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Message: 9625 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 20:24:59

Subject: Re: Weighting

From: Carl Lumma

>>>> >bservation One: The extent and intensity of the influence of a
>>>> magnet is in inverse proportion to its ratio to 1.
>

>Can you give us Partch's definition of "magnet". 


I don't see a succinct def., but I gather "root" or "key center"
might be proxies.


>And "in inverse proportion to its ratio to 1" makes no sense
>whatsoever. For a start "to 1" is completely redundant.


It's just Partch keeping intervals and pitches separate.  He
didn't have slash/colon notation.  :)  Interestingly, though,
is that he seems to assume (at least in this part of the book)
that slash means interval, and to get a pitch he refers to 'the
identity lying [interval] away from unity'.  And it seems to
me that intervals are more common than pitches (at least around
here) and slashes more natural than colons.  Therefore it could
be argued that we standardized incorrectly, despite Lou Harrison
et al.


>And it would make a lot more sense if it said "in inverse
>proportion to the size of the numbers in the ratio".

Yes.
 

>> Anyway, Partch is saying you can create a dissonance by using a
>> complex interval that's close in size to a simple one.  I
>> translate his Observations into the present context thus...
>> 
>> 'The size (in cents) of the 'field of attraction' of an interval
>> is proportional to the size of the numbers in the ratio, and
>> the dissonance (as opposed to discordance) becomes *greater* as
>> it gets closer to the magnet.'
>

>Since I don't know what he, or you, mean by a "magnet" I can only
>comment on the first part of this purported translation. And I find
>that it is utterly foreign to my experience, and I think yours.


Whoops!  I meant *inversely proportional*.  Sorry!!


>Did you accidentally drop an "inversely".

Mea culpa.

>i.e. we can safely assume that
>Partch is only considering ratios in othe superset of all his JI
>scales, so things like 201:301 do not arise. i.e. he's ignoring
>TOLERANCE and only considering COMPLEXITY. So surely he means that as
>the numbers in the ratio get larger, the width of the field of
>attraction gets smaller.

Yes.

>To me, that's an argument for why TOP isn't necessarily what you
>want.


The entropy minima are wider for simple ratios, but that doesn't
mean that error is less damaging to them.  What it does mean is
that you're less likely to run afoul of extra-JI effects when
measuring error from a rational interval when that interval is
simple.

One of my first posts to the tuning list was about how measuring
deviation from "JI" can get you into trouble...

-Carl

Message: 9626 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 21:38:41

Subject: Re: Back to the 5-limit cutoff

From: Carl Lumma

>> >ut Yes, true.  Increasing my tolerance for complexity 
>> simultaneously increases my tolerance for error, since this is Max().
>

>I have no idea why you say that. However, when I said "more of one", 
>I didn't mean "more tolerance for one", I simply meant "higher values 
>of one".


If I have a certain expectation of max error and a separate
expectation of max complexity, but I can't measure them directly,
I have to use Dave's formula, I wind up with more of whatever I
happened to expect less of.  Dave's function is thus a badness
function, since it represents both error and complexity.

-Carl

Message: 9627 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 02:52:23

Subject: Re: I guess Pajara's not #2

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 

>> That would be excellent, and then I could make a graph for 

> Dave . . .
> 
> Do I need to push any farther towards either less accurate or more 
> complex temperaments to make everyone happy?


No; I'm thinking that, if anything, it's near the *center* where log-
flat badness may suffer from some paucity, for our exploratory 
purposes.

Please remember to post the results as a delimited table, with one 
temperament per row. That way, most of us will be able to graph them 
easily.

Message: 9628 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 18:58:04

Subject: Re: Back to the 5-limit cutoff (was: 60 for Dave)

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:


> It'll generally be different people, though. For a given 
individual, 
> very low error doesn't make complexity any easier to tolerate than 
> merely tolerable error. Nevertheless, I think we should use 
something 
> close to a straight line, slightly convex or concave to best fit 
> a "moat".


Recall that any goofy, ad-hoc weirdness may need to be both explained 
and justified.

Message: 9629 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 20:27:56

Subject: Re: Back to the 5-limit cutoff

From: Carl Lumma

>>>>> >'m arguing that, along this particular line of thinking, 
>>>>> complexity does one thing to music, and error another, but
>>>>> there's no urgent reason more of one should limit your
>>>>> tolerance for the other . . .
>>>> 

>>>> Taking this to its logical extreme, wouldn't we abandon badness
>>>> alltogether?
>>>> 
>>>> -Carl
>>> 

>>> No, it would just become 'rectangular', as Dave noted.
>> 

>> I didn't follow that.
>

>Your badness function would become max(a*complexity, b*error), thus 
>having rectangular contours.


More of one can here influence the tolerance for the other.  Thus
this doesn't fulfill your suggestion, let alone take it to its
logical extreme.


>> Maybe you could explain how it explains
>> how someone who sees no relation between error and complexity
>> could possibly be interested in badness.
>

>Dave and I abandoned badness in favor of a "moat".


"Badness" to me is any combination of complexity and error, which
I took your moat to be.  Maybe I should wait until you publish a
graph or something more public.

-Carl

Message: 9630 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 02:55:01

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:


> Schismic and kleismic/hanson start being useful (barely) around 12 
notes,
> but the tiny size of schismic steps beyond 12 notes is a drawback 
until you
> get to around 41 notes when the steps are a bit more evenly spaced.


Others may feel differently. Schismic-17 is a favorite of Wilson and 
others and closely resembles the medieval Arabic system; Helmholtz 
and Groven used 24 and 36 notes, respectively. Justin White seemed to 
be most interested in the 29-note version.

Message: 9631 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 19:01:26

Subject: Re: 7-limit horagrams

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
>> 
> 
Yahoo groups: /tuning_files/files/Erlich/seven... * [with cont.] 
> 
> I dumped the awful name "quartiminorthirds" in favor 
of "valentine", 
> which you may recall from over a year ago:
> 
> Yahoo groups: /tuning-math/files/Paul/dualzoom... * [with cont.] 


Sounds OK; let's hope it doesn't choke Dave.

Message: 9632 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 20:29:48

Subject: Re: The true top 32 in log-flat?

From: Carl Lumma

>>>>>>> >OP generators [1201.698520, 504.1341314]
>>>> 

>>>> So how are these generators being chosen?  Hermite?
>>> 

>>> No, just assume octave repetition, find the period (easy) and then 
>>> the unique generator that is between 0 and 1/2 period.
>>> 
>>>> I confess

>>>> I don't know how to 'refactor' a generator basis.
>>> 

>>> What do you have in mind?
>> 

>> Isn't it possible to find alternate generator pairs that give
>> the same temperament when carried out to infinity?
>

>Yup! You can assume tritave-equivalence instead of octave-
>equivalence, for one thing . . .


And can doing so change the DES series?

-Carl

Message: 9633 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 21:39:57

Subject: Re: The true top 32 in log-flat?

From: Carl Lumma

At 09:33 PM 2/1/2004, you wrote:

>--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>>>>>>>>>>> TOP generators [1201.698520, 504.1341314]
>>>>>>>> 

>>>>>>>> So how are these generators being chosen?  Hermite?
>>>>>>> 

>>>>>>> No, just assume octave repetition, find the period (easy)
>>>>>>> and then the unique generator that is between 0 and 1/2
>>>>>>> period.
>>>>>>> 
>>>>>>>> I confess

>>>>>>>> I don't know how to 'refactor' a generator basis.
>>>>>>> 

>>>>>>> What do you have in mind?
>>>>>> 

>>>>>> Isn't it possible to find alternate generator pairs that give
>>>>>> the same temperament when carried out to infinity?
>>>>> 

>>>>> Yup! You can assume tritave-equivalence instead of octave-
>>>>> equivalence, for one thing . . .
>>>> 

>>>> And can doing so change the DES series?
>>> 

>>> Well of course . . . can you think of any octave-repeating DESs
>>> that are also tritave-repeating?
>> 

>> Right, so when trying to explain a creepy coincidence between
>> complexity and DES cardinalities, might not we take this into
>> account?
>

>Sure . . . some of the ones that 'don't work' may be working for 
>tritave-DESs rather than octave-DESs, is that what you were thinking?

Yep!

-Carl

Message: 9634 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 03:12:14

Subject: Back to the 5-limit cutoff (was: 60 for Dave)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 

>> Yes. I like that idea too. But by "a wide swath" don't you mean one
>> that it's easy to put a simple smooth curve thru?
> 
> Right.
> 

>> And you must have
>> some general idea of which way this intergalactic moat must curve.
> 

> In the 5-limit linear case, it would be really easy to do this if we 
> didn't want to go out to the complexity of schismic and 
> kleismic/hanson (the only argument that would arise would be whether 
> the father and beep couple should be in or out, leading to a grand 
> total of 11 or 9 5-limit LTs).


There's no doubt in my mind that father and beep should be out. I
notice Herman agrees that these are not of interest as approximations
of 5-limit JI.

And yes, I can see what you mean, when I look at
Yahoo groups: /tuning_files/files/Erlich/dave3... * [with cont.] 
(I've decided I like to see both axes linear)
There's a wide moat cutting off these 9.


> Unfortunately, the low error of 
> schismic has proved tantalizing enough for a few musicians to 
> construct instruments capable of playing the large extended scales 
> that its approximations require.


Yes. Schismic and kleismic must be in.


> If we consider such complexity 
> justifiable, it seems we should be interested in 15 to 17 5-limit 
> LTs, 

> or 17 to 19 if we include father and beep. The couple residing 
> in "the middle of the road" is 2187;2048 and 3126;2916. With Herman, 
> we could split the difference and select only the better of the pair, 
> 2187;2048


But they are so close in both error and complexity. If our error and
complexity measures are any good at all then it's either both or none.

 (Dave, have you *heard* Blackwood's 21-equal suite?) . . . 

No I haven't. But hearing it wouldn't tell me if its complexity was
objectionable. I'd have to think about composing in it or playing it
for that. I suppose the fact that Blackwood did so, suggests it's OK,
but only if it was actually that 5-limit linear temperament that
Blackwood used, and not some other or more general way of viewing of
21-ET.

And I'll take your word that it sounds ok, but are you certain that
its consonances only depend on the actual 2187;2048 linear mapping?


> I don't think anyone's talked about the 20480;19683 system. But if 
> schismic, and certainly if semisixths, is not too complex to be a 
> useful alternative to strict JI, why shouldn't this system merit some 
> attention from musicians too?


That's easy to answer. It's error is way larger than either schismic
or kleismic, and I wouldn't mind omitting semisixths. Has anyone
expressed a strong desire to include it (78125;78732).


> I don't think near-JI [e.g. schismic] triads sound 
> enough better than chords in this system (which are purer than those 
> of augmented, porcupine, or diminished) to merit a much higher 
> allowed complexity to generate them linearly.


They are only slightly purer than augmented, and about 50% worse than
meantone. This certainly doesn't allow 20480;19683 "in" when it has
complexity near that of schismic.

First you say it would be nice not to have to admit anything as
complex as schismic. Then you say schismic has to be admitted because
people have found schismic useful because of its very low error. Then
you say we should admit stuff as complex as schismic but with far
greater errors. Doesn't seem consistent to me.

I think I know what you're trying to do. I can see that the only
really wide moats/channels happen to depart from the X axis in a
direction that is almost due north and go quite a long way in that
direction before curving west. But I don't think this is acceptable
from a psychological standpoint. Straight lines I could definitely
live with. Convex curves (as viewed from the side opposite the origin)
I find more difficult to accept. The farthest I'd be willing to go in
the convex direction would be  k*error^2 + comp^2 < x and the curve
should only just admit 24;25 and 128;135 (neutral thirds and pelogic)
and only just admit schismic (32768;32805), and possibly just admit
Negri (16384;16875).

However, I can see that if we plot that curve then if there's anything
that's only just outside it we'll feel obliged to adjust the curve to
include it, and repeat this until there are none "just outside the
curve", and then I can see that we will probably end up including
semisixths and the "middle of the road pair" (actually large error
_and_ high complexity) 2187;2048 and 3126;2916. 

I could live with these 18. Although I note that a straight line can
be drawn that includes all but these last two, and a convex curve with
powers intermediate between 1 and 2 could do so more cleanly.

On the dave3.gif plot, the formula would be something like

(5000*error)^2 + comp^2 < 35^2


> A problem with our plan to have versions of these badness curves for 
> sets of temperaments of different dimensions is that moving to a 
> higher-dimensional tuning system would theoretically increase the 
> badness by an infinite factor.


I don't understand?


> But in practice you never use an 
> infinite swath of the lattice so eventually any complex enough 5-
> limit linear temperament becomes indistinguishable from a planar 
> system.
> 
> I'm speaking like you now, Dave! ;)


In the above paragraph? In what way? I don't follow.

Certainly you are now agreeing that log-flat badness with additional
cutoffs on error and complexity is not the best way to choose a
reduced list. What made you change you mind? Was it seeing them plotted?

Message: 9635 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 19:09:10

Subject: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:


> I played a lot with the 29 note scale, and it worked fine. 


So why is 29 notes of schismic good, and 27 notes of ennealimmal bad? 
I think both temperaments are so strong they should be included. 
Ennealimmal still seems to me to be the obvious cutoff point in terms 
of complexity. Having at least one septimal temperament which is more 
or less JI makes sense also.

Message: 9636 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 20:30:16

Subject: Re: 7-limit horagrams

From: Carl Lumma

>>>> >eautiful!  I take it the green lines are proper scales?
>>>> 
>>>> -C.
>>> 

>>> Guess again (it's easy)!
>> 

>> Obviously not easy enough if we've had to exchange three
>> messages about it.
>> 
>> -Carl
>

>Then you can't actually be looking at the horagrams ;)


Why not just explain things rather than riddling your users?

-Carl

Message: 9637 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 21:43:12

Subject: Re: 7-limit horagrams

From: Carl Lumma

>I'm ssorry.
>
>Green-black-green-black-green-black-green-black-green-black-green-
>black . . .
>
>Wasn't that your idea in the first place?


I think so, and this is one of the patterns (including even, odd,
prime, proper ...) I ruled out.  Look at decimal.bmp.  10 & 14
are both green.

-Carl

Message: 9638 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 03:17:52

Subject: Re: I guess Pajara's not #2

From: Dave Keenan

--- 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:
>> 

>>> That would be excellent, and then I could make a graph for 

>> Dave . . .
>> 
>> Do I need to push any farther towards either less accurate or more 
>> complex temperaments to make everyone happy?
> 

> No; I'm thinking that, if anything, it's near the *center* where log-
> flat badness may suffer from some paucity, for our exploratory 
> purposes.

Agreed.

> Please remember to post the results as a delimited table, with one 
> temperament per row. That way, most of us will be able to graph them 
> easily.

Agreed.


-- Dave Keenan

Message: 9639 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 19:13:22

Subject: Re: 114 7-limit temperaments

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> You're using temperaments to construct scales, aren't you?


Not me, for the most part. I think the non-keyboard composer is 
simply being ignored in these discussions, and I think I'll stand up 
for him.


> Oh, yes, I think the 9-limit calculation can be done by giving 3 a 
> weight of a half.  That places 9 on an equal footing with 5 and 7, 
and I 
> think it works better than vaguely talking about the number of 
> consonances. 


You also get a lattice which is two symmetrical lattices glued 
together that way.

Message: 9640 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 20:31:18

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>>>>> >uch distinctions may be important for *scales*, but for 
>>>>> temperaments, I'm perfectly happy not to have to worry about
>>>>> them. Any reasons I shouldn't be?
>>>> 

>>>> You're using temperaments to construct scales, aren't you?
>>> 

>>> Not necessarily -- they can be used directly to construct music, 
>>> mapped say to a MicroZone or a Z-Board.
>>> 
>>>  * [with cont.]  (Wayb.)
>> 

>> ???  Doing so creates a scale.
>> 
>> -Carl
>

>A 108-tone scale?


"Scale" is a term with a definition.  I was simply using it.  You
meant (and thought I meant?) "diatonic", or "diatonic scale", maybe.

-Carl

Message: 9641 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 03:38:46

Subject: Re: Back to the 5-limit cutoff (was: 60 for Dave)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
>> wrote:
>> 

>>> Yes. I like that idea too. But by "a wide swath" don't you mean 
one
>>> that it's easy to put a simple smooth curve thru?
>> 
>> Right.
>> 

>>> And you must have
>>> some general idea of which way this intergalactic moat must 
curve.
>> 

>> In the 5-limit linear case, it would be really easy to do this if 
we 
>> didn't want to go out to the complexity of schismic and 
>> kleismic/hanson (the only argument that would arise would be 
whether 
>> the father and beep couple should be in or out, leading to a 
grand 
>> total of 11 or 9 5-limit LTs).
> 

> There's no doubt in my mind that father and beep should be out. I
> notice Herman agrees that these are not of interest as 
approximations
> of 5-limit JI.
> 
> And yes, I can see what you mean, when I look at
> Yahoo groups: /tuning_files/files/Erlich/dave3... * [with cont.] 
> (I've decided I like to see both axes linear)
> There's a wide moat cutting off these 9.
> 

>> Unfortunately, the low error of 
>> schismic has proved tantalizing enough for a few musicians to 
>> construct instruments capable of playing the large extended 
scales 
>> that its approximations require.
> 

> Yes. Schismic and kleismic must be in.
> 

>> If we consider such complexity 
>> justifiable, it seems we should be interested in 15 to 17 5-limit 
>> LTs, 
> 
>> or 17 to 19 if we include father and beep. The couple residing 
>> in "the middle of the road" is 2187;2048 and 3126;2916. With 
Herman, 
>> we could split the difference and select only the better of the 
pair, 
>> 2187;2048
> 

> But they are so close in both error and complexity. If our error and
> complexity measures are any good at all then it's either both or 
none.


*** But Dave, the pair is stradding the smooth curve that passes 
through the center of the moat everywhere else. How is one to decide? 
I prefer this arbitrariness for just 2 temperaments when it makes the 
rest so unarbitrary . . .


>  (Dave, have you *heard* Blackwood's 21-equal suite?) . . . 
> 
> No I haven't. But hearing it wouldn't tell me if its complexity was
> objectionable.


I think hearing it makes a *great* case that it's complexity is 
unobjectionable.

 I'd have to think about composing in it or playing it

> for that. I suppose the fact that Blackwood did so, suggests it's 
OK,
> but only if it was actually that 5-limit linear temperament that
> Blackwood used, and not some other or more general way of viewing of
> 21-ET.


He modulates his 'diatonic' music around a circle of 7 fifths -- thus 
availing himself of 2048;2187. He might not have used its "native 
scale", but yes, he used the temperament.


> And I'll take your word that it sounds ok, but are you certain that
> its consonances only depend on the actual 2187;2048 linear mapping?


In my view, linear mappings depend on consonances, not vice 
versa . . .

Anyway, see above. The diatonic scale is so unusual in 21-equal that 
(or since) the syntonic comma is *negative*, so the usual diatonic 
play has to deal with this particular "quirk", certainly not 
tempering it out . . .


>> I don't think anyone's talked about the 20480;19683 system. But 
if 
>> schismic, and certainly if semisixths, is not too complex to be a 
>> useful alternative to strict JI, why shouldn't this system merit 
some 
>> attention from musicians too?
> 

> That's easy to answer. It's error is way larger than either schismic
> or kleismic, and I wouldn't mind omitting semisixths. Has anyone
> expressed a strong desire to include it (78125;78732).


But the moat!


>> I don't think near-JI [e.g. schismic] triads sound 
>> enough better than chords in this system (which are purer than 
those 
>> of augmented, porcupine, or diminished) to merit a much higher 
>> allowed complexity to generate them linearly.
> 

> They are only slightly purer than augmented, and about 50% worse 
than
> meantone. This certainly doesn't allow 20480;19683 "in" when it has
> complexity near that of schismic.


I'm arguing that, along this particular line of thinking, complexity 
does one thing to music, and error another, but there's no urgent 
reason more of one should limit your tolerance for the other . . .


> First you say it would be nice not to have to admit anything as
> complex as schismic. Then you say schismic has to be admitted 
because
> people have found schismic useful because of its very low error. 
Then
> you say we should admit stuff as complex as schismic but with far
> greater errors. Doesn't seem consistent to me.


I drew a moat, stepped beyond it, and drew a new moat to accomodate 
the expansion.
 

> I think I know what you're trying to do. I can see that the only
> really wide moats/channels happen to depart from the X axis in a
> direction that is almost due north and go quite a long way in that
> direction before curving west. But I don't think this is acceptable
> from a psychological standpoint. Straight lines I could definitely
> live with. Convex curves (as viewed from the side opposite the 
origin)
> I find more difficult to accept.


See above . . .


>> A problem with our plan to have versions of these badness curves 
for 
>> sets of temperaments of different dimensions is that moving to a 
>> higher-dimensional tuning system would theoretically increase the 
>> badness by an infinite factor.
> 

> I don't understand?


What's the area of a volume?


>> But in practice you never use an 
>> infinite swath of the lattice so eventually any complex enough 5-
>> limit linear temperament becomes indistinguishable from a planar 
>> system.
>> 
>> I'm speaking like you now, Dave! ;)
> 

> In the above paragraph? In what way? I don't follow.

It's this:

> Certainly you are now agreeing that log-flat badness with additional
> cutoffs on error and complexity is not the best way to choose a
> reduced list.


That's what I meant.


> What made you change you mind? Was it seeing them plotted?


I have thousands of your posts swimming around my head at all 
times :) The real reason -- because I want them to get used for 
***music*** (which is why speaking to people like herman is 
useful . . .;)

Message: 9642 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 19:52:03

Subject: Schismic and stuff (was: Re: 60 for Dave (was: 41 "Hermanic" 7-limit linear temperaments)

From: Graham Breed

Gene Ward Smith wrote:


> So why is 29 notes of schismic good, and 27 notes of ennealimmal bad? 
> I think both temperaments are so strong they should be included. 
> Ennealimmal still seems to me to be the obvious cutoff point in terms 
> of complexity. Having at least one septimal temperament which is more 
> or less JI makes sense also.


29 notes of schismic happens to fit a 7+5 keyboard well.

Ennealimmal may well work, although it is more complex.  Shouldn't it be 
36 notes for the first 7-limit MOS, and 45 for the first 9-limit one?

My 9-limit ranking is currently:

1) Ennealimmal

2) Magic

3) Dominant Seventh

4) Meantone

5) Pajara

6) Miracle

7) Schismic

all of which may have their uses.  Then comes a 41&58 with mapping [(1 1 
-5 -1), (0 2 25 13)] which I don't know the name for and some more 
meantone variants.

I'd go for magic, schismic or miracle.  Nothing obviously stands out 
like miracle does in the 11-limit.


                  Graham



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Message: 9643 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 04:22:20

Subject: Re: Back to the 5-limit cutoff (was: 60 for Dave)

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:


>>> oThe couple residing 
>>> in "the middle of the road" is 2187;2048 and 3126;2916. With 
> Herman, 

>>> we could split the difference and select only the better of the 
> pair, 
>>> 2187;2048
>> 

>> But they are so close in both error and complexity. If our error and
>> complexity measures are any good at all then it's either both or 
> none.
> 

> *** But Dave, the pair is stradding the smooth curve that passes 
> through the center of the moat everywhere else. 


Which smooth curve? Which moat? I note that there are no islands (or
stepping stones) in moats. :-)


> How is one to decide? 
> 
> I prefer this arbitrariness for just 2 temperaments when it makes the 
> rest so unarbitrary . . .


Can you easily re-plot dave3.jpg showing a quarter-ellipse

(k*err)^2 + comp^2 = x^2 

that passes thru 24;25 (neutral thirds) and 78125;78732 (semisixths)?

This is not intended to represent the centre of the moat but one edge
of it.


> I'm arguing that, along this particular line of thinking, complexity 
> does one thing to music, and error another, but there's no urgent 
> reason more of one should limit your tolerance for the other . . .


I disagree. I think people will only tolerate more complexity if it
gives them less errolr and vice versa. You seem to be arguing for a
rectangular region where 

max(k*err, comp) < x

But I'm willing to move far enough in this direction to admit of
elliptical regions.

They would address Carl's points about a doubling of complexity being
more than twice as inconvenient, and JdL using error-squared for
mistuning-pain.


> The real reason -- because I want them to get used for 
> ***music*** (which is why speaking to people like herman is 
> useful . . .;)

Amen.

Message: 9644 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 04:40:50

Subject: Re: The true top 32 in log-flat?

From: Paul Erlich

There's something VERY CREEPY about my complexity values. I'm going 
to have to accept this as *the* correct scaling for complexity (I'm 
already convinced this is the correct formulation too, i.e. L_1 norm, 
for the time being) . . .


> 2.
>> Meantone (Huygens)
>> 

>> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
>> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>> TOP generators [1201.698520, 504.1341314]
>> bad: 21.551439 comp: 3.562072 err: 1.698521
> 

> 11.7652 -> bad = 235.1092


11.7652 -> 12 -? chromatic scale


> 3.
>> Miracle
>> 

>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>> TOP generators [1200.631014, 116.7206423]
>> bad: 29.119472 comp: 6.793166 err: .631014
> 

> 21.1019 --> bad = 280.9843


21.1019 --> 21 -? blackjack scale


> 7.
>> Magic
>> 

>> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
>> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>> TOP generators [1201.276744, 380.7957184]
>> bad: 23.327687 comp: 4.274486 err: 1.276744
> 

> 15.5360 -> bad = 308.1642


15.5360 --? 16 -? magic-16 MOS scale


> 8.
>> Beep
>> 

>> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
>> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>> TOP generators [1194.642673, 254.8994697]
>> bad: 23.664749 comp: 1.292030 err: 14.176105
> 

> 4.7295 -> bad = 317.0935


4.7295 --> 5 -? beep-5 MOS scale


> 9.
>> Pajara
>> 

>> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
>> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>> TOP generators [598.4467109, 106.5665459]
>> bad: 27.754421 comp: 2.988993 err: 3.106578
> 

> 10.4021 -> bad = 336.1437


10.4021 --> 10 -? symmetrical or omnitetrachordal decatonic scale


> 12.
>> Diminished
>> 

>> [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
>> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
>> TOP generators [298.5321149, 101.4561401]
>> bad: 37.396767 comp: 2.523719 err: 5.871540
> 

> 7.917 -> bad = 368.02


7.917 --> 8 -? octatonic scale


> 16.
>> Father
>> 

>> [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
>> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
>> TOP generators [1185.869125, 447.3863410]
>> bad: 33.256527 comp: 1.534101 err: 14.130876
> 

> 5.2007 -> bad = 382.2


5.2007 --> 5 -? father-5 MOS scale


> 20.
>> Tripletone
>> 

>> [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
>> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
>> TOP generators [399.0200131, 92.45965769]
>> bad: 48.112067 comp: 4.045351 err: 2.939961
> 

> 12.125 -> bad = 432.24


12.125 --> 12 -? 12-note DE with "augmented" symmetry


> 21.
>> {21/20, 28/27}
>> 

>> [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
>> TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
>> TOP generators [1214.253642, 509.4012304]
>> bad: 42.300772 comp: 1.722706 err: 14.253642
> 

> 5.5723 -> bad = 442.58


5.5723 --> 5 -? 5-note MOS


> 27.
>> Dicot
>> 

>> [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
>> TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
>> TOP generators [1204.048159, 356.3998255]
>> bad: 42.920570 comp: 2.137243 err: 9.396316
> 

> 7.2314 -> bad = 491.37


7.2314 --> 7 -? Mohajira


> 29.
>> Injera
>> 

>> [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
>> TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
>> TOP generators [600.8889070, 93.60982493]
>> bad: 42.529834 comp: 3.445412 err: 3.582707
> 

> 11.918 -> bad = 508.85


11.918 --> 12 -? 12-note Injera DE or omnitetrachordal scale


> 30.

>> {25/24, 81/80} Jamesbond?
>> 
>> [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
>> TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
>> TOP generators [172.7759159, 86.69241190]
>> bad: 58.637859 comp: 2.493450 err: 9.431411
> 

> 7.4202 -> bad = 519.28


7.4202 -> 7 -? 7-note equal-tempered scale


> 32.
>> Pelogic
>> 

>> [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
>> TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
>> TOP generators [1209.734056, 532.9412251]
>> bad: 39.824125 comp: 2.022675 err: 9.734056
> 

> 7.426 -> bad = 536.78


7.426 --> 7 --> 7-note "P.E.Logic Diatonic" scale

Creepy, isn't it?

Message: 9645 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 04:44:43

Subject: Re: Back to the 5-limit cutoff (was: 60 for Dave)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:


> Can you easily re-plot dave3.jpg showing a quarter-ellipse
> 
> (k*err)^2 + comp^2 = x^2 
> 
> that passes thru 24;25 (neutral thirds) and 78125;78732 
(semisixths)?
> 
> This is not intended to represent the centre of the moat but one 
edge
> of it.


Tough assignment . . . you?


>> I'm arguing that, along this particular line of thinking, 
complexity 
>> does one thing to music, and error another, but there's no urgent 
>> reason more of one should limit your tolerance for the other . . .
> 

> I disagree. I think people will only tolerate more complexity if it
> gives them less errolr and vice versa.


It'll generally be different people, though. For a given individual, 
very low error doesn't make complexity any easier to tolerate than 
merely tolerable error. Nevertheless, I think we should use something 
close to a straight line, slightly convex or concave to best fit 
a "moat".

Message: 9646 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 04:50:04

Subject: Re: The true top 32 in log-flat?

From: Paul Erlich

Sorry I skipped this:

4.

> Hemiwuerschmidt
>
> [16, 2, 5, -34, -37, 6] [[1, -1, 2, 2], [0, 16, 2, 5]]
> TOP tuning [1199.692003, 1901.466838, 2787.028860, 3368.496143]
> TOP generators [1199.692003, 193.8224275]
> bad: 31.386908 comp: 10.094876 err: .307997


31.212 -> bad = 300.04

31.212 --> 31 -? 31-note proper MOS



--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> There's something VERY CREEPY about my complexity values. I'm going 
> to have to accept this as *the* correct scaling for complexity (I'm 
> already convinced this is the correct formulation too, i.e. L_1 
norm, 
> for the time being) . . .
> 
>> 2.
>>> Meantone (Huygens)
>>> 

>>> [1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
>>> TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
>>> TOP generators [1201.698520, 504.1341314]
>>> bad: 21.551439 comp: 3.562072 err: 1.698521
>> 

>> 11.7652 -> bad = 235.1092
> 

> 11.7652 -> 12 -? chromatic scale
> 
>> 3.
>>> Miracle
>>> 

>>> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
>>> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
>>> TOP generators [1200.631014, 116.7206423]
>>> bad: 29.119472 comp: 6.793166 err: .631014
>> 

>> 21.1019 --> bad = 280.9843
> 

> 21.1019 --> 21 -? blackjack scale
> 
>> 7.
>>> Magic
>>> 

>>> [5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
>>> TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
>>> TOP generators [1201.276744, 380.7957184]
>>> bad: 23.327687 comp: 4.274486 err: 1.276744
>> 

>> 15.5360 -> bad = 308.1642
> 

> 15.5360 --? 16 -? magic-16 MOS scale
> 
>> 8.
>>> Beep
>>> 

>>> [2, 3, 1, 0, -4, -6] [[1, 2, 3, 3], [0, -2, -3, -1]]
>>> TOP tuning [1194.642673, 1879.486406, 2819.229610, 3329.028548]
>>> TOP generators [1194.642673, 254.8994697]
>>> bad: 23.664749 comp: 1.292030 err: 14.176105
>> 

>> 4.7295 -> bad = 317.0935
> 

> 4.7295 --> 5 -? beep-5 MOS scale
> 
>> 9.
>>> Pajara
>>> 

>>> [2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
>>> TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
>>> TOP generators [598.4467109, 106.5665459]
>>> bad: 27.754421 comp: 2.988993 err: 3.106578
>> 

>> 10.4021 -> bad = 336.1437
> 

> 10.4021 --> 10 -? symmetrical or omnitetrachordal decatonic scale
> 
>> 12.
>>> Diminished
>>> 

>>> [4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
>>> TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
>>> TOP generators [298.5321149, 101.4561401]
>>> bad: 37.396767 comp: 2.523719 err: 5.871540
>> 

>> 7.917 -> bad = 368.02
> 

> 7.917 --> 8 -? octatonic scale
> 
>> 16.
>>> Father
>>> 

>>> [1, -1, 3, -4, 2, 10] [[1, 2, 2, 4], [0, -1, 1, -3]]
>>> TOP tuning [1185.869125, 1924.351908, 2819.124589, 3401.317477]
>>> TOP generators [1185.869125, 447.3863410]
>>> bad: 33.256527 comp: 1.534101 err: 14.130876
>> 

>> 5.2007 -> bad = 382.2
> 

> 5.2007 --> 5 -? father-5 MOS scale
> 
>> 20.
>>> Tripletone
>>> 

>>> [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
>>> TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
>>> TOP generators [399.0200131, 92.45965769]
>>> bad: 48.112067 comp: 4.045351 err: 2.939961
>> 

>> 12.125 -> bad = 432.24
> 

> 12.125 --> 12 -? 12-note DE with "augmented" symmetry
> 
>> 21.
>>> {21/20, 28/27}
>>> 

>>> [1, 4, 3, 4, 2, -4] [[1, 2, 4, 4], [0, -1, -4, -3]]
>>> TOP tuning [1214.253642, 1919.106053, 2819.409644, 3328.810876]
>>> TOP generators [1214.253642, 509.4012304]
>>> bad: 42.300772 comp: 1.722706 err: 14.253642
>> 

>> 5.5723 -> bad = 442.58
> 

> 5.5723 --> 5 -? 5-note MOS
> 
>> 27.
>>> Dicot
>>> 

>>> [2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
>>> TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
>>> TOP generators [1204.048159, 356.3998255]
>>> bad: 42.920570 comp: 2.137243 err: 9.396316
>> 

>> 7.2314 -> bad = 491.37
> 

> 7.2314 --> 7 -? Mohajira
> 
>> 29.
>>> Injera
>>> 

>>> [2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
>>> TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
>>> TOP generators [600.8889070, 93.60982493]
>>> bad: 42.529834 comp: 3.445412 err: 3.582707
>> 

>> 11.918 -> bad = 508.85
> 

> 11.918 --> 12 -? 12-note Injera DE or omnitetrachordal scale
> 
>> 30.

>>> {25/24, 81/80} Jamesbond?
>>> 
>>> [0, 0, 7, 0, 11, 16] [[7, 11, 16, 20], [0, 0, 0, -1]]
>>> TOP tuning [1209.431411, 1900.535075, 2764.414655, 3368.825906]
>>> TOP generators [172.7759159, 86.69241190]
>>> bad: 58.637859 comp: 2.493450 err: 9.431411
>> 

>> 7.4202 -> bad = 519.28
> 

> 7.4202 -> 7 -? 7-note equal-tempered scale
> 
>> 32.
>>> Pelogic
>>> 

>>> [1, -3, -4, -7, -9, -1] [[1, 2, 1, 1], [0, -1, 3, 4]]
>>> TOP tuning [1209.734056, 1886.526887, 2808.557731, 3341.498957]
>>> TOP generators [1209.734056, 532.9412251]
>>> bad: 39.824125 comp: 2.022675 err: 9.734056
>> 

>> 7.426 -> bad = 536.78
> 

> 7.426 --> 7 --> 7-note "P.E.Logic Diatonic" scale
> 
> Creepy, isn't it?

Message: 9647 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 19:15:29

Subject: Re: Weighting

From: Carl Lumma

>> >bservation One: The extent and intensity of the influence of a
>> magnet is in inverse proportion to its ratio to 1.
>

>Hmm, that's fairly impenetrable.  But it does say "extend" and
>"inverse proportion".


Um, it says "extent"... :)


>> "To be taken in conjunction with the following"
>> 
>> Observation Two: The intensity of the urge for resolution is in
>> direct proportion to the proximity of the temporarily magnetized
>> tone to the magnet.
>

>So it's only about resolution?


The observations are given in a discussion of chord progressions.

Does this mean you don't have a copy of _Genesis_?  Wait, let me
guess: Gene and Dave don't either.  God almighty.

Anyway, Partch is saying you can create a dissonance by using a
complex interval that's close in size to a simple one.  I translate
his Observations into the present context thus...

'The size (in cents) of the 'field of attraction' of an interval
is proportional to the size of the numbers in the ratio, and
the dissonance (as opposed to discordance) becomes *greater* as
it gets closer to the magnet.'

He obviously isn't considering approximations here.  In fact he
says:

"There is undoubtedly a point, in the case of a magnetized tone
extremely close to a magnet, where the two would be so compounded
that the urge would be dampened or lost, but such intervals do
not exist in Monophony."

There's also Observation Three:

"The insensity of the urge for resolution in a satellite, or
magnetized tone, is in direct proportion to the smallness of
the numbers of its ratio to the unity of desired perfection.
For example, if 1/1 is considered the Otonality of desired
perfection, the urge for resolution in a satellite related to
it by the ratio 4/3 is stronger than one related to it by the
ratio 5/3, if the two situations where these ratios might be
involved could ever be exactly parallel.  Neither these
situations nor any two situations in the psychological
phenomenon are ever exactly parallel, since 4/3 is affected
by its proximity (in the ratio 16/15) to a strong magnet (the
identity in 5/4 relation to the unity), whereas 5/3 is affected
by a greater proximity (in the ratio 21/20) to a weaker magnet
(the identity in 7/4 relation to its unity) or by a much lesser
proximity (in the ration 10/9) to a stronger magnet (the
identity in 3/2 relation to its unity)."

Here he seems to be glossing an outline for harmonic entropy.
But I see nothing about how fast entropy would increase with
error.


>Carl:

>> ?  The more complex ones already have the highest entropy.  You mean
>> they gain the most entropy from the mistuning?  I think Paul's saying
>> the entropy gain is about constant per mistuning of either complex
>> or simple putative ratios.
>

>Oh no, the simple intervals gain the most entropy.  That's Paul's 
>argument for them being well tuned.  After a while, the complex 
>intervals stop gaining entropy altogether, and even start losing it.
>At that point I'd say they should be ignored altogether, rather than 
>included with a weighting that ensures they can never be important. 
>Some of the temperaments being bandied around here must get way beyond 
>that point.  Actually, any non-unique temperament will be a problem.


This brings up the point: error from JI is not ultimately something
we should use to evaluate temperaments.  We really have to use
harmonic entropy (NB Gene).


>What I meant is that, because the simple intervals have the least 
>entropy to start with, they still have the least after mistuning, 
>although they're gaining it more rapidly.


Only the gain/loss, and the estimated relative frequency of occurrence
of the internal in music, are considerations in my book.  If we just
use harmonic entropy directly, we don't have to worry about the rate
at which gain/loss happens relative to the identity and the error and
and and...


>Carl:

>> I was thinking about this last night before I passed out.  If you
>> tally the number of each dyad at every beat in a piece of music and
>> average, I think you'd find the most common dyads are octaves, to be
>> followed by fifths and so on.  Thus if consonance really *does*
>> deteriorate at the same rate for all ratios as Paul claims, one
>> would place less mistuning on the simple ratios because they occur
>> more often.  This is, I believe, what TOP does.
>

>It depends on the music, of course.  My decimal counterpoint tends to 
>use 4:6:7 a lot because it's simple, and not much of 6:5.  So tuning
>for such pieces would be different to TOP, which assumes a different
>pattern of intervals.


Temperament could ultimately be customized for a piece, and that's
what John deLaubenfels does so well -- he tempers in every domain.
But we're interested in publishing general-purpose tunings here.
I think some simple (n*d or log(n*d)) weighting is appropriate for
a general solution.

-Carl

Message: 9648 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 05:32:45

Subject: Re: The true top 32 in log-flat?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> There's something VERY CREEPY about my complexity values. I'm going 
> to have to accept this as *the* correct scaling for complexity (I'm 
> already convinced this is the correct formulation too, i.e. L_1 norm, 
> for the time being) . . .
...
> Creepy, isn't it?

Woah! Yes.

Message: 9649 - Contents - Hide Contents

Date: Sun, 01 Feb 2004 19:15:59

Subject: Re: 114 7-limit temperaments

From: Carl Lumma

>>>> >ou're using temperaments to construct scales, aren't you?
>>> 

>>> Not me, for the most part. I think the non-keyboard composer is 
>>> simply being ignored in these discussions, and I think I'll stand
>>> up for him.
>> 

>> How *are* you constructing scales, and what does it have to do
>> with keyboards?
>

>Often I'm not constructing them because I'm not using them.


Care to explain how you can compose without the existence of
scales?

-Carl

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