear temperament does look a bit like pelog:
>>> temper.getLinearTemperaments(temper.getLimitedETs(swas_oct))[1]
2/9, 224.8 milli-equivalence generator basis: (1.0, 0.2247677727873047) mapping by period and generator: [(1, 0), (1, 2), (2, 1), (3, -2)] mapping by steps: [(5, 4), (7, 6), (11, 9), (13, 10)] highest interval width: 4 complexity measure: 4 (5 for smallest MOS) highest error: 0.015133 (15.133 milli-equivalences) The generator is half of pelog's. To get pelog itself, the cutoff for equal temperments has to be raised to allow 7-equal in. Then, pelog narrowly makes the top 10:
>>> temper.getLinearTemperaments(temper.getLimitedETs(swas_oct,
cutoff=0.6))[9] 4/9, 445.2 milli-equivalence generator basis: (1.0, 0.44517628148326549) mapping by period and generator: [(1, 0), (1, 1), (4, -4), (3, -1)] mapping by steps: [(7, 2), (10, 3), (16, 4), (18, 5)] highest interval width: 5 complexity measure: 5 (7 for smallest MOS) highest error: 0.019492 (19.492 milli-equivalences) unique The situation improves slightly when you add in some harmonic partials. With 3:1, and allowing for a bit of inconsistency, pelog is the fourth best.
>>> swas2 = temper.PrimeDiamond(4) >>> swas2.primes = [temper.log2(x) for x in (2.76, 3, 4.72, 5.92)] >>> temper.getLinearTemperaments(temper.getLimitedETs(swas2,
cutoff=0.6), worstError=0.1)[3] 4/9, 440.3 milli-equivalence generator basis: (1.0, 0.44030328510322092) mapping by period and generator: [(1, 0), (1, 1), (2, -1), (4, -4), (3, -1)] mapping by steps: [(7, 2), (10, 3), (11, 3), (16, 4), (18, 5)] highest interval width: 5 complexity measure: 5 (7 for smallest MOS) highest error: 0.025266 (25.266 milli-equivalences) And with 3:1 and 5:1, it moves down to 7
>>> swas3 = temper.PrimeDiamond(5) >>> swas3.primes = [temper.log2(x) for x in (2.76, 3.0, 4.72, 5.0, 5.92)] >>> temper.getLinearTemperaments(temper.getLimitedETs(swas3,
cutoff=0.6))[6]
7/16, 440.3 milli-equivalence generator
basis:
(1.0, 0.44030328510322092)
mapping by period and generator:
[(1, 0), (1, 1), (2, -1), (4, -4), (1, 3), (3, -1)]
mapping by steps:
[(9, 7), (13, 10), (14, 11), (20, 16), (21, 16), (23, 18)]
highest interval width: 7
complexity measure: 7 (9 for smallest MOS)
highest error: 0.025266 (25.266 milli-equivalences)
The scale recorded for this gamelan doesn't fit the theoretical
large-small pattern for pelog very well either.
Graham
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