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Message: 4175 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 02:17:35

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 
wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 

>>> I'd rather they were listed in increasing order of "badness", 
>> assuming 

>>> "badness" actually means something, like badness. Then 
if 
>> I'm looking 

>>> for the best temperament whose error is in a particular 
range 
>> of sizes 

>>> I'll just go down the list until I find the first one _in_ that 
> range. 
>> 

>> sounds like a terrible idea.
> 

> Would you mind saying why?


because the systems of similar complexity should be next to 
each other, so a musician who's interested in a particular 
complexity range can immediately compare and contrast the 
systems in that range.


> I'm just as likely to be looking for the best temperament whose 
> generator size is in a particular range


why? seems overly specific. who is likely to have that as their 
priority?


> or whose period is a 
> particular fraction of an octave,

ditto.


temperaments are just mappings of ji. it's more important, 
compositionally, *which* commas vanish than what the 
generator is or what the period is. of course a fifth generator is 
preferable and that's why i cling to my unpopular idea of a 
*weighted* complexity calculation.


> or whose rms error is in a 
> particular range,


you can always introduce more error by other means. really one 
wants the tunings that *minimize* the error within the musician's 
preferred sphere of 'complexity'.
 

> Why favour any one of these (and thereby 
> make the others much more difficult to find) by sorting the list 
on 
> it?


hopefully this makes it clear. maybe this is why you didn't 
understand my liking of log-flat badness -- it's because i've 
assumed this is how you'd present the tunings.

Message: 4176 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 04:54:25

Subject: Re: 32 best 5-limit linear temperaments redux

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> by the way, my now-famous heursitic for complexity would sort 
> the 5-limit temperaments by the size of the numerators (or 
> denominators, or n*d) of the commas. kees van prooijen 
> webpages seem to suggest clearly that this should be 
> expressible as a distance measure of some kind. no one else 
> seems interested in pursuing this obversation, however :(


Height functions can be thought of as distance measures, but I'm not getting your point.

Message: 4177 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 10:17:17

Subject: Re: Everybody's 26 best 5-limit temperaments?

From: dkeenanuqnetau

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

>> See the spreadsheet at
>> http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.]  (Wayb.)
> 

> Great.  This is the best.
> 
> I can't figure out how to change ^3 to ^2 for all the "O" rows.  Is
> it difficult to add steps^2 badness?

Done.

http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.]  (Wayb.)


But you don't know if any are missing, since Gene hasn't supplied 
us with a search based on flat badness < 160.

By the way. I checked Graham's 5limit.txt against this spreadsheet. 
There weren't any that aren't in the spreadsheet (although it helped 
me correct some mistakes where I didn't actually have the generator in 
lowest terms).

It looks like we may have a list we can all agree on.

Message: 4178 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 02:48:13

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

>>> That's exactly what I _don't_ want.  I want to know the size of
>>> the chain I need to complete my map.
>> 

>> why insist so vehemently on completeness? you can do 
>> wonderful musical things with an incomplete map.
> 

> I agree, but then it's just a good temperament at a different
> limit.

not necessarily.

> If we use the concept of limit, then we should.

?


i thought the paper was going to concern {2,3,5}, {2,3,7}, {2,5,7}, 
{3,5,7}, and {2,3,5,7}. isn't that right?

Message: 4179 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 04:57:02

Subject: Re: Dave's 18 best 5-limit temperaments

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 
wrote:
> 

>> Thanks for that Gene. Much appreciated. Why no names?
> 

> Because I just gave computer output.
> 

>> But I don't understand why we lost the following that _were_ 
in your 
>> list of 32:
>> 
>> semisuper
>> parakleismic
>> hemithird
> 

> I suspect it's because the computation broke down in the 

middle and I restarted it from where it seemed to have gotten; I'll 
try again and see what happens.

> 

>> I know why you didn't find the two half-meantones and the 
two 
>> half-kleismics. Because you don't consider them to be 5-limit 
linear 
>> temperaments. 
> 

> Not this time--I made no attempt to exclude them, but it found 
> (81/80)^2 instead of the half-meantones.


hmm? i thought dave said 6561:6400 was half-meantone. 
someone needs to write a gentle introduction to contortion, just 
to list all the issues in one place.

Message: 4180 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 10:19:15

Subject: Re: 8-tone Euclidean-reduced scale

From: dkeenanuqnetau

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

>> 39375739
> 

> Anybody look at the steps in 46?
> 
> (0 3 12 15 22 27 34 37 46)
> 
> -Carl


Now _that's_ spooky!

Message: 4181 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 21:00:05

Subject: Re: 32 best 5-limit linear temperaments redux

From: Carl Lumma

>--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

>>> carl, if 'identity' is defined as 'consonant interval', then the 
>>> *only* thing going in here is the individual gens per identity. 
>>> that's all. E13-F13 is the major sixth or minor third.
>> 

>> Oh.  Well, I'm not sure how that's significant, since in regular
>> temperaments it will always be the difference of the 3 and 5
>> mappings.
>

>right -- so what's your objection?


None anymore, except that people aren't considering max-absolute
complexity and ms error, with an exponent on complexity that makes
the list finite or one that makes it log-flat supplemented by a
sharp cutoff.

-Carl

Message: 4182 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 14:23:14

Subject: Re: some output from Graham's cgi

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Graham wrote:

>You mean the Pajaras?

Yes.

>I presume you have some kind of miracles already.
>Currently in 72-equal, although
>I'd prefer them with the 11:8-just generator of 116.755 cents.


Yes, the 72-equal ones. I will add the just 11/8 ones.


>I'd rather you include a template for the keyboard mapping, so that it 
can
>be filled with any generator size. But I don't think that'll work, as the
>mapping files aren't geared for open-ended tunings. 


No, that's right.


>Also 58 from multiple-29 should be there. 


Ok, will add.


>I was thinking of the searches and optimizations.


I didn't plan to add the searches. There are other things I'd like more 
to add. The optimizations can be done, although I'm not sure whether 
it's general enough to handle cases like 58 from multiple-29.


>For people who have Scala, simple instructions for generating the scales 
>are probably better than static files, so that they can play around with 
>these things. 


Ok, I'll see about adding that.


>Although for people who don't have Scala, the lists of cents
>might be useful.


Yes, I think your top list should at least be available in files.
Preferably with file names near each other in lexical order.


>I've counted 95 files in my copy of the archive including the word
>'meantone'. Do you want that many for every temperament we come up with?


No thanks :-)

Manuel

Message: 4183 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 03:08:33

Subject: Re: Dave's 18 best 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Here's a list of temperaments with keenan badness less than 15, 
where
> keenan badness is g exp(sqrt(r/7.4)) :
...


Thanks for that Gene. Much appreciated. Why no names?

That uncovered two more middle-of-the-road temperaments that didn't 
appear in your list of 32 and that I didn't have in my spreadsheet.

6561/6400 (twin meantone)
1990656/1953125

But I don't understand why we lost the following that _were_ in your 
list of 32:

semisuper
parakleismic
hemithird

I know why you didn't find the two half-meantones and the two 
half-kleismics. Because you don't consider them to be 5-limit linear 
temperaments. Can you point me to a definition of 5-limit or linear 
temperament (or 5-limit linear temperament) that excludes them, other 
than yours?. 

But I don't understand why you didn't find twin kleismic 
(244140625/241864704).

The fact that we lost some and didn't find twin kleismic, leads me to 
suspect there may still be some we haven't seen yet, that have keenan 
badness < 15.

Have you got too low a cutoff on complexity? Like 9 or 10 gens rms? Of 
course, with my badness measure, you don't need any cutoff on 
complexity or error, but I understand you might need them to limit the 
amount of computation. We should go to at least 30 gens and 50 cents 
to be sure it's ultimately only my badness cutting them off.

There should be _at_least_ 26 temperaments (or 22 without the 
half-meantones and half-kleismics) on a list with keenan badness < 15.

By the way, if you were to set your cuttoffs to

smith badness  < 861
rms complexity < 13.2 gens
rms error      < 28.9 cents

you would include in your list, all those on my list (kb<15) so far, 
with a minimum of what I consider junk.

Message: 4184 - Contents - Hide Contents

Date: Mon, 11 Mar 2002 04:59:25

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:

> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 

>> by the way, my now-famous heursitic for complexity would 
sort 
>> the 5-limit temperaments by the size of the numerators (or 
>> denominators, or n*d) of the commas. kees van prooijen 
>> webpages seem to suggest clearly that this should be 
>> expressible as a distance measure of some kind. no one 
else 
>> seems interested in pursuing this obversation, however :(
> 

> Height functions can be thought of as distance measures, but 
>I'm not getting your point.


a _lattice_ distance function.

Message: 4185 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 08:00:02

Subject: Re: Dave's 23 best 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> 1990656/1953125 Extends to the 1029/1024^126/125 = 
> [9,5,-3,-21,30,-13] system, and needs a name.)
> 
> map   [[0, 9, 5], [1, 1, 2]]
> 
> generators   77.96498962   1200
> 
> keenan   12.03289099   rms   2.983295872   g   6.377042156


How about "quarter major thirds"?


> 16875/16384 (Extends to the 225/224^49/48 = [4,-3,2,13,-8,-14] 

system, and needs a name.)

> 
> map   [[0, -4, 3], [1, 2, 2]]
> 
> generators   126.2382718   1200
> 
> keenan   12.16857021   rms   5.942562596   g   4.966554810


I've called it "quarter fourths" in the past, but it could also be 
"third of major thirds".

I can certainly live without those 4 beyond pelogic, i.e. hemithird 
and the three new unnamed ones (which can probably remain unnamed). So 
that would leave 19.

Message: 4186 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 09:21:57

Subject: Re: skhismic

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, March 12, 2002 3:24 AM
> Subject: [tuning-math] Re: skhismic
>
>
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 

>> it's spelled this way sometimes. pronounce it. there's
>> no reason one should think it might be spelled "shismic". 
> 

> I used to spell it that way, but quit because everyone
> seemed to spell it differently. Still use "sk" to mean
> 2^(1/612), though.
> 
> how do you pronounce "schism"? like "shism"? naah . . .
> 
> Usage Note: The word schism, which was originally spelled
> scisme in English, is traditionally pronounced (szm).
> However, in the 16th century the word was respelled with
> an initial sch in order to conform to its Latin and Greek
> forms. From this spelling arose the pronunciation (skzm).
> Long regarded as incorrect, it became so common in both
> British and American English that it gained acceptability
> as a standard variant. Evidence indicates, however, that
> it is now the preferred pronunciation, at least in American
> English. In a recent survey 61 percent of the Usage Panel
> indicated that they use (skzm), while 31 percent said they
> use (szm). A smaller number, 8 percent, preferred a third
> pronunciation, (shzm).



thanks for that note, Gene ... with my penchant for history,
i found it very interesting.  here's a little history on
the musical use of the terms "schisma" and "skhisma":



the earliest reference i know of is in Boethus, _De institutione
musica, liber III_; here's Freidlin's text of the original:
BOEMUS3 TEXT * [with cont.]  (Wayb.)



>> De minoribus semitonio intervallis. 
>> 
>> VIII. Philolaus igitur haec atque his minora spatia
>> talibus definitionibus includit. Diesis, inquit, est
>> spatium, quo maior est sesquitertia proportio duobus
>> tonis. Comma vero est spatium, quo maior est sesquioctava
>> proportio duabus diesibus, id est duobus semitoniis
>> minoribus. Schisma est dimidium commatis, diaschisma
>> vero dimidium dieseos, id est semitonii minoris. 



Bower's translation, _Fundamentals of Music, book 3_, p 97 ,
with my interpolated mathematical explanation:  


>> 8. Concerning intervals smaller than a semitone
>> 
>> Philolaus incorporates these [the two sizes of
>> Pythagorean semitones, which we today call _limma_
>> and _apotome_] and intervals smaller than these in
>> the following definitions.
>> 
>> the _diesis_, he says, is the interval by which a
>> sesquitertian [= 4:3] is larger than two tones [= (9/8)^2].


     2  3    ratio    ~cents        name

   [ 2 -1]    4:3    498.0449991   sesquitertian
 - [-6  4]   81:64   407.8200035   two tones
 -----------------------------------------------
   [ 8 -5]  256:243   90.2249957   diesis


our term for this interval is the _limma_.
Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)




>> the _comma_ is the interval by which the sequioctave
>> ratio [= 9/8] is larger than two dieses -- that is,
>> larger than two minor semitones.

 

      2   3       ratio       ~cents        name

   [- 3   2]       9:8      203.9100017   sesquioctave
 - [ 16 -10]   65536:59049  180.4499913   two dieses
 -----------------------------------------------------
   [-19  12]  531441:524288  23.4600104   comma


this is the same interval which we today call the
"Pythagorean comma".



>> the _schisma_ is half a comma.


     2    3     ~ratio     ~cents

 [-19/2 12/2]  ~148:147  11.73000519


so Philolaus's _schisma_ is very different from the
interval we know by that term today.  the modern usage
of "skhisma" or "schisma" is due to Ellis.

note the spelling with a "c" by Boethius, and also
presumably Philolaus. (none of Philolaus's own work
exists today.)



>> The _diaschisma_ is half a _diesis_ -- that is,
>> half a minor semitone.


   2    3     ~ratio    ~cents

 [8/2 -5/2]   ~39:38  45.11249784


_diaschisma_ is another term that was given a different
meaning by Ellis -- ratio 2048:2025 = ~19.55 cents; see:
Internet Express - Quality, Affordable Dial Up... * [with cont.]  (Wayb.)


note Margo Schulter's observation that medieval (and
perhaps earlier) theorists recognized that Philolaus's
"diaschisma" is nearly equal to 2 commas, which is what
gave rise to the idea that "1 tone = 9 commas", a concept
which persisted in common practice as late as Mozart's
lifetime, and could be used in connection with both
53edo (quasi-Pythagorean and quasi-JI) and 55edo (meantone).
Yahoo groups: /tuning/messages/29025?expand=1 * [with cont.] 




Ellis, who is responsible for the modern usage of the term
"skhisma" or "schisma", deliberately chose the spelling with
a "k", in order to avoid any connotation or reference to
the religious views associated with the "Great Schism".


(i've looked around in Ellis's translation of Helmholtz
for this info but can't find it; perhaps Ellis mentioned
it in another paper he published, which i'd read in a
library but of which i don't have a copy.)



here's a quote from a webpage with an excellent explanation
of the Great Schism
The 11th to the 13th Centuries: Innocent III a... * [with cont.]  (Wayb.)


(note that much that is said here has a bearing on the future
development of music and music-theory, because of:  1) the
increasing secularization of European society -- establishment
of universities at Paris, Padua, Naples, Prague, Vienna,
Heidelberg and Cologne, at least the first of which became
an extremely important center for developments in music, and
2) the rediscovery in the West of the ancient Greek writings.
Both of these were direct results of the sack of Constantinople
in 1204.)



>> The Church had, by the time of Innocent III [ruled as 
>> Pope 1198-1216], taken on the organisational role of the
>> Crusades with all its political and economic ramifications.
>> Crusades were to be launched against heretics at the
>> discretion and direction of the presiding Pontiff and
>> were used as a means of imposing the rule and will of the
>> Church on the unbeliever. Augustinian teaching that
>> justified the use of torture and death as legal instruments
>> to be used by the Church to convert the heretic became
>> widely accepted. This acted as a prelude to the legitimisation
>> of the Inquisition, which was to receive papal approval
>> under Gregory IX in 1233. Heresy was to be punished for
>> the spiritual "good" of the individual as well as for the
>> preservation and enhancement of the status of the Church
>> and State - an attitude and mentality equally accepted
>> by future Western reformers such as Calvin and Luther.
>> Such was to be the patrimony and inheritance of the Crusades.
>> 
>> An even darker shadow was cast over Innocent's pontificate
>> by his involvement in the Fourth Crusade, which led to
>> schism between Eastern and Western Christendom in the
>> eleventh century, an event which is one of the greatest
>> calamities in the history of the Church. The main aim of
>> the Crusades was to try to free the Holy Land from the
>> Seljuk Turks who had conquered Jerusalem in 1071. This
>> was not accomplished, but rather in its consequences it
>> seriously undermined the powers of resistance of the
>> Christian East to the advance of Islam. It also encouraged
>> the excessive growth of papal power in the West, and this
>> over-centralisation of Church government resulted in many
>> abuses and provoked widespread discontent. Thus the
>> Reformation itself, which split the West into two hostile
>> camps, was one of its results flowing from the split
>> between East and West
>> ...
>> Roman Catholics have accused the East of an obstinate
>> refusal to accept the leadership of the Pope, and of
>> undue submissiveness towards the secular power. The Orthodox,
>> in return, have hurled against Western Christians charges
>> of arrogance and pride, and have insisted that both Latins
>> and Protestants have wilfully departed from the sound
>> tradition of the early Church and perverted their religion
>> by arbitrary and harmful innovations.
>> 
>> Many controversial books have been written on this subject;
>> but if the simple question is asked, "What was the cause of
>> the Schism between Rome and Constantinople, and when exactly
>> did it occur?", too often no clear answer is forthcoming.
>> The absence of an agreed statement on such a vital issue,
>> one which has so profoundly and so disastrously affected
>> the life of all Christians, is puzzling indeed. Yet an
>> explanation of it is to be found in the study of the
>> political and ecclesiastical events which led to the break
>> of communion between East and West.
>> 
>> Though conflict, disagreement and tensions in politics and
>> theological interpretation existed from the fifth century
>> onwards, this gradual process of open hostility and bitter
>> hate reached its climax between the ninth and thirteenth
>> century. It is often thought that the lasting split in the
>> Church must have been caused by some major doctrinal
>> disagreement. The history of the schism does not confirm
>> this opinion. The growing alienation between the Christian
>> East and West was provoked by political competition, petty
>> quarrels and personal rivalries. It was a slow movement;
>> for the Church organism vigorously resited these attacks
>> of destructive forces. The final blow to the unity of the
>> Church was inflicted by no heresy, but by the drunken and
>> undisciplined mob of Crusaders who sacked Constantinople
>> in 1204 and massacred its Christian population.
>> ...
>> This day, April 13, 1204 marks the end of the fellowship
>> between Eastern and Western Christians. The split was brought
>> about, not by quarrelsome theologians or ambitious prelates,
>> as is usually suggested, but by the greed and lust of those
>> men who, in the name of the Prince of Peace, had embarked
>> upon a war of aggression and conquest.



i stick with "skhisma" out of respect for Ellis's sensitivity
in this matter, but it's necessary to note that nearly everyone
else today uses the "schisma" spelling.



-monz



 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.]  (Wayb.)

Message: 4187 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 10:44:15

Subject: Re: Dave's 23 best 5-limit temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> so 7625597484987:7629394531250 would be the first major one we're 
> cutting off? it's of course important as the 5-limit aspect of 
> ennealimmal. you might argue that ennealimmal is a higher-limit 
> consideration.

Absolutely.

> but will the current criteria dave is using preclude 
> its inclusion when we get to higher limits? i certainly hope not. we 
> don't want to go through this same subjective process over again for 
> every 'limit'.


I doubt it. I'll probably lose interest by then. ;-)

Message: 4188 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 08:03:10

Subject: Re: 32 best 5-limit linear temperaments redux

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

>> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

>>> ok. gene, once again, this means that in the 'gens' calculation, 
> the 

>>> number of generators in the 3:1 should be multiplied by log(3), 
> the 

>>> number of generators in the 5:3 should be multiplied by log(5), 
> the 

>>> number of generators in the 5:1 should be multiplied by log(5).
>> 

>> And to make the result meaningful (i.e. comparable to the 
> unweighted 

>> values) then after you take the RMS of these weighted values you 
>> should divide by sqrt(log(3)^2+log(5)^2+log(5^2)).
> 

> why do you want them to be comparable to the unweighted values?


Obviously it doesn't matter as far as choosing lists, but I like for a 
human (e.g. me) to be able to look at the error values and have them 
mean something. i.e. to actually be in cents. So when you see 5 you 
know kinda what a 5c mistuning sounds like.

Message: 4189 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 10:47:29

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


> See my latest version spreadsheet 
> http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.]  (Wayb.)
> 
> I've fixed the names.


i don't think this is correct, dave:

"Flat_badness = complexity^2*error with cutoffs"

that's not flat at all, right, gene?

Message: 4190 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 20:15:02

Subject: Re: 32 best 5-limit linear temperaments redux

From: paulerlich

--- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <a6jdeq+58k4@e...>
> paulerlich wrote:
> 

>> ok. gene, once again, this means that in the 'gens' calculation, 
the 
>> number of generators in the 3:1 should be multiplied by log(3), 
the 
>> number of generators in the 5:3 should be multiplied by log(5), 
the 
>> number of generators in the 5:1 should be multiplied by log(5). 
this 
>> will cause temperaments generated by the fifth to look better 
than 
>> they currently do, relative to those that aren't. this is 
important 
>> since augmented (especially) and diminished (a little less so) 
are 
>> far harder for the ear to understand than meantone, even when all 
are 
>> *tuned* in 12-equal, and the badness values would no longer put 
>> meantone as the 'best'.
> 

> When was it decided these temperaments were "far harder for the ear 
to 
> understand"?  Even if so, augmented is more complex than meantone 
if you 
> measure by the simplest MOS to contain a consonant chord (8 

compared to 5 

> notes).


augmented is 6, diminished is 8 . . .

Message: 4191 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 08:06:55

Subject: Re: amt

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> 
>> 1600000/1594323 amt
>> 

>> map   [[0, -5, -13], [1, 3, 6]]
>> 
>> generators   339.5088258   1200
>> 
>> keenan   11.64300516   rms   .3831037874   g   9.273618495
> 

> This extends to the 7-limit system 5120/5103^4375/4374 =
> [5,13,-17,-76,41,9], with a 28/99 generator. This could be called 

"amt" also, unless someone has a better idea for a name--which 
probably would not be hard.

How was the name amt arrived at. Is it an abbreviation for something? 
It could be called "fifth of eleventh".

Message: 4192 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 10:52:52

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:


> Oh I think I realise why you ask, since you wouldn't want it by your 
> own badness measure. You think I'm violating my own badness measure. 


Nope.
My 5-limit badness is 774, not something to set the world on fire, but
it did pass your measure, as you say. I want a name for it because my
7-limit badness score is 173, so we have a decent 5-limit
temperament which heats up in the 7-limit.

Message: 4193 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 20:26:05

Subject: the most famous application of augmented temperament

From: paulerlich

404 Not Found * [with cont.]  Search for http://home1.gte.net/furlott/coltrane.htm in Wayback Machine

Voice Leading in the Music of John Coltrane * [with cont.]  (Wayb.)

Message: 4194 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 00:23:32

Subject: heuristic...

From: Carl Lumma

Paul,

Do you have the original expository message number handy,
or any good words to search for, or do you know what list
you posted it to, or around when?

-Carl

Message: 4195 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 10:55:02

Subject: Re: Dave's 23 best 5-limit temperaments

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> sounds like 'bastoni' (italian bread). fits in well with 'injera' 
> (ethiopian bread). actually this works well, since 225/224^49/48 
> (bastoni) is in the same 'aisle' as 81/80^50/49 (injera).


Next time I'm stuck for a name, I'll think bread.

Message: 4196 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 12:34:30

Subject: Re: amt

From: Carl Lumma

>> >t's an acronym for "acute minor third", from its generator.
>

>Lets call it that then, since AMT isn't pronouncable and why save one 
>syllable just to make it more obscure.


Amt sure is pronouncable.  I must say I'm quite amt about the keeping
the name.  :)

Actually, I don't care what you call it.

Except "acute minor thirds".  That's a terrible name:

1. Three words.  Temperaments should have cool, single-word
names.

2. I find it perverse to name temperaments by their relation to
diatonic intervals.  I guess this counts against Amt too.

-Carl

Message: 4197 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 08:26:03

Subject: A proposed list of 5-limit not-quite-Just-thingies

From: dkeenanuqnetau

I propose we go with either these 19 (in weighted-complexity order)

neutral thirds
meantone
pelogic
augmented
porcupine
diminished
diaschismic
small diesic
quarter fourths
kleismic
twin meantone
half meantone-fourth
half meantone-fifth
minimal diesic
schismic
wuerschmidt
tiny diesic
orwell
amt

or if you want a bit more complexity add

seventh of major third
semisuper
parakleismic
twin schismic
half schismic-fourth
half schismic-fifth

bringing the total up to 26.

To see what they mean and see how they might be arrived at using 
Gene's badness, modified to use Paul's weighted complexity, see the 
latest incarnation of my spreadsheet.

http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Type Ok * [with cont.]  (Wayb.)

They require an error cutoff of 35 cents and the short list has a 
weighted complexity cutoff of 10 while the long list has 12.

Any objections? Short list or long?

Message: 4198 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 10:56:13

Subject: Re: A proposed list of 5-limit not-quite-Just-thingies

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> i don't think this is correct, dave:
> 
> "Flat_badness = complexity^2*error with cutoffs"
> 
> that's not flat at all, right, gene?


Not for 5-limit.

Message: 4199 - Contents - Hide Contents

Date: Tue, 12 Mar 2002 12:35:47

Subject: Re: Dave's 23 best 5-limit temperaments

From: Carl Lumma

>chromatic unison vectors.


You're suggesting this as a name for a temperament???

-Carl

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