>
> Thanks. I've read Monz's summary of Woolhouse's book which I've
found very
> helpful. I was just wondering, though, why he doesn't take the
square root
> when calculating Root-Mean-Square. Is it unneccessary?
if you´ve minimized the mean-square, you´ve also minimized the root-
mean-square.
I also see a little
> problem with the final calculation for rms for meantone. If
multiplication
> is commutative, you could take the results to ALSO mean (3/2)/(7/26)
> ^(81/80) which is obviously a wrong result. Clarification, anyone?
Thanks.
i don´t have the woolhouse in front of me, but the correct formula is
(3/2)/(81/80)^(7/26)
>
>
>
> "Gene Ward
Smith
> <genewardsmith@ju To: tuning-
math@xxxxxxxxxxx.xxx
> no.com>" cc: (bcc: Paul
G Hjelmstad/US/AMERICAS)
> <genewardsmith Subject: [tuning-
math] Re: Minimax generator
>
> 01/03/2003
12:56
>
PM
> Please respond
to
> tuning-
math
>
>
>
>
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>"
> <genewardsmith@j...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>>>
>>> Would someone explain "minimax" generator (I understand rms
generator)
>>
>> Let's take meantone for an example. If we define the three linear
> functions
>>
>> u1 = x - l2(3)
>>
>> u2 = 4x - 4 - l2(5)
>>
>> u3 = 3x -4 - l2(5/3)
>>
>> using "l2" to mean log base 2, then the rms generator can be found
>> by minimizing u1^2+u2^2+u3^2. On the other hand, we can minimize
>> |u1| + |u2| + |u3| instead (which gives 1/4 comma meantone.) In
fact,
>> for any p>1, we can minimize |u1|^p + |u2|^p + |u3|^p; if for
instance
>> p is 4, this gives us essentially 1/3 comma meantone (.33365
comma.)
>> If we consider
>
> It turns out this is so high because of round off error. I redid the
> calculation using 100 digits of accuracy, and got:
>
> p=2 7/26 comma meantone
>
> p=4 7/26 comma meantone again
>
> p=6 .26295498 comma meantone, about 5/19 comma
>
> p=8 .25942006 comma meantone, about 7/27 comma
>
> p=10 .25739442 comma meantone, about 9/35 comma
>
> p=1 and p=infinity, 1/4 comma meantone
>
>
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Message: 5905 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 20:36:55
Subject: Re: Poptimal generators
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>>> By "this endeavour" I meant specifically the mathematical
modelling
>> of
>>> perceptual optimality of generators for musical temperaments,
not
>>> mathematic or statistics in general. I also only
said "possibly".
>> I'm
>>> happy to be corrected.
>>
>> consider yourself corrected :)
>
> So who, before you, has championed the use of mean-absolute error to
> find optimum generators?
i'm not championing them, but i've seen them suggested several times.
i think carl lumma was one of those people. it seems that carl,
george, and some others feel that even a single just interval can
profoundly affect the sound of a chord -- in which case weighting the
largest errors most isn't pointing you in the right direction.
>> nope. it´s just that an infinite number of tunings, a continuous
>> range of generators, can be considered poptimal for a given
>> temperament -- thus "absolutely and ideally perfect" cannot be
taken
>> seriously, and was obviously poking fun of similar claims by
people
>> like lucy.
>
> Ah! Well the problem was that I didn't understand that a continuous
> range was being referred to. Gene was quoting specific rational
> fractions of an octave for the generators. And if I didn't
understand
> that, I suspect a lot of other people didn't either.
you must have skipped over most of gene's original message. why do
you think we're having this discussion about absolute error criteria
in the first place? it's because gene suggested using the entire
range of p-norms, with p from 2 to infinity, and i suggested moving
the lower limit down to 1.
> Hope you're having a great trip apart from the trains.
tonight i'm stuck in london -- plane screwups this time.
Message: 5906 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 00:40:27
Subject: Re: Minimax generator
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>
> Thanks. I've read Monz's summary of Woolhouse's book which I've
found very
> helpful. I was just wondering, though, why he doesn't take the
square root
> when calculating Root-Mean-Square. Is it unneccessary?
If you only want to find an optimum generator it is unnecessary, in
fact you don't even need to take the mean, just minimise the sum of
the squares of the errors. But if you want to express the resulting
overall error in a perceptually meaningful fashion, e.g. for comparing
different temperaments, then you need to take the root of the mean of
the squares, so it has dimensions of log-of-frequency-ratio, and
therefore (usually) units of cents.
> I also see a little
> problem with the final calculation for rms for meantone. If
multiplication
> is commutative, you could take the results to ALSO mean (3/2)/(7/26)
> ^(81/80) which is obviously a wrong result. Clarification, anyone?
Thanks.
I find it easier to follow when expressed in the logarithmic frequency
domain (e.g. in cents) as 702.0c - 21.5c * 7/26
Message: 5907 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 20:40:09
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> you´re talking prime limit, which is fine for the mapping,
>> as usual.
>
> I'm talking 'put whatever you want in the map'.
>
>> but for the optimization of the generator size, we need a
>> list of consonances to target.
>
> Why not optimize the generator size for the map, and let
> it target the consonances? Presumably because in some
> tunings the errors for say 3 and 5 will cancel on consonances
> like 5:3.
i'm not following you, or where you differ from what's "standard"
around here . . . why don't you post a complete calculation for the
meantone case, or if you wish, some other, more contrived case . . .
Message: 5908 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 00:51:44
Subject: Re: Poptimal generators
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> By "this endeavour" I meant specifically the mathematical modelling
> of
>> perceptual optimality of generators for musical temperaments, not
>> mathematic or statistics in general. I also only said "possibly".
> I'm
>> happy to be corrected.
>
> consider yourself corrected :)
So who, before you, has championed the use of mean-absolute error to
find optimum generators?
> nope. it´s just that an infinite number of tunings, a continuous
> range of generators, can be considered poptimal for a given
> temperament -- thus "absolutely and ideally perfect" cannot be taken
> seriously, and was obviously poking fun of similar claims by people
> like lucy.
Ah! Well the problem was that I didn't understand that a continuous
range was being referred to. Gene was quoting specific rational
fractions of an octave for the generators. And if I didn't understand
that, I suspect a lot of other people didn't either.
Hope you're having a great trip apart from the trains.
Message: 5909 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 20:43:06
Subject: Re: thanks manuel
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:
>> to start with, it would be good enough to simply have all the
>> consonant intervals (say within a given odd limit) show up as
>> diagonal lines -- the rest of the chart can be all white or all
black
>> for now . . . the point is you could visually tweak the scale with
an
>> eye toward approximating this consonance here and that consonance
>> there . . . don´t know of a better way to achieve this goal than an
>> applet like this!
>
> Ok I understand. It probably won't be much work to expand the triad
> player to do this.
>
> Manuel
the triad player? . . . what's weird is that the idea above deals
with dyads, while i had a similar idea actually dealing with triads --
plotting all the triads in the scale on top of the snowflake, and
seeing how the points (scale's triads) move around as you fiddle with
the scale (possibly with an eye towards approximating a number of
otonal triads) . . .
Message: 5910 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 02:24:51
Subject: Re: Poptimal generators
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <
> nope. it´s just that an infinite number of tunings, a continuous
> range of generators, can be considered poptimal for a given
> temperament -- thus "absolutely and ideally perfect" cannot be taken
> seriously, and was obviously poking fun of similar claims by people
> like lucy.
Are we still on this? But Paul did get the point, which is that we have a *=
range* of absolutely, completely, 100% perfect systems which everyone must a=
nd shall use. I was saying my 100% perfect systems were are perfect as anyon=
e's, so I should be taken as seriously as anyone else making such a claim. T=
hat means I am just as serious as anyone else for an uncountably infinite nu=
mber of different 100% perfect systems, which is pretty danged serious if yo=
u can count that high.
> paul (stuck in mannheim right now due to trains not running on time)
Have a burger at the Burger King. When I lived in Heidelberg, it had two Ma=
cDonalds but no Burger King. I tried the one in Mannheim. Cold. :(
Message: 5911 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 21:34:32
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>> >hy not optimize the generator size for the map, and let
>> it target the consonances? Presumably because in some
>> tunings the errors for say 3 and 5 will cancel on consonances
>> like 5:3.
>
>i'm not following you, or where you differ from what's
"standard" around here . . .
As I say, I don't know how much differing from what's
standard. Calculations are seldom posted here at the
undergrad level.
As usual, I'm trying to figure things out by synthesizing
something and asking about it. When I hit something that
works, I keep it.
>why don't you post a complete calculation for the meantone
>case, or if you wish, some other, more contrived case . . .
Map for 5-limit meantone...
2 3 5
gen1 1 1 -2
gen2 0 1 4
Complexity for each identity...
2= 1
3= 2
5= 6
Let's weight by 1/base2log(i)...
2= 1.00
3= 1.26
5= 2.58
Now gen1 and gen2 are variables, and minimize...
error(2) + 1.26(error(3)) + 2.58(error(5))
I don't know how to do such a calculation, or even
if it's guarenteed to have a minimum. It would
give us minimum-badness generators, not minimum
error gens.
The log2(i) weighting is only off the top of my head.
One could imagine no weighting. One could imagine
weighting so steep we could find the optimal generators
for harmonic limit infinity.
If this does cause us to miss temperaments with good
composite consonances like 5:3, we can go back to
minimizing the error of all the intervals in the givin
limit, and keep the summed graham complexity.
-Carl
Message: 5912 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 02:52:21
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>you´re talking prime limit, which is fine for the mapping,
>as usual.
I'm talking 'put whatever you want in the map'.
> but for the optimization of the generator size, we need a
> list of consonances to target.
Why not optimize the generator size for the map, and let
it target the consonances? Presumably because in some
tunings the errors for say 3 and 5 will cancel on consonances
like 5:3.
-Carl
Message: 5913 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 22:22:39
Subject: Re: Poptimal generators
From: Carl Lumma
>> >o who, before you, has championed the use of mean-absolute
>> error to find optimum generators?
>
>i'm not championing them, but i've seen them suggested several
>times. i think carl lumma was one of those people.
I don't remember it, but I won't deny it. Lately I've been
behind RMS, though I'm interested in the possiblity of a
universal thinger such as the heuristic or the poptimal stuff.
>you must have skipped over most of gene's original message. why
>do you think we're having this discussion about absolute error
>criteria in the first place? it's because gene suggested using
>the entire range of p-norms, with p from 2 to infinity, and i
>suggested moving the lower limit down to 1.
How does this sort of thing compare with the heuristic approach?
-Carl
Message: 5914 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 14:29:05
Subject: Re: A common notation for JI and ETs
From: David C Keenan
At 01:16 AM 9/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
>> In case anyone has already looked at the .bmp for my latest
>suggestion
>> regarding symbolising the 5'-comma (5-schisma) up and down:
>>
>> It was riddled with vertical alignment errors so I've had another
>> -- Dave Keenan
>> Brisbane, Australia
>
>I've looked at it and thought about it for a couple of days.
>
>Good points:
>
>1) The +-5' symbols can easily be used in conjunction with any
>existing symbol.
>2) They clearly indicate (by vertical position) the line or space of
>the note being modified.
>3) There is also a helpful indication (a difference in vertical
>position in addition to slope) of whether the 5' is plus or minus.
>
>Comment on point 3: The slope is most meaningful if the new symbols
>are placed to the left, as in the upper staff (which you also favor).
Yes. I'm happy to eliminate right-hand accents from consideration.
>Problems:
>
>1) The +-5' symbols are detached from the others, so are too easy to
>overlook (particularly if this is the only thing modifying a natural
>note).
I see this as a good point. It really doesn't matter much if a performer
misses a 2 cent modification. On a flexible-pitch instrument many would not
be able to do anything about them anyway. They just aren't that accurate.
On a fixed pitch instrument there will presumably not be two notes
available that are only 2 cents apart.
It would be much worse if the performer couldn't interpret the symbol at
all (or quickly enough) because the conjoined 5' modification made it look
unfamiliar.
>2) Since they are detached from the others, we technically have two
>new modifying symbols used together, so the double-symbol version of
>the notation might now become a triple-symbol version -- something to
>think about.
Yes, technically 3 symbols, but in reality no different to adding an accent
to a roman character. Because they are so close together and because the
accent is small relative to the character, it is perceived as a single
character.
We can even refer to these small slanting lines as acute and grave.
>Since looking at this I also tried something else, which I have added
>to this file (on the third staff):
>
>
Yahoo groups: /tuning- * [with cont.]
>math/files/secor/notation/Schisma.gif
>
>Note: If you don't see 4 staves in the figure, then click on the
>refresh button on your browser to ensure that you're looking at the
>latest version of the file.
>
>I tried small arrowheads to indicate the 5' down and up symbols. In
>the 3rd staff I attached them to the point of an existing sagittal
>symbol; for the up-arrow I removed the pixel at the end of the shaft
>to clarify the symbol. The big advantage here is that we would avoid
>having detached symbol elements.
>In the 4th staff (up to the first double bar) I placed the arrows to
>the left of existing sagittal symbols, but they could just as easily
>be placed to the right, or on either side, depending on where they
>would look or fit best.
>
>Wherever you put them, I think that these small arrowheads are easier
>to see than those tiny slanted lines,
Based on making symbols proportional to their size in cents relative to
strict Pythagorean, the 5' symbol should only have about 6 pixels because
the 19 comma flag has 10 and corresponds to 3.4 cents. The small arrowheads
(or circumflex and caron) contain 8 pixels.
>and they give a better
>indication of direction of alteration.
That's true. To some degree acute and grave reproduce the problem of the
Bosanquet comma slash that your arrow shaft solved. But I think this is
greatly mitigated by the up and down displacement and because they relate
to the arrow shaft on the symbol that they are modifying.
I now agree that they should only be placed to the left so that the grave
symbol retains its linguistic meaning of low or falling pitch and acute -
high or rising pitch.
Full arrowheads already have a sagittal association with the prime 11
whereas the slanted lines preserve the association with 5.
Code Charts (PDF Version) * [with cont.] (Wayb.) is useful to check for clashes with existing
music symbols. I didn't find any except the "marcato" symbol (downward
arrowhead) which appears below (not beside) the notehead.
I just tried adding very short shafts to the acute and grave to make their
direction clearer but this makes them look totally like separate symbols
rather than accents, in fact it makes them look like 5-comma (not 5'-comma)
symbols intended for grace notes.
>While I was writing this I got a couple of other ideas that use 5'
>flags, so I quickly added them on the fourth staff. I lowered the
>short straight -5' flag to the same vertical position that we seem to
>be agreeing on to see how that would look and made 3 symbols that way.
I agree with including a bare shaft when a 5' accent mark would otherwise
occur on its own.
>Then, after the next double bar, I used the small arrowheads as right
>flags and tried some symbols that way. (The 5:7 comma is also there
>for comparison.) After I looked at them for a little while, I
>decided to move the 5' flags one pixel to the right, so that they are
>almost, but not quite touching the rest of the sagittal symbol (to
>avoid confusion with a concave right flag). I think that this last
>group is my preference in that:
>
>1) The 5' flags are clear and logical;
>2) The 5' symbol elements aren't off by themselves, therefore don't
>get overlooked;
>3) Their vertical positions are well placed;
>4) They aren't larger than concave flags.
Well, I'd go along with kerning the acute nearer to (the left of) the
symbol being modified, when that symbol has a left flag (as in the
pythagorean comma symbol), but I'd still prefer that the 5' symbols were
defined as separate symbols in the font, for what are, I hope, obvious
reasons, and I'd still prefer that the unkerned distance was two pixels
(such as in the diaschisma symbol).
Pythag comma '/|
Diaschisma `/|
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 5915 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 20:14:01
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
> Working down the ratio popularity list, of those we don't yet have
a symbol
> for:
>
> There are two 125 commas of interest
> 125-diesis 125:128 41.06 c .//| exact, no symbol without
5'
> 125'-diesis 243:250 49.17 c /|) or (|~
Now that we've agreed on the 5' comma symbols, may I suggest that the
ascii symbols for -5' and +5' be ' and ` respectively, regardless of
the direction of alteration of the main symbol (particularly since
the actual accents don't appear aligned with the point of the arrow
in the actual symbols)? I think that the period and comma are too
difficult to remember, especially the way you've done the 125-diesis
above (which is different than before), and I think `//| and '\\!
should be clear enough for a 125-diesis up and down, respectively.
For the 125' diesis, many divisions (including 171, 217, 224, 270,
282, 342, 388, and 612) would allow either /|) or (|~, but 53, 99,
and 494 all require /|), while 311 allows neither. So I believe
that /|) is the clear choice.
> two 49 commas
> 49-diesis 48:49 35.69 c ~|)
> 49'-diesis 3963:4096 54.53 c (/| or |))
For the 49 comma ~|) is obviously the right size.
The 49' diesis should be 3969:4096. More on this one below.
> one 7:25 comma
> 7:25-comma 224:225 7.71 c '|( exact, no symbol without
5'
>
> two 5:49 commas
> 5:49-comma 321489:327680 33.02 c (|
> 5:49'-diesis 392:405 56.48 c '(/| or '|)) exact, no symbol w/o
5'
>
> Perhaps we should ditch the (/| symbol entirely and use |)) for the
31'
> comma since |)) is the more obvious symbol for the 49'-diesis.
For the 31' comma only the divisions that have any semblance of
consistency up to the 31 limit would have any practical bearing on
this decision. For 270 and 311 |)) is required, while for 217, 388,
and 653 either one is valid; 494 requires (/|, but is not 1,7,31,49-
consistent. It looks like |)) takes it. But this would require
other symbols for 23 and 24deg494; any ideas?
--George
Message: 5916 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 20:32:37
Subject: Re: Notating Kleismic
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
>> However, if you _were_ using kleismic for 7-limit with the least
>> complex 7's, you would probably be dissatisfied with a 53-ET based
>> notation.
>
> Why? Neither one is using the best value of the "7" of the et in
question, and in both cases the 7-limit intervals are much more out
of tune than the 5-limit intervals. It is cheesy no matter how you
notate it, and I don't see why it is any *more* dissapointing for 53
than for 72.
Since 7 is +22 and 11 is -21 in the series of ~5:6's in 53, 72, and
125, these are the normal positions for the kleismic temperament. So
we should be using symbols of a different nature to notate a lot of
the tones closer to the origin, which would seem to call for the 5^2
symbol, //|. The notation for 125 uses not only this, but also both
the 7-comma and 11-diesis symbols if the tuning is extended far
enough to take in ratios of 7 and 11, so 125 seems like a logical
choice to me.
--George
Message: 5917 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 20:50:28
Subject: Fwd: Re: A common notation for JI and ETss
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
>>> Are there any ETs in which we should now prefer )|( over some
other
>> symbol
>>> given that it now has such a low prime-limit or low product
>> complexity?
>>>
>
> I'll just note that neither of us have answered the above yet, in
case the
> way I edited things might have made it look like the following was
> answering it, which of course it is not.
There are none that I see for this as a 7':11' comma (or whatever
we're going to call it). It has a dual role with the 7+5+19 comma in
212, 224, 311, 342, 612, and 624, where )|( has already been agreed
on or is the obvious choice. And it is not valid as the 7':11' comma
in either 217 or 494.
--George
Message: 5918 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 22:23:49
Subject: Re: Notating Kleismic
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:
>> Why? Neither one is using the best value of the "7" of the et in
> question, and in both cases the 7-limit intervals are much more out
> of tune than the 5-limit intervals. It is cheesy no matter how you
> notate it, and I don't see why it is any *more* dissapointing for 53
> than for 72.
>
> Since 7 is +22 and 11 is -21 in the series of ~5:6's in 53, 72, and
> 125, these are the normal positions for the kleismic temperament.
We're not on the same page here, so it's no wonder we haven't come to an agreement on notation. The above system I have been calling "Catakleismic",
reserving "Kleismic" for the system which approximates 7/4 by three
minor third generators, and hence uses the
875/864 comma. Since this is how "Kleismic" was described when
introduced, I think we need to stick to that name.
I agree that Catakleismic, with kernel generated by
[225/224, 4375/4374], is best notated by 72-et. I am proposing that
Kleismic, with kernel generated by [49/48, 126/125] be notated by
53-et.
Message: 5919 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 22:26:04
Subject: Re: Notating Kleismic
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
>>> However, if you _were_ using kleismic for 7-limit with the least
>>> complex 7's, you would probably be dissatisfied with a 53-ET based
>>> notation.
>>
>> Why? Neither one is using the best value of the "7" of the et in
> question, and in both cases the 7-limit intervals are much more out
> of tune than the 5-limit intervals. It is cheesy no matter how you
> notate it, and I don't see why it is any *more* dissapointing for 53
> than for 72.
You're right. I hadn't realised that 72-ET's 7 wasn't kleismic's least
complex 7.
> Since 7 is +22 and 11 is -21 in the series of ~5:6's in 53, 72, and
> 125, these are the normal positions for the kleismic temperament. So
> we should be using symbols of a different nature to notate a lot of
> the tones closer to the origin, which would seem to call for the 5^2
> symbol, //|. The notation for 125 uses not only this, but also both
> the 7-comma and 11-diesis symbols if the tuning is extended far
> enough to take in ratios of 7 and 11, so 125 seems like a logical
> choice to me.
This agrees better with the ET-independent notation I gave too.
So
/ // ///
would become
/| //| (|)
Message: 5920 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 23:12:46
Subject: Re: Notating Pajara
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> This is the system with wedgie [2, -4, -4, 2, 12, -11] which we used
to call Paultone. It has [1/2, 5/56] as poptimal in both the 7-limit
and the 9-limit, and my recommendation is that the 56-et be used to
notate it. The alternative is 22, but with all due respect for Paul's
favorite division, 56 is in much better tune. As a way of tuning a
22-tone MOS and playing Decatonic, it's something Paul might try if he
hasn't already.
>
I believe 56-ET would notate a chain of 22 just the same as 22-ET
would, since the only symbol required is the 5-comma symbol /|. But
beyond that, I suspect that the best 7 of 56-ET may not be the 7 of
pajara/paultone.
Message: 5921 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 23:47:08
Subject: Re: Notating Pajara
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:
> I believe 56-ET would notate a chain of 22 just the same as 22-ET
> would, since the only symbol required is the 5-comma symbol /|. But
> beyond that, I suspect that the best 7 of 56-ET may not be the 7 of
> pajara/paultone.
Eeek,
you're right; the val for Pajara is [56,89,130,158]. This still gives
5-limit harmony quite a bit better than that of the 22-et, but 22 has
a better 7/4 and 7/6, and the same 7/5, and should be considered. Does
Paul want to weigh in?
Message: 5922 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 23:49:42
Subject: Achieving consensus?
From: Gene Ward Smith
Can we reach it for the following list, which I don't think is too
controversial?
"Dominant seventh"
[1, 4, -2, -16, 6, 4] [36/35, 64/63] 12
"Diminished"
[4, 4, 4, -2, 5, -3] [36/35, 50/49] 12
"Blackwood"
[0, 5, 0, -14, 0, 8] [28/27, 49/48] 15
"Augmented"
[3, 0, 6, 14, -1, -7] [36/35, 128/125] 33
"Kleismic"
[6, 5, 3, -7, 12, -6] [49/48, 126/125] 53
"Tripletone"
[3, 0, -6, -14, 18, -7] [64/63, 126/125] 27
"Hemifourths"
[2, 8, 1, -20, 4, 8] [49/48, 81/80] 19
"Meantone"
[1, 4, 10, 12, -13, 4] [81/80, 126/125] 31
"Injera"
[2, 8, 8, -4, -7, 8] [50/49, 81/80] 26
"Double Wide"
[8, 6, 6, -3, 13, -9] [50/49, 875/864] 26
"Porcupine"
[3, 5, -6, -28, 18, 1] [64/63, 250/243] 22
"Magic"
[5, 1, 12, 25, -5, -10] [225/224, 245/243] 41
"Nonkleismic"
[10, 9, 7, -9, 17, -9] [126/125, 1728/1715] 89
"Semisixths"
[7, 9, 13, 5, -1, -2] [126/125, 245/243] 46
"Orwell"
[7, -3, 8, 27, 7, -21] [225/224, 1728/1715] 84
"Miracle"
[6, -7, -2, 15, 20, -25] [225/224, 1029/1024] 72
"Supermajor Seconds"
[3, 12, -1, -36, 10, 12] [81/80, 1029/1024] 31
"Schismic"
[1, -8, -14, -10, 25, -15] [225/224, 3125/3087] 94
"Superkleismic"
[9, 10, -3, -35, 30, -5] [875/864, 1029/1024] 41
"Semififths"
[2, 8, -11, -48, 23, 8] [81/80, 6144/6125] 31
"Catakleismic"
[6, 5, 22, 37, -18, -6] [225/224, 4375/4374] 72
"Ennealimmal"
[18, 27, 18, -34, 22, 1] [2401/2400, 4375/4374] 612
Message: 5923 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 00:27:30
Subject: Re: Poptimal generators
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:
>> you must have skipped over most of gene's original message. why
>> do you think we're having this discussion about absolute error
>> criteria in the first place? it's because gene suggested using
>> the entire range of p-norms, with p from 2 to infinity, and i
>> suggested moving the lower limit down to 1.
>
> How does this sort of thing compare with the heuristic approach?
In
general, requiring poptimal generators seems to be a little
over-precisein its demands; however combining the 7-limit with the
9-limit helped. Paul's idea would relax things some, but I don't
really trust p to go all the way down to 1 myself. I think we have
good candidates for most of the really significant 7-limit
temperaments, but the 11 and 5 limit stuff (which I've looked at but
not posted) again seems to be a bit problematic.
Message: 5924 - Contents - Hide Contents
Date: Fri, 10 Jan 2003 01:10:31
Subject: Heptadecal temperament
From: Gene Ward Smith
I mentioned that I've looked at 5 and 11 limit poptimal generators. One thing I discovered is that generators can be "universal" (stuck
on one value of p; dF/dp = 0 in the objective function F) so there is
no use trying to prove it can't happen.
The example is the temperament deriving from the heptadecal comma,
(25/24)^17 ~ 2, which is not to be confused with the minortone comma,
(10/9)^17 ~ 6, though if you use the 901-et you can have both and
marry the17 Revolution to 53-et besides. It gives a microtemperament
which can be described as 17-equal divisions, separated by intervals
of exactly sqrt(3) for the optimum tuning. Since 485/612 is only
0.00289 cents sharper than sqrt(3), there is no practical reason not
to useit to notate Heptadecal, but it belligerently persists in
wanting sqrt(3) for all values of p.
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