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Message: 11425 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 18:21:35

Subject: Re: names and definitions: meantone

From: monz

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


> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:


>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s..
.> 
>> wrote:
>> 
>> re: 
>> Frame Index for Tuning Dictionary * [with cont.]  (Wayb.)
>> meantone
> 


> It doesn't seem to be up.




looks OK to me.  did you hit "refresh/reload"?

Gene, when you wrote, for example,
"7 limit poptimal: 86/205", you *did* mean
"7-limit poptimal *generator*", right?



-monz
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Message: 11426 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 02:27:54

Subject: Re: Beep and bug

From: Gene Ward Smith

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



> What I'm saying is that's the one that would logically get the name 
> "bug" if we extend the 5-limit names to higher limits.



But I want to steal either "beep" or "bug" for this musical analysis
purpose, leaving aside the question of actually using it as a temperament.
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Message: 11427 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 08:25:46

Subject: Re: Atomic temperament as a basis for a valhalla-level notation

From: Gene Ward Smith

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



> If you really had to do 2460-ET you'd have to invent a new kind of 
> accent mark that is defined as some comma (specifically a schismina) 
> somewhere in the 0.3 to 1 cent range, preferably around 0.6 or 0.7 
> of a cent, that doesn't vanish in the 12-fold schismas (atomic) 
> temperament.



I don't know that I really have to do it, but it seems worthwhile to
notate if you are on a quest for ultra notations, which I gathered
from what George said you were. We already have our nominals in a very
familiar form, say Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#. It would
be possible to find 23 limit intervals corresponding to n schismas for
various values of n, and then see how Sagittal would propose to notate
those intervals. This would, it seems to me, give a good start towards
notating atomic. Of course another approach would be to notate the
5-limit and then use the result for the 23-limit.
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Message: 11428 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 18:24:27

Subject: need definition: copoptimal

From: monz

someone please write a definition of copoptimal for
inclusion into the Encyclopaedia.


-monz
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Message: 11429 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 08:29:35

Subject: Re: names and definitions: schismic

From: Gene Ward Smith

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

name: garibaldi, 41&53
wedgie: <<1 -8 -14 -15 -25 -10||
mapping: [<1 2 -1 -3|, <0 -1 8 14|]
7&9 limit copoptimal generator: 39/94
TM basis: {225/224, 3125/3087}
MOS: 12, 17, 29, 41, 53

Left off 225/224.
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Message: 11430 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 18:26:03

Subject: need definition: wedgie

From: monz

can someone please write an *understandable* definition
of wedgie, for inclusion into the Encyclopaedia?

i have a copy of a list posting there for right now,
but it just illustrates and doesn't define.

the definitions on Gene's website i'm sure are the
most precise ones, but they're incomprehensible to me.



-monz
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Message: 11431 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 03:34:56

Subject: Re: Naming temperaments

From: Dave Keenan

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


>> Right. But why _is_ this 11/27 and not 20/49?
> 


> It could be either one: a 27-note scale with an approximate 11/27 
> generator or a 49-note scale with an approximate 20/49 generator. 
But a 
> 27-note scale is simpler, and it's good to have an idea of the 
simplest 
> useful scale associated with a temperament. Note that in the 11-
limit 
> these represent different temperaments:
> 
> 11/27 [<1, 2, 6, 2, 1|, <0, -1, -9, 2, 6|]
> 20/49 [<1, 2, 6, 2, 10|, <0, -1, -9, 2, -16|]



It's not only different limits that cause this sort of problem, it's 
also different optimisation criteria at the same limit. TOP and 
minimax-beat-rate are two extremes that will sometimes give 
generators so different as to correspond to different ET/MOS/DEs. 
Any time the generator happens to be very close to some small-
denominator fraction of an octave this will be a problem, e.g. with 
5-limit Diminished, as I mentioned in another post in this thread.

Is there any way we are ever going to agree on which octave fraction 
is most representative of the temperament. It seemed like Gene was 
nailing that down somewhat with p-optimal, but then along came TOP 
(or was it copoptimal? -- I have no idea what that is).



> 4/46 gives you [<2, 3, 5, 7|, <0, 1, -2, -8|], not pajara (see 
> A third kind of linear temperament * [with cont.]  (Wayb.)).



Oops. Sorry. That should have been 8/46. But even so, that's only 
good for "5-limit pajara" or diaschismic.

My question was whether we all agree we should use the smallest 
possible value of the generator ( the one that's less than half the 
period) in these octave-fraction-type names? I note that Paul is not 
doing this in his paper when he gives generators in cents. He is 
using whatever falls out of a simple algorithm for deriving the 
mapping from a set of vanishing commas.

I note that the number of periods per octave can be obtained as the 
GCD of numerator and denominator, since we won't be reducing an 
octave fraction like 8/46.



> Pajara is historically associated with 22-ET, of course. But you 
can 
> think of the denominator as representing the size of a typical MOS 
scale 
> associated with the temperament, rather than an ET. In that case, 
the 
> minimum is 10 steps, which matches Paul's decatonic scale.



As I've said elsewhere, in the two-ET/MOS/DE method (the two-
cardinalities method?) of naming, I'd like the two numbers to give 
the denominators of two convergents (or semi-convergents) of the 
generator as an octave fraction, such that one is near the minimum 
useful generator size and the other is near the maximum. 

It would be ideal if you could also obtain the typical MOS/DE 
cardinality by subtracting these two numbers, and obtain a good 
approximation of an optimum generator by adding them. 

For example, calling meantone the "12&19-LT" works perfectly. 12-ET 
and 19-ET are very near the extreme generator values re "harmonic 
waste". The typical MOS cardinality is 19-12 = 7, and a near optimum 
occurs at 12+19 = 31-ET.

I think at least one of the two numbers should be a convergent, i.e. 
it should give the cardinality of a Rothenberg-proper MOS/DE for 
most optimum generator sizes.



>>> if you've heard of 
>>> "superpyth", you take one look at the name and say "Ah yes, 
*that* 
>>> temperament". You might not immediately recognize "7/31" 
>> 


>> or "9&22", but 
>> 
>> Why wouldn't you immediately recognise one of these, if that's 
what 
>> you've been used to seeing it called.
> 


> Combinations of numbers aren't especially easy to remember. It 
would be 
> like using ZIP codes to refer to cities in the US, instead of 
names; 
> they all look alike.



Don't you recognise your own zipcode and those of people you 
regularly send mail to, and similarly phone numbers, although these 
have many more digits than we're talking about here. It's more like 
Australian postcodes, which are only four digits. But not even as 
bad as that. 

They are generally pairs of only 2 digit numbers taken from a small 
set of n-limit consistent ETs, whose numbers already have many 
associations for us. So it's really just associating pairs of 
already familiar things that we're already used to representing as 
numbers. And they have the enormous advantage that they are not 
totally opaque jargon to a newcomer, as are names like sensipent, 
orson, amity, subchrome, wurschmidt, compton.

At least when I see a postcode I've never seen before, I can 
immediately tell what state it's in, and sometimes I can figure out 
some towns I know that it must be near.



>>> probably anyone who's familiar with it knows the name "orwell". 
>> 


>> That's a tautology.
> 


> The pronoun "it" refers to the temperament represented by "7/31" 
and 
> "9&22", which happens to be named "orwell". I think it's a fairly 
safe 
> assumption that most people who've heard of this temperament will 
> recognize that name.



Only if they have been reading the tuning lists, and even then they 
may only know that they seen the name but have no way of picking it 
out of the jargon-diarrhoea that we're swimming in.

Imagine if we hadn't learnt that George Secor originally 
discovered "Miracle", and George Secor turned up on the list for the 
firs time now. How long would it take him to realise we were talking 
about his temperament every time we write "miracle". compared to if 
we were instead calling it the "31&41-LT" or the "7/72-oct-LT"?



> (Certainly "19/84" is more familiar, but it implies 
> a greater degree of complexity, and could easily be overlooked by 
people 
> who don't care for highly complex scales.)



Yes. 84-ET is outside of most people's familiarity zone. But suppose 
someone who had independently discovered that temperament turned up 
on the lists. How long would it take them to figure out "orwell" as 
opposed to "subminor thirds".

Gene complains that some of these descriptive names are "a 
mouthfull". So what? How many times a day do you find yourself 
having to say or type them? Anyway, what's a few extra keystrokes 
for one person in exchange for a whole lot of extra understanding on 
the part of a whole lot of readers.



>>> So you might as well use the name in combination with one 
>>> or more of the other methods, especially if it's a less familiar 
>> 
>> one 
>> 


>>> like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.
>> 
>> 


>> So who needs the "lemba". It adds absolutely nothing, for me. 
For 
>> some reason it suggests "unleavened bread" to me. Huh?
> 


> Anyone who's vaguely heard of it, but doesn't know much about it. 
You 
> might not know how to recognize a beech tree if you see one, or 
how it 
> differs from other trees, but you probably know that the 
word "beech" 
> represents a kind of tree. I don't know the mapping 


of "nonkleismic" off 


> the top of my head, and probably wouldn't recognize it if I saw it 
(it's 
> [<1, -1, 0, 1|, <0, 10, 9, 7|]), but I do recall it being (in 
theory) a 
> good temperament. "8/31" might give some idea of its usefulness, 
but 
> doesn't distinguish it from the many other "n/31" temperaments. So 
even 
> a questionable name like "nonkleismic" has some use. But "myna" is 
> better because it links it with the starling family of 
temperaments. 



I have been looking at "myna" in Paul's table and I never once 
associated it with "starling". To me it was just another random 
assembly of syllables. And even if I had, the name starling gives me 
no clues as to the identity of _that_ temperament.



> Along those lines, Gene's "Japanese monster" names also provide 
useful 
> hints to similarities between temperaments.



Sure, it's good to indicate similarities, but it still remains just 
an isolated clump of related somethings, with no clue as to what 
they are.

The point is, we can do a lot better. We can actually have names 
that give someone a clue, even when they have not been initaiated 
into the Smith-Erlich-Miller mysteries.

And in a situation where the names are a priori meaningless in 
musical terms, and so any one is good as another, why the heck do 
you guys have the need to keep changing them!!!!? It's tempting to 
assume it's sheer arrogance or egotism, such as I was (more or less) 
accused of when I wanted to use descriptive names in my 
Microtempered Guitar article for Xenharmonikon. So I included the 
cryptic/meaningless names as well, and now half of them are probably 
obsolete.



> Why does anything have a name? Why do we talk about "major thirds" 
when 
> we could call them 5/4's?



In that case, I assume it's historical. But notice that in both 
cases the term doesn't come from outer-space, but is descriptive, 
using words/symbols with something like their existing meanings.



> Language works by naming things; the problem 
> is that the study of linear and higher-dimensional temperaments is 
so 
> new that we haven't settled on the best names for things. So in 
the 
> meantime, names which may end up being changed will have to be 
> supplemented by numerical keys of one kind or another.



I totally disagree. I don't see any point to using musically-
meaningless names or eponyms that may have to be changed, except for 
the few most commonly discussed or used temperaments.

Paul Erlich recently pointed out that we use names for colours, not 
wavelength numbers. And I responded that, while most people can 
distinguish thousands of colours, they only use simple words for 
about 20, and use combinations of these, and adjectives like dark 
light etc.



> I'm leaning toward the fractional generator + period notation for 
> unfamiliar temperaments, with wedgies for those few that can't 
easily be 
> symbolized in this way. But I still find names easier to remember, 
and I 
> don't want to discourage the naming of temperaments that look like 
they 
> might be useful.



OK. Well It seems we aren't that far apart in our thoughts on this, 
but based on the colours thing, I wouldn't like to see non-
descriptive names for more than about the best 20. We've already got 
more than 50, and we've only got to the 7-limit.
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Message: 11432 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 08:45:43

Subject: Re: Atomic temperament as a basis for a valhalla-level notation

From: Gene Ward Smith

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

It would


> be possible to find 23 limit intervals corresponding to n schismas for
> various values of n, and then see how Sagittal would propose to notate
> those intervals. This would, it seems to me, give a good start towards
> notating atomic. Of course another approach would be to notate the
> 5-limit and then use the result for the 23-limit.



Here are some 11-limit intervals corresponding to certain quantities
of schismas in atomic. If we had symbols for these, it would be a start.

1: {32805/32768, 703125/702464, 5632/5625}
2: {441/440, 1375/1372, 6250/6237}
3: {3136/3125, 5120/5103}
4: {225/224}
5: {176/175, 896/891}
7: {4000/3969, 126/125}
9: {99/98, 100/99}
10: {2048/2025}
11: {81/80, 3125/3087, 245/242}
14: {64/63}
16: {56/55}
18: {50/49}
20: {45/44}
21: {128/125}
25: {36/35}
29: {405/392}
32: {648/625}
39: {256/245}
43: {360/343}
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Message: 11433 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 18:33:56

Subject: Re: names and definitions: schismic

From: Gene Ward Smith

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


> Gene Ward Smith wrote:
> 


>> Check the TOP tuning before blowing your top. This temperament is a
>> straightforward extension of 5-limit schismic; the TOP tunings are
>> very close. Garibaldi, which Paul wants to name something other than
>> schismic anyway, has a different tuning--instead of slightly flat,
>> like 171 or 118, the fifth of 94 is slightly sharp.
> 


> I can guess the TOP tuning, but why?  I don't give a shit about TOP.  I 
> never tune to the theoretical optima.  I can't even get the 7-limit 
> tetrads on my keyboard.



Someone who is tuning schismic to get highly accurate 5-limit
intonation will be tuning using sligtly flatted fifths--in the range
1/8 to 1/9-schisma schismic. This, historically, is what it has been
used for. Someone wanting that degree of accuracy, and who has tuned
up a 53 note MOS, will find that 39 of these slightly flattened fifths
will give a better 7 than will 14 of the slightly sharpened fourths;
those will be a decent enough 6.8 to 7 cents sharp, and certainly
useable, but the other 7s will be between 1 and 1.5 cents flat, which
is much better. Optimizing for the 7s will then do little damage to
5-limit harmony, since it is a matter of moving from 1/9 to 1/10
schisma flat. The fifth of 94-et is 1/11 schisma *sharp*, and not at
all optimal for 5-limit harmony; 94 would never be considered for
5-limit schisma.
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Message: 11434 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 03:47:07

Subject: Re: Naming temperaments

From: Dave Keenan

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


> At 06:37 PM 7/16/2004, Dave Keenan wrote:


>>> How do you choose a period/generator representation?
>> 


>> You don't have to. You just base it directly on the map -- any 
map 
>> that's valid. i.e. the generator doesn't have to be in lowest 
>> (period-reduced terms).
> 


> So there'll be multiple names for each temperament?



No. The name ends up the same.



>> For any of the ET/MOS/DE-based names you need to choose specific 
>> values of period and generator. In most cases, different kinds of 
>> optima do not change the period and generator enough to make much 
>> difference, but I just found that while minimax and RMS versions 
of 
>> 5-limit Diminished can be described as 12&16-LT or 12&28-LT, the 
TOP 
>> version cannot. It could be described as 12&20-LT or 12&32-LT but 
20 
>> and 32 are not 5-limit consistent, so the best you can do
>> is 8&12-LT.
> 


> I think that's why Gene is proposing to use ETs that represent the
> extreme ranges of the generator.



Sure, I like that idea, but what you consider extreme, depends on 
what you consider optimal.



>> The map tells you how many periods to the octave. That's all you 
>> need to know about the period to know whether the temperament is 
>> twin or triple etc.
> 


> What about temperaments that map 2 through a combination of both
> the "period" and generator?



Aw c'mon Carl. Gimme a break. :-)

None of the other methods mentioned in this thread (except giving 
the full wedgie or a full mapping) can handle that either.



> Hmm... I thought one could refactor these maps is several
> annoying ways.  Thus, the reason for something called hermite
> normal form -- whatever that is.



Sure, but it makes no difference, assuming we agree to use the 
smallest of the equivalent generators (the one that's less than half 
the period) and describe that generator as a fraction of the diamond 
ratio that requires the fewest (absolute number of) generators (and 
zero periods). Just as we describe the period as a fraction of the 
octave.

But I certainly am assuming that the map has no generators in prime-
2, as is the case for 99.9% of maps we've ever talked about.
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Message: 11435 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 10:09:04

Subject: Re: names and definitions: schismic

From: Graham Breed

Gene Ward Smith wrote:



> name: schismic, 118&171
> wedgie: <<1 -8 39 -15 59 113||
> mapping: [<1 2 -1 19|, <0 -1 8 -39|]
> 7&9 limit copoptimal generator: 732/1763
> TM basis: {4375/4374, 32805/32768}
> MOS: 12, 17, 29, 41, 53, 65, 118, 171



What, *this* is the default septimal schismic?  But it's the absurdly 
complex one!  If it has to have a name, make it "microschismic".



> name: garibaldi, 41&53
> wedgie: <<1 -8 -14 -15 -25 -10||
> mapping: [<1 2 -1 -3|, <0 -1 8 14|]
> 7&9 limit copoptimal generator: 39/94
> TM basis: {225/224, 3125/3087}
> MOS: 12, 17, 29, 41, 53



This should be "schismic".  It's consistent with all those ETs except 17 
(which isn't 7-limit consistent in itself).



> name: schism, 12&17
> wedgie: <<1 -8 -2 -15 -6 18||
> mapping: [<1 2 -1 2|, <0 -1 8 2|]
> 7 limit poptimal generator: 27/65
> 9 limit poptimal generator: 22/53
> TM basis: {64/63, 360/343}
> MOS: 12, 17, 29, 41, 53



That name's confusingly similar to "schismic".  It also looks like a 
white elephant.  And certainly don't call it "12&17" because this is 
ambiguous with the 7/9-limit optimal mapping of 17-equal, which happens 
to give an all-round better temperament in this case.



> 11 limit
> 
> name: garibaldi, 41&53
> wedgie: <<1 -8 -14 23 -15 -25 33 -10 81 113||
> mapping: [<1 2 -1 -3 13|, <0 -1 8 14 -23|]
> poptimal generator: 95/229
> TM basis: {225/224, 385/384, 2200/2187}
> MOS: 12, 17, 29, 41, 53, 94
> 
> name: garybald, 29&41
> wedgie: <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
> mapping: [<1 2 -1 -3 -4|, <0 -1 8 14 18|]
> poptimal generator: 63/152
> TM basis: {100/99, 225/224, 245/242}
> MOS: 12, 17, 29, 41, 70



More confusingly similar names.


                         Graham
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Message: 11436 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 18:36:37

Subject: Re: names and definitions: schismic

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:



> what do you guys think?  should i put Gene's data into
> my "schismic" Encyclpaedia webpage, and correct as we
> go along?  or should i wait until there's been more of
> a consensus here?



The data is more important than the names; I'd put the pages up and
wait to see if some kind of consensus emerged, or if some compromise
could be found.
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Message: 11437 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 03:49:53

Subject: Re: Extreme precison (Olympian) Sagittal

From: Dave Keenan

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


> I was just hoping for a statement of the problem which did not 
require
> me to read things I have to squint at and still can't make out. 
What
> are the barbs, arcs, scrolls, boathooks and accent marks supposed 
to
> do, in numerical terms?



I'll get that for you eventually Gene. Sorry for the delay.
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Message: 11438 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 09:37:20

Subject: Re: names and definitions: schismic

From: Gene Ward Smith

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


> Gene Ward Smith wrote:
> 


>> name: schismic, 118&171
>> wedgie: <<1 -8 39 -15 59 113||
>> mapping: [<1 2 -1 19|, <0 -1 8 -39|]
>> 7&9 limit copoptimal generator: 732/1763
>> TM basis: {4375/4374, 32805/32768}
>> MOS: 12, 17, 29, 41, 53, 65, 118, 171
> 


> What, *this* is the default septimal schismic? 



Check the TOP tuning before blowing your top. This temperament is a
straightforward extension of 5-limit schismic; the TOP tunings are
very close. Garibaldi, which Paul wants to name something other than
schismic anyway, has a different tuning--instead of slightly flat,
like 171 or 118, the fifth of 94 is slightly sharp.

 But it's the absurdly 


> complex one!  If it has to have a name, make it "microschismic".



I propose to keep a little consistency and order in the naming of
these things, by making the tunings in the various limits correspond.
Dave likes that because he likes things being systematic, and Paul
does not seem eager to give the same name to different limits at all,
so presumably at least having the tunings correspond would be
important to him.



>> name: garibaldi, 41&53
>> wedgie: <<1 -8 -14 -15 -25 -10||
>> mapping: [<1 2 -1 -3|, <0 -1 8 14|]
>> 7&9 limit copoptimal generator: 39/94
>> TM basis: {225/224, 3125/3087}
>> MOS: 12, 17, 29, 41, 53
> 


> This should be "schismic".  It's consistent with all those ETs
except 17 
> (which isn't 7-limit consistent in itself).



I did name it that at first, and then changed it because the tunings
were different; morever Paul wants to name it garibaldi anyway. I
think it makes sense not to claim this is really schismic, given that
the fifth is actually sharp and schismic wants it to be very slightly
flat, by 2/9 of a cent.



>> name: schism, 12&17
>> wedgie: <<1 -8 -2 -15 -6 18||
>> mapping: [<1 2 -1 2|, <0 -1 8 2|]
>> 7 limit poptimal generator: 27/65
>> 9 limit poptimal generator: 22/53
>> TM basis: {64/63, 360/343}
>> MOS: 12, 17, 29, 41, 53
> 


> That name's confusingly similar to "schismic".  



The name is supposed to tell you it is like schismic, but clearly is
not--which it does. Confusing schism with schismic seems prettyn
confused, but recalling they are related should be easy.

It also looks like a 


> white elephant.



Well, it's been sitting on my 7-limit list, down there at Number 89,
and people keep finding uses for temperaments far down on that list;
lemba is Number 82 and Herman loves it, along with superpelog (Number
107.) Anyway I don't think it is any more of a white elephant than
grackle. 

  And certainly don't call it "12&17" because this is 


> ambiguous with the 7/9-limit optimal mapping of 17-equal, which happens 
> to give an all-round better temperament in this case.



So how is the optimal 17-equal temperament defined? I called it 12&17
because the "standard" 17 using closest approximations to primes gives
us this.



>> name: garibaldi, 41&53
>> wedgie: <<1 -8 -14 23 -15 -25 33 -10 81 113||
>> mapping: [<1 2 -1 -3 13|, <0 -1 8 14 -23|]
>> poptimal generator: 95/229
>> TM basis: {225/224, 385/384, 2200/2187}
>> MOS: 12, 17, 29, 41, 53, 94
>> 
>> name: garybald, 29&41
>> wedgie: <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
>> mapping: [<1 2 -1 -3 -4|, <0 -1 8 14 18|]
>> poptimal generator: 63/152
>> TM basis: {100/99, 225/224, 245/242}
>> MOS: 12, 17, 29, 41, 70
> 


> More confusingly similar names.



For temperaments which happen to be the same in the 7-limit.
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Message: 11439 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 18:38:32

Subject: Re: names and definitions: meantone

From: Gene Ward Smith

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


> Gene Ward Smith wrote:
> 


>> name: meantone, 12&19
>> comma: 81/80
>> mapping: [<1 2 4|, <0 -1 -4|]
>> poptimal generator: 34/81
>> MOS: 5, 7, 12, 19, 31, 50, 81
> 


> This MOS list is valid for quarter-comma meantone and Kornerup's golden 
> meantone (among others), but as we've seen, various tunings of the same 
> temperament can have different MOS structures. In particular, meantone 
> with a 23/55 generator/period ratio (Mozart's tuning) has a 43-note MOS 
> (12L+31s), but not one with 50 notes. 1/5-comma meantone technically
has 
> a 43-note MOS, but it's so close to 43-ET that the step sizes are 
> roughly equal.



True enough; I went with the proposition that MOS for the optimized
tunings were the proper ones to list. Trying to list all of them gets
a little out of hand, and could even be confusing.
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Message: 11440 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 03:58:51

Subject: Re: Naming temperaments

From: Dave Keenan

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


> Dave Keenan wrote:


>> e.g. Miracle is "semi 7:8's" or "semi supermajor seconds". 
> 


> This could be potentially useful up to a point; certainly there's 
a 
> mnemonic value in names like "semisixths". But I don't see how 
this can 
> be generalized to the 7-limit and higher without being arbitrary.
> Which 
> LT gets to be called "fourths" -- dominant (5&12), meantone 
(12&19), 
> superpyth[agorean] (5&22), flattone (19&26), or schismic (12&29)? 
You 
> could make good arguments at least for dominant, meantone, and 
schismic; 
> then you need to figure out how to name the others. "Major thirds" 
could 
> be either muggles (16&19) or magic (19&22), and so on.



Right. This is where we use adjectives like wide and narrow applied 
to the generator (but only where these don't imply a different n-
limit ratio entirely).

And the fallback method is to rank them by some badness measure 
(probably most reasonable badness measures will agree on the ranking 
of temperaments having the same ratio approximated by their 
generator), and then the best one gets to have no adjective and the 
others are called "complex", "supercomplex" or "inaccurate", "super-
inaccurate", as the case may be. Is their a shorter word 
for "inaccurate"?



>> This is used up to some point where the LT is so complex you 
just 
>> describe the generator in cents. e.g. What used to be called 
>> Aritoxenean is the 12-fold 15 cent LT.
>> 
>> This at least works up to 11-limit.
> 


> But giving the generator in cents doesn't determine a unique 
mapping; 
> you can derive one from a rational generator/period ratio if you 
make 
> some assumptions, but an arbitrary value in cents could represent 
more 
> than one temperament. An LT with a 316.5 cent generator can be 
mapped as 
> [<1, 0, 1, 2|, <0, 6, 5, 3|] or [<1, 0, 1, -3|, <0, 6, 5, 22|]. 
With 
> rational generators and the naming conventions I've described, you 
can 
> unambiguously describe the first mapping as 5/19 and the second as 
19/72.



I only proposed using cents when the temp is so complex that the 
diamond ratio with the fewest generators has say 5 or more 
generators in it. That's getting pretty complex. How many 12-fold 15-
cent temperaments do you know of?
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Message: 11441 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 10:43:49

Subject: On the Mappings of Primes to Degrees of Equal Tunings

From: Kalle Aho

Hi, 

the expression "n-equal" used to refer to the division of pure octave 
into n equal parts. So it was completely self-evident that standard 
mapping of primes was defined as round(n*log(p)/log(2)) where n is 
the number of steps to the octave and p is the mapped prime. This 
simply gave the best approximations to the primes. 

Since the discovery of TOP tuning paradigm prime number 2 has lost 
some of its specialness. The octave is no longer automatically 
assumed to be just. Because TOP equal tunings are defined by their 
mapping of primes there are many flavours of n-equal tunings. But I 
think some of these mappings are somehow more natural than others. 
Let me show what I mean.

The 7-limit standard mapping of primes to 12-tone equal division of 
2:1 is [12 19 28 34] (or <12 19 28 34] if you want). Now let's see if 
the division of 3:1 into 19 equal parts gives the same mapping. Yes 
it does. The same holds for 28-tone equal division of 5:1 and 34-tone 
equal division of 7:1. The primes 2, 3, 5 and 7 kind of "agree" about 
the mapping. 

Def. mapping m from a set P of primes to integers (or degrees of an 
equal tuning) is natural for P if m(p)=round(m(q)*log(p)/log(q)) for 
all p and q belonging to P. 

Another example: there is no natural mapping of P={2, 3, 5, 7} to 
integers where m(2)=16 because round(16*log(3)/log(2))=25 and round
(16*log(7)/log(2))=45 but round(25*log(7)/log(3))=44. 

The name "natural" for these mappings is just a suggestion and if you 
think it sucks please come up with a better one. Maybe something 
derived from the verb "agree" would be better. 
 
Some natural mappings:

P={2, 3, 5}

3 5 7
4 6 9
5 8 12
7 11 16
8 13 19
9 14 21
10 16 23
12 19 28
15 24 35
16 25 37
18 29 42
19 30 44

P={2, 3, 7}

4 6 11
5 8 14
9 14 25
10 16 28 (contorsion) 
12 19 34
14 22 39
17 27 48
18 29 51
19 30 53
22 35 62
24 38 67
26 41 73
27 43 76

P={2, 3, 5, 7}

4 6 9 11
9 14 21 25
10 16 23 28
12 19 28 34
18 29 42 51
19 30 44 53
22 35 51 62
26 41 60 73
27 43 63 76
29 46 67 81
31 49 72 87
41 65 95 115

Notice that 5, 15 and 16 are missing. 

P={2, 3, 5, 7, 11}

9 14 21 25 31
22 35 51 62 76
29 46 67 81 100
31 49 72 87 107
41 65 95 115 142
46 73 107 129 159
49 78 114 138 170
58 92 135 163 201
63 100 146 177 218
72 114 167 202 249
80 127 186 225 277
87 138 202 244 301


Kalle Aho
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Message: 11442 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 18:55:34

Subject: Re: names and definitions: meantone

From: Gene Ward Smith

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



> This MOS list is valid for quarter-comma meantone and Kornerup's golden 
> meantone (among others), but as we've seen, various tunings of the same 
> temperament can have different MOS structures. 



One possible answer to this question which works for meantone, but not
anything else, is to use 1/4-comma meantone. Meantone is provided with
a standard, classical definition of its generator, so we can do this,
but it won't work for anything else. The semiconvergents of the
1/4-comma fifth give us 5, 7, 12, 19, 31, 50, 81, 112, 143, 205 ...
for MOS, and it would certainly be possible to view these as standard.
It matches the list I gave up to the point I stopped.
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Message: 11443 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 12:09:04

Subject: Re: names and definitions: schismic

From: Graham Breed

Gene Ward Smith wrote:



> Check the TOP tuning before blowing your top. This temperament is a
> straightforward extension of 5-limit schismic; the TOP tunings are
> very close. Garibaldi, which Paul wants to name something other than
> schismic anyway, has a different tuning--instead of slightly flat,
> like 171 or 118, the fifth of 94 is slightly sharp.



I can guess the TOP tuning, but why?  I don't give a shit about TOP.  I 
never tune to the theoretical optima.  I can't even get the 7-limit 
tetrads on my keyboard.



> I propose to keep a little consistency and order in the naming of
> these things, by making the tunings in the various limits correspond.
> Dave likes that because he likes things being systematic, and Paul
> does not seem eager to give the same name to different limits at all,
> so presumably at least having the tunings correspond would be
> important to him.



I thought Dave called a stop to the jargon explosion.  Paul may decide 
to give redundant names to everything, but let's not suddenly change the 
meaning of "schismic".  It should be perfectly simple to construct a 
consistent naming scheme that calls "schismic" "schismic".

See here:

Schismic temperaments * [with cont.]  (Wayb.)

I've been calling this "schismic" for the past 5 years, on a site that 
has been indexed and archived.  The spelling's in flux, but it's too 
late to change the meaning now.



> I did name it that at first, and then changed it because the tunings
> were different; morever Paul wants to name it garibaldi anyway. I
> think it makes sense not to claim this is really schismic, given that
> the fifth is actually sharp and schismic wants it to be very slightly
> flat, by 2/9 of a cent.



Schismic doesn't "want" anything.  I always preferred sharp fifths 
anyway, even in the 5-limit.  The Pythagorean commas get too small 
otherwise.  Schismic is quite often described with Pythagorean tuning. 
It's rarely given enough notes for this microtempered mapping to make sense.



> The name is supposed to tell you it is like schismic, but clearly is
> not--which it does. Confusing schism with schismic seems prettyn
> confused, but recalling they are related should be easy.



The more closely things are related, the more chance there is of 
confusion if they have similar names.  If somebody talked about "schism" 
temperament I'd assume they meant schismic.  Especially as that has a 
variety of spellings already.  I certainly wouldn't pick up on the 
missing "ic" denoting an obscure 7-limit mapping.



> Well, it's been sitting on my 7-limit list, down there at Number 89,
> and people keep finding uses for temperaments far down on that list;
> lemba is Number 82 and Herman loves it, along with superpelog (Number
> 107.) Anyway I don't think it is any more of a white elephant than
> grackle. 



Number 89???  And how many of these are you planning to name?  I see 
you've already broken your resolution about reminding us what the names 
are supposed to refer to.



> So how is the optimal 17-equal temperament defined? I called it 12&17
> because the "standard" 17 using closest approximations to primes gives
> us this.



The one with the lowest 7- or 9-limit minimax (they agree in this case).




>> More confusingly similar names.
> 


> For temperaments which happen to be the same in the 7-limit.



Yes, so who's going to remember which one the trailing "i" is supposed 
to refer to?


                   Graham




________________________________________________________________________
________________________________________________________________________



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<*> To visit your group on the web, go to:
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<*> To unsubscribe from this group, send an email to:
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Message: 11444 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 19:19:17

Subject: MOS/semiconvergents for various meantone fifths

From: Gene Ward Smith

1/11 comma 5, 7, 12, 19, 31, 43, 55, 67, 79, 91, 103, 115 ...
1/6 comma 5, 7, 12, 19, 31, 43, 55, 67, 122, 189, ...
1/5 comma 5, 7, 12, 19, 31, 43, 74, 117, 160, 203, ...
2/11 comma 5, 7, 12, 19, 31, 43, 55, 98, 153, 208, ...
1/4 comma 5, 7, 12, 19, 31, 50, 81, 112, 143, 174, ...
3/11 comma 5, 7, 12, 19, 31, 50, 81, 131, 181, 231, ...
2/7 comma 5, 7, 12, 19, 31, 50, 69, 119, 169, 288, ...
1/3 comma 5, 7, 12, 19, 31, 50, 69, 88, 107, 126, ...
2/5 comma 5, 7, 12, 19, 26, 45, 64, 109, 154, 199, ...
1/2 comma 5, 7, 12, 19, 26, 33, 59, 92, 125, 217, ...
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Message: 11445 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 04:30:49

Subject: Re: Naming temperaments

From: Gene Ward Smith

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



> I only proposed using cents when the temp is so complex that the 
> diamond ratio with the fewest generators has say 5 or more 
> generators in it.



Speaking of jargon, what is a diamond ratio?
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Message: 11446 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 13:50:17

Subject: Re: names and definitions: orwell

From: Herman Miller

family name: orwell
period: octave
generator: 5/22, 7/31, 9/40, 12/53, 17/75, 19/84
TOP period = 1199.532657, generator = 271.4936472
wedgie: <<7, -3, 8, -21, -7, 27||
mapping: [<1, 0, 3, 1|, <0, 7, -3, 8|]
11-limit versions:
[<1, 0, 3, 1, 3|, <0, 7, -3, 8, 2|]
[<1, 0, 3, 1, -4|, <0, 7, -3, 8, 33|] (19/84)

Typical MOS: 9, 13, 22, 31, 53

9-note MOS:
-4  Eb+ (16/15)
-3  F#- (5/4)
-2  A-< (35/24)
-1  B>  (12/7)
0   D   (1/1)
+1  F<  (7/6)
+2  G+> (48/35)
+3  Bb+ (8/5)
+4  C#- (15/8)
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Message: 11447 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 05:13:37

Subject: Re: Naming temperaments

From: monz

hi Dave and Herman (and everyone else),


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

wrote:


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

> As I've said elsewhere, in the two-ET/MOS/DE method
> (the two-cardinalities method?) of naming, I'd like the
> two numbers to give the denominators of two convergents
> (or semi-convergents) of the generator as an octave fraction,
> such that one is near the minimum useful generator size
> and the other is near the maximum. 
>
> <snip most of a long and interesting post>
>


>> I'm leaning toward the fractional generator + period
>> notation for unfamiliar temperaments, with wedgies for
>> those few that can't easily be symbolized in this way.
>> But I still find names easier to remember, and I 
>> don't want to discourage the naming of temperaments
>> that look like they might be useful.
> 


> OK. Well It seems we aren't that far apart in our 
> thoughts on this, but based on the colours thing, 
> I wouldn't like to see non-descriptive names for more
> than about the best 20. We've already got more than 50,
> and we've only got to the 7-limit.




i'm convinced that those of you who are putting forth
the argument against the cute verbal names are just
frustrated at the inability to keep up with the pace
of developments in tuning theory recently.

verbal names are really easy to remember, and short
concise numerical descriptions convey a lot of data,
so why not just use both?

i've always been a huge fan of redundant coding.
it makes life easy.  so what if there is already one
name for something? natural linguistic processes are
always coining new names for old things.

i pursue the goal of including individual entries for
all these different temperaments and numerical descriptions
in the Encyclopaedia.  that way a reader can simply look
up any name that's unfamiliar.  i'm just buried with
work and haven't kept up with the lists for a long time
until recently, and would have to study a bit to be
able to write those pages.

if anyone else would like to contribute to this project,
please just post your efforts here and i'll make the
posts into webpages.  i'll start the project by sending
a bunch of posts with the temperament names in the
subject lines.

if no-one objects to this, then by all means please
feel free to contribute more names as individual threads.
doing this will also coalesce a lot of related data
together for the archives of this list.



-monz
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Message: 11448 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 14:26:36

Subject: Re: need definition: wedgie

From: Herman Miller

monz wrote:



> can someone please write an *understandable* definition
> of wedgie, for inclusion into the Encyclopaedia?
> 
> i have a copy of a list posting there for right now,
> but it just illustrates and doesn't define.
> 
> the definitions on Gene's website i'm sure are the
> most precise ones, but they're incomprehensible to me.



The easiest way to get these is to start with a pair of vals, such as a 
tuning map. Take meantone as an example. Here's a map that represents 
the meantone temperament:

[<1, 2, 4, 7|, <0, -1, -4, -10|]

This map is based on using a fourth as a generator; if you use a fifth, 
the map is different.

[<1, 1, 0, -3|, <0, 1, 4, 10|]

To form the wedgie, take each possible combination of two prime numbers: 
2 and 3, 2 and 5, 2 and 7, 3 and 5, 3 and 7, 5 and 7. By convention, the 
elements of the wedge product are calculated in this order: all the 
combinations with 2 first, in numerical order, then the remaining ones 
with 3, 5, 7, and so on. To calculate an element of the wedge product 
involving primes "a" and "b", multiply the period map of prime "a" by 
the generator map of prime "b", then subtract the product of the 
generator map of prime "a" times the period map of prime "b". In other 
words, if val1 represents the period map (e.g. <1, 2, 4, 7|) and val2 
represents the generator map (<0, -1, -4, -10|), the element of the 
wedgie is given by

val1[a] val2[b] - val2[a] val1[b]

Example: calculating the wedgie of meantone:
[<1, 2, 4, 7|, <0, -1, -4, -10|]

1st element (2, 3): (1 * -1) - (0 * 2) = -1
2nd element (2, 5): (1 * -4) - (0 * 4) = -4
3rd element (2, 7): (1 * -10) - (0 * 7) = -10
4th element (3, 5): (2 * -4) - (-1 * 4) = -4
5th element (3, 7): (2 * -10) - (-1 * 7) = -13
6th element (5, 7): (4 * -10) - (-4 * 7) = -12

Result: <<-1, -4, -10, -4, -13, -12||.

By convention, if the first number is negative, the wedgie is normalized 
by multiplying each element by -1. So the normalized wedgie for meantone 
is <<1, 4, 10, 4, 13, 12||.

Note that you get the same result if you start from the fifth-based map. 
The nice thing about wedge products is that all tuning maps that temper 
out the same commas end up with the same wedgie to represent them.

You can also wedge commas together, but the resulting wedgie needs to be 
reversed and some of the signs negated. Say you want to wedge the 
meantone comma 81;80 with the starling comma 126;125. You start with the 
monzo representation |-4 4 -1 0> for 81;80 and |1 2 -3 1> for 126;125. 
After wedging them together, the result is ||12, -13, 4, 10, -4, 1>>. In 
order to convert this to the standard form, first reverse it: <<1, -4, 
10, 4, -13, 12||, then multiply the 2nd and 5th elements by -1: <<1, 4, 
10, 4, 13, 12||.

[I don't recall the exact rules for determining which signs to negate, 
and since it's easier to start with a pair of vals, I usually don't 
bother with wedging commas.]
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Message: 11449 - Contents - Hide Contents

Date: Sat, 17 Jul 2004 05:23:52

Subject: Re: Naming temperaments

From: Gene Ward Smith

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



> The notation "12&16" doesn't need to be restricted to tunings with
exact 
> octaves; it's just a convenient shorthand for a temperament with map 
> [<4, 6, 9|, <0, 1, 1|]. Certainly, to get from the name "12&16" to the 
> map [<4, 6, 9|, <0, 1, 1|], you need to adopt the convention of exact 
> octave ET's, but the values of the period and generator can be any 
> values that are consistent with the map. But it seems that what you're 
> saying is that 12+16 isn't an MOS scale in TOP diminished; it skips
from 
> 12+8 to 12+20.



I'm not sure how you are using the 12+16 notation. My definition for
it says to look at the continued fraction for 12/16, which has a
convergent of 1/1 to the ratio 3/4. The ratio of numerators is 1/3,
and of denominators is 1/4; taking the mediant gives us 2/7 of a
period as a generator, or 2/28 of an octave. This, of course, doesn't
tell us how to map any primes other than 2, but we can easily enough
find plausible choices, and most especially diminished in 28-equal.
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