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Message: 4975 - Contents - Hide Contents

Date: Fri, 07 Jun 2002 16:13:00

Subject: Re: A common notation for JI and ETs

From: genewardsmith

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


> I thought that we were notating everything that we possibly could.  
> Who knows what the tuning scavengers might want to use?


Good idea--I recently found some interesting uses for the 108-et.

Message: 4976 - Contents - Hide Contents

Date: Fri, 07 Jun 2002 20:16:01

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote [#4412]:

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote 
[#4405]:

>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>>> 140:  )|  |~  /|  )|\  /|~  /|)  (|~  (|\  )||  ||~  ||\  )

||\  /||~  /||\

>>> 
>>> This is the simplest set I could come up with that uses both /| 
and ||\.
>> 

>> I'll leave the second half-apotome out of it for now. It seems we 
have 4
>> options:
>> 140:  )|  |~  /|  )|\  (|   /|)  (|~    6 flags
>> 140:  )|  |~  /|  )|)  (|   /|)  (|~    5 flags
>> 140:  )|  |~  /|  )|\  /|~  /|)  (|~    6 flags monotonic flags 
per symb
>> 140:  )|  |~  /|  )|)  /|~  /|)  (|~    5 flags monotonic flags 
per symb
>> 
>> I prefer the last one, and with mirror complements it would be
>> 
>> 140:  )|  |~  /|  )|)  /|~  /|)  (|~  (|\  ~||\  (||(  ||\  ~||  
||( /||\
>> 
>> Note that with mirror complements, (|\ is the same as (||\.
> 

> In 70-ET )|\ is 2deg, whereas )|) is 1deg, so I prefer the former.


Let me correct myself:  in 70-ET |) will be the 13-5 comma of 2deg, 
so (| will then be the 13'-7 comma of 2deg.  So )|) must also be 
2deg70.

My reason for choosing |~ for 2deg140 is that the flag is also used 
for 7deg (where this is no other choice), which serves to limit the 
total number of flags.  However, this flag (taken as either a 19'-19 
or a 23 flag) is not really a very good choice for either 1deg70 or 
2deg140; the 17 flag ~| would be better for both.  In 70-ET both 5 
and 7 are awful, and whatever inconsistency we find for 17 is only 
with respect to 5 and 7.  Taken separately, 17 is one of the best 
factors in 70-ET, and it's also better than either 19 or 23 in 140-ET.

Using the 17 flag, I got the following easy-to-remember sequence of 
symbols:

140 (70 ss.):  )|  ~|  /|  )|)  ~|)  /|)  (|~  (|\  )||  ~||  ||\  )
||)  ~||)  /||\

The two disadvantages with this are that it uses 7 flags and has two 
pairs of laterally confusable symbols.  So I would be more inclined 
to go with one of yours.

Of the four options you gave, I agree with your choice of the last 
one (5 flags monotonic flags per symb), but with non-mirroring double-
shaft symbols:

140 (70 ss.):  )|  |~  /|  )|)  /|~  /|)  (|~  (|\  )||  ||~  ||\  )
||)  /||~  /||\  5 flags monotonic flags per symb

--George

Message: 4977 - Contents - Hide Contents

Date: Fri, 07 Jun 2002 00:20:40

Subject: Re: Fwd: scala questions

From: genewardsmith

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

> All;
> 
> I've created a tuning-math list on freelists.org...
> 
> Welcome to FreeLists - Free, No-hassle Mailing... * [with cont.]  (Wayb.)


All I get is an error message.

Message: 4978 - Contents - Hide Contents

Date: Fri, 07 Jun 2002 20:37:50

Subject: Carl's new list

From: genewardsmith

I don't think it's going to work unless it becomes easier to subscribe,
read, and post to. I was going to post this remark there, but havn't
figured out how. :(

Message: 4979 - Contents - Hide Contents

Date: Fri, 07 Jun 2002 13:54:22

Subject: Re: Carl's new list

From: Carl Lumma

>I don't think it's going to work unless it becomes easier to subscribe,
>read, and post to. I was going to post this remark there, but havn't
>figured out how. :(


I got a notification that you're subscribed, so you just send mail to
tuning-math@xxxxxxxxx.xxx.

-Carl

Message: 4980 - Contents - Hide Contents

Date: Sat, 08 Jun 2002 15:35:28

Subject: Re: A 7-limit best list

From: genewardsmith

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

Could
you put this list either into a full period matrix format or give the
wedgie? I'm finding the conversion aggravating, and I think
temperaments should *always* be given in one of the two forms above.

Message: 4981 - Contents - Hide Contents

Date: Sat, 08 Jun 2002 18:04:13

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Hi George,

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

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote 


>> In fact, (and I've been making gentle noises about this 

possibility 
> for a

>> some time now), I'm willing to throw away everything we agonised 
> over with

>> regard to rational complements and instead adopt a simple system 
> that

>> applies automatically to all ETs and rational tunings.
>> 
>> I propose that the complement of a|b is always b||a, except that 

the
>> complement of |//| (natural) is /||\ and the complement of /|\ 

> is /|\ if it

>> represents the same number of steps and (|) if it represents a 
> different

>> number of steps.
>> 
...
>> Why do I want to do this despite some obvious disadvantages? 
> Because I

>> realised while trying to consistently notate the whole n*12-ET 
> family, that

>> it required us to repeat the whole somewhat arbitrary process we 
> went thru

>> for rational tunings, to find complements with minimum offsets. 

And 
> what's

>> more, that this process would have to be repeated for every such 
> family or

>> small range of fifth-sizes across the whole range of ETs. For 
> example the

>> n*29-ET family is the next largest, followed by n*17. And every 

such
>> family, or small range of fifth sizes, would have a completely 
> different

>> complement mapping. The cognitive load for anyone who uses more 
> than two

>> such systems would be enormous.
>> 
>> Now for those obvious disadvantages:
>> 
>> 1. The second shaft does not have a fixed comma value.
>> 
>> This doesn't seem very important to me? 
>> 
>> 2. We lose the association of flag size with rational comma size 

in 
> the
>> second half-apotome.
>> 
>> This is the biggie. It can be remedied to some degree by 
> redesigning the

>> double-shaft (and X-shaft) symbols so their concave flags are 

wider 
> than

>> their wavy flags which are wider than their straight and convex 
> flags.

>> However it will be difficult to make single flag symbols bigger 

than
>> double-flag ones.
>> 
>> What other disadvantages have I omitted?
> 

> One very big one that I will state below, when you give a couple of 
> examples.
> 
>> Advantages:
>> 

>> Simple to remember. 
>> Covers all tunings. 
>> Flags are more strongly associated with particular primes because 
> the flags

>> don't change when the comma is complemented.
>> No new flag types ever need to be introduced merely to handle 
> complements.

>> Doesn't require /| and ||\ as a special case.
..
>> Except for the mirror complement thingy that we need to thrash out 
> now.
> 

> In effect, mirroring gives the flags negative values, with the zero 
> point being the apotome, which itself is notated as an 
> exception, /||\ ,when its proper mirror should be ||.  For the 
> simpler ET's that use no concave or wavy flags, I don't see much of 

a 
> problem, since the symbol arithmetic usually works in spite of the 
> mirroring.  But as soon as you introduce concave or wavy flags, 
> particularly in two-flag symbols, the symbol arithmetic goes crazy.



As you imply above, it's just different arithmetic, not necessarily crazy.



> If we can get mirroring in the lower-numbered ET's by means of the 
> complementation that we already worked out, then that's a 

commendable 
> goal.  But please, let's not dump the concept of consistent symbol 
> arithmetic in the process.



So do we agree to always use mirrorred complements when they agree with
both kinds of arithmetic? as in 72-ET version 3. But I haven't yet given up
on the idea of using them everywhere.



> If you feel that the best choice of single-shaft symbols is in some 
> instances compromised by the need to have double-shaft complements, 
> then I'll work with you to address that problem.


OK. Thanks.


> After all that we went through figuring out the rational 

complements, 


I think we agreed that the effort expended in designing something should
never be a consideration in whether to abandon it if something better is
possible.


> I can't see replacing that with something in which the order of 
> symbols in the second half-apotome makes very little sense?


But it makes perfect sense. Just different sense to what you have become
used to over the past months. It is always an exact mirror image of the
first half-apotome. The same pairs of symbols are _always_ complements of
each other, in _every_ tuning. No exceptions.

We should try to think like someone coming to sagittal notation for the
first time. Or maybe we should actually ask a few people in that boat (e.g.
Ted Mook). Would they rather have complements that were a complete
no-brainer (just flip horizontally and add a stroke [except for !//|
(natural) and /|\ ]), or would they rather have to learn the order of twice
as many symbols for each new tuning. Clearly, under the current system the
second half-apotome cannot easily be derived from the first, or we wouldn't
be presenting various options and arguing over their merits.


>  All 
> to "fix" a problem involving not-quite-matched symbols /| and ||\ in 
> a few ET's?  I say: "forget it."


This is not the only reason for the mirror proposal. See the list of
advantages I gave above for mirrored apotomes and read their converse as
_disadvantages_ of the current system. 

It's just that, so long as I thought the half-apotomes were _always_ going
to match (except for natural), and therefore folks only needed to learn the
first half-apotome of any new tuning, then the mirror solution was, for me,
hovering just below the the threshold of being considered.

So far I have identified 4 solutions to be considered for the second apotome.

1. Always matched, except that !//| has various pseudo-matches [e.g. (|) or
(|\ ] when /|\ is not an exact half-apotome.
2. Always mirrored, except !//| is always the pseudo-mirror of /||\, and
/|\ is always the pseudo-mirror of (|) when /|\ is not an exact half-apotome.
3. Mostly matched. As for 1, except that /| and |\ are mirrored instead of
matched when matching disagrees with an arithmetic that says that /| + ||\
= /| + ||\ = /||\
4. Mostly mirrored. As for 2, except that mirroring is replaced by matching
if the mirroring does not agree with an arithmetic where the second shaft
represents the addition of an 11' comma.

I thought we had agreed on 1, but it seems you actually had in mind 3, and
presumably vice versa. Now, pending more argument from you (or anyone
else), my order of preference is 1,2,4,3.

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 4982 - Contents - Hide Contents

Date: Sat, 08 Jun 2002 19:09:30

Subject: Best 7 proposal

From: genewardsmith

A search on rms < 25, weighted complexity < 17, and weighted badness < 350
will give us all of Dave's list (finally turned back into wedgies) and
ennealimmal besides, so I propose to do that one.

Message: 4983 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 01:11:10

Subject: New 7-limit best list

From: genewardsmith

This is ordered by weighted complexity, low to high, and uses the
cutoffs I suggested: weighted badness 350, weighted complexity 17, and
rms error 25 cents. On each line is listed the period matrix, followed
by the non-octave generator which goes with it.

[[2, 0, 3, 4], [0, 2, 1, 1]]   950.9775006
[[1, 0, -4, 6], [0, 1, 4, -2]]   1902.225978
[[4, 0, 3, 5], [0, 1, 1, 1]]   1885.698207
[[5, 8, 0, 14], [0, 0, 1, 0]]   2789.386745
[[3, 0, 7, -1], [0, 1, 0, 2]]   1889.740309
[[2, 0, -5, -4], [0, 1, 3, 3]]   1928.512337
[[2, 0, -5, -4], [0, 1, 3, 3]]   1928.512337
[[1, 0, 7, -5], [0, 1, -3, 5]]   1873.109081
[[1, 2, 2, 3], [0, 4, -3, 2]]   -125.4687958
[[1, 0, 1, 2], [0, 6, 5, 3]]   316.6640534
[[3, 0, 7, 18], [0, 1, 0, -2]]   1911.279336
[[1, 0, -4, 2], [0, 2, 8, 1]]   947.2576878
[[1, 0, -4, -13], [0, 1, 4, 10]]   1896.647968
[[2, 0, -8, -7], [0, 1, 4, 4]]   1893.651026
[[2, 1, 3, 4], [0, 4, 3, 3]]   325.6113679
[[1, 2, 3, 2], [0, 3, 5, -6]]   -162.3778142
[[1, 0, -12, 6], [0, 1, 9, -2]]   1910.384820
[[1, 0, 2, 5], [0, 5, 1, -7]]   377.6398806
[[1, 1, 5, 4], [0, 2, -9, -4]]   356.3080310
[[1, 0, -4, 17], [0, 1, 4, -9]]   1893.456080
[[1, 0, 2, -1], [0, 5, 1, 12]]   380.5064474
[[1, 9, 9, 8], [0, 10, 9, 7]]   -890.0485289
[[1, 6, 8, 11], [0, 7, 9, 13]]   -756.3796144
[[1, 0, 3, 1], [0, 7, -3, 8]]   271.3263633
[[1, 1, 3, 3], [0, 6, -7, -2]]   116.5729472
[[1, 1, 2, 3], [0, 9, 5, -3]]   77.70708740
[[1, 1, 0, 3], [0, 3, 12, -1]]   232.1235474
[[1, 0, 15, 25], [0, 1, -8, -14]]   1902.140161
[[1, 4, 5, 2], [0, 9, 10, -3]]   -321.8581275
[[1, 3, 8, 6], [0, 4, 16, 9]]   -425.9591136
[[1, 1, 0, 6], [0, 2, 8, -11]]   348.3528923
[[2, 0, 11, 31], [0, 1, -2, -8]]   1903.737092
[[1, 1, 1, 2], [0, 8, 18, 11]]   88.14540670
[[1, 4, -3, -3], [0, 5, -11, -12]]   -580.4242148
[[1, 1, -1, 3], [0, 3, 17, -1]]   234.4104087
[[2, 1, 9, -2], [0, 2, -4, 7]]   652.8935173
[[1, 0, 1, -3], [0, 6, 5, 22]]   316.7238784
[[1, 15, 4, 7], [0, 16, 2, 5]]   -1006.090063
[[1, 0, 1, 4], [0, 12, 10, -9]]   158.7324720
[[1, 4, 2, 2], [0, 15, -2, -5]]   -193.2841226
[[2, 1, 5, 2], [0, 6, -1, 10]]   216.7129478
[[12, 19, 0, -22], [0, 0, 1, 2]]   2784.052566
[[1, 2, 2, 3], [0, 13, -10, 6]]   -38.46612668
[[1, 3, 6, -2], [0, 5, 13, -17]]   -339.4147298
[[1, 1, -5, -1], [0, 2, 25, 13]]   351.4712147
[[9, 1, 1, 12], [0, 2, 3, 2]]   884.3341826

Message: 4984 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 21:20:42

Subject: Re: A 7-limit best list

From: Carl Lumma

>I did say audible. Over on the main list, they are sniffing at the scale I 
>gave in 612-et, but logically, should they?


Most probably not.  I haven't seen that thread yet, but it doeesn't sound
like there's a reason for us to sniff about giving it in JI, either.

-Carl

Message: 4985 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 01:23:34

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

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

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> 
> Could you put this list either into a full period matrix format or 

give the wedgie? I'm finding the conversion aggravating, and I think 
temperaments should *always* be given in one of the two forms above.

I'm sorry. I just gave enough info for anyone to recognise and 
distinguish them, and assumed you would have your software _generate_ 
the more detailed list, using the cutoffs I gave. I eagerly await your 
next list. I fear it will still contain some temperaments that are of 
no real interest to musicians.

My spreadsheet is at 
Yahoo groups: /tuning-math/files/Dave/7LimTemp... * [with cont.] 

Message: 4986 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 21:21:27

Subject: gd update

From: Carl Lumma

I've added several scales (now there are 36), updated the Scala files,
added a mean variety column to the spreadsheet.

Excel 2000 Spreadsheet:
 * [with cont.]  (Wayb.)

Zip archive of the Scala files used:
 * [with cont.]  (Wayb.)

Explanation of the columns in the spreadsheet:
 * [with cont.]  (Wayb.)

Examples of the "diatonic harmony" metric:
 * [with cont.]  (Wayb.)

I've been trying to scan for scales that share a common rank-order
matrix.  The only ones I know of right now are rothenberg and
balzano-20.  That's right, Rothenberg, then Balzano, then Dan Stearns
discovered this scale, all for different reasons.  For R., it was
a search of subsets of 31-tET with high stability and efficiency.
For Balzano -- well, I won't go into it here.  Stearns points out that
there are 11:13:15, 1/(15:13:11) chords on scale degrees 1-3-5.  It
ranks high on my list because the 8ths are either 5:3 and 7:4.  You'll
get more out of the 31-tET version for those, but the 20-tET version
gets you higher Lumma stability.

As far as ranking the scales, the top ten according to the "diatonic
harmony" property are:

08_octatonic
10_blackwood
06_hexatonic
07_diatonic
07_qm(2)
07_hungarian-minor
06_super7
05_pentatonic
10_sym-major
10_pent-major

That's not bad, I'd say.  The next two are:

09_balzano-20 / 09_rothenberg
08_nova

This excludes Paul Hahn's trichordal scales and my subset-13 scale,
having their high scores from the 12:7, which is arguably not
singable in two-part harmony.

For the next version, I hope to improve the modal transposition
property (s - i isn't very good), find some method of creating a
combined score out of the four, and robustly check for rank-order
doubles.

-Carl

Message: 4987 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 04:36:46

Subject: 7-limit list in non-Dave format

From: genewardsmith

Decimal
[4, 2, 2, -1, 8, -6]   [[2, 0, 3, 4], [0, 2, 1, 1]]

bad   184.7010517   g   2.777313996   rms   23.94525150

1/2   950.9775006


Dominant seventh
[1, 4, -2, -16, 6, 4]   [[1, 0, -4, 6], [0, 1, 4, -2]]

bad   197.3456024   g   3.128478105   rms   20.16328150

1   1902.225978


Diminished
[4, 4, 4, -2, 5, -3]   [[4, 0, 3, 5], [0, 1, 1, 1]]

bad   189.2082747   g   3.144366918   rms   19.13699259

1/4   1885.698207


Quintal
[0, 5, 0, -14, 0, 8]   [[5, 8, 0, 14], [0, 0, 1, 0]]

bad   171.2126805   g   3.290247187   rms   15.81535241

1/5   2789.386745


Augmented
[3, 0, 6, 14, -1, -7]   [[3, 0, 7, -1], [0, 1, 0, 2]]

bad   224.2088808   g   3.675273386   rms   16.59867843

1/3   1889.740309


Pajara
[2, -4, -4, 2, 12, -11]   [[2, 0, 11, 12], [0, 1, -2, -2]]

bad   169.1429833   g   3.938677761   rms   10.90317755

1/2   1908.814330


Hexadecimal
[1, -3, 5, 20, -5, -7]   [[1, 0, 7, -5], [0, 1, -3, 5]]

bad   296.7523121   g   3.995964429   rms   18.58450012

1   1873.109081


Tertiathirds
[4, -3, 2, 13, 8, -14]   [[1, 2, 2, 3], [0, 4, -3, 2]]

bad   225.9663420   g   4.305717081   rms   12.18857055

1   -125.4687958


Kleismic
[6, 5, 3, -7, 12, -6]   [[1, 0, 1, 2], [0, 6, 5, 3]]

bad   234.2754881   g   4.368916409   rms   12.27380956

1   316.6640534


Tripletone
[3, 0, -6, -14, 18, -7]   [[3, 0, 7, 18], [0, 1, 0, -2]]

bad   173.7904007   g   4.631825456   rms   8.100678834

1/3   1911.279336



[2, 8, 1, -20, 4, 8]   [[1, 0, -4, 2], [0, 2, 8, 1]]

bad   293.7346084   g   4.811111338   rms   12.69007837

1   947.2576878


Meantone
[1, 4, 10, 12, -13, 4]   [[1, 0, -4, -13], [0, 1, 4, 10]]

bad   103.8247475   g   5.322447240   rms   3.665035228

1   1896.647968


Injera
[2, 8, 8, -4, -7, 8]   [[2, 0, -8, -7], [0, 1, 4, 4]]

bad   320.3524287   g   5.343650829   rms   11.21894132

1/2   1893.651026


Double wide
[8, 6, 6, -3, 13, -9]   [[2, 1, 3, 4], [0, 4, 3, 3]]

bad   334.6430427   g   5.746952448   rms   10.13226624

1/2   325.6113679


Porcupine
[3, 5, -6, -28, 18, 1]   [[1, 2, 3, 2], [0, 3, 5, -6]]

bad   231.9225067   g   5.836211169   rms   6.808961862

1   -162.3778142


Superpythagorean
[1, 9, -2, -30, 6, 12]   [[1, 0, -12, 6], [0, 1, 9, -2]]

bad   246.9834642   g   6.207109365   rms   6.410458352

1   1910.384820



[5, 1, -7, -19, 25, -10]   [[1, 0, 2, 5], [0, 5, 1, -7]]

bad   347.0112248   g   6.305725722   rms   8.727168682

1   377.6398806


Neutral thirds
[2, -9, -4, 16, 12, -19]   [[1, 1, 5, 4], [0, 2, -9, -4]]

bad   265.2522216   g   6.517068879   rms   6.245315858

1   356.3080310


Flattone
[1, 4, -9, -32, 17, 4]   [[1, 0, -4, 17], [0, 1, 4, -9]]

bad   329.1049270   g   6.557956330   rms   7.652394368

1   1893.456080


Magic
[5, 1, 12, 25, -5, -10]   [[1, 0, 2, -1], [0, 5, 1, 12]]

bad   190.6152791   g   6.786228469   rms   4.139050792

1   380.5064474


Small diesic
[10, 9, 7, -9, 17, -9]   [[1, 9, 9, 8], [0, 10, 9, 7]]

bad   181.6005942   g   7.395689110   rms   3.320167332

1   -890.0485289


Semisixths (tiny diesic)
[7, 9, 13, 5, -1, -2]   [[1, 6, 8, 11], [0, 7, 9, 13]]

bad   278.2627568   g   7.420887650   rms   5.052931030

1   -756.3796144


Orwell
[7, -3, 8, 27, 7, -21]   [[1, 0, 3, 1], [0, 7, -3, 8]]

bad   142.6121910   g   7.421511799   rms   2.589237496

1   271.3263633


Miracle
[6, -7, -2, 15, 20, -25]   [[1, 1, 3, 3], [0, 6, -7, -2]]

bad   94.80434091   g   7.609147969   rms   1.637405196

1   116.5729472


Quartaminorthirds
[9, 5, -3, -21, 30, -13]   [[1, 1, 2, 3], [0, 9, 5, -3]]

bad   179.6938179   g   7.655669978   rms   3.065961726

1   77.70708740


Supermajor seconds
[3, 12, -1, -36, 10, 12]   [[1, 1, 0, 3], [0, 3, 12, -1]]

bad   214.5541544   g   7.742330569   rms   3.579262150

1   232.1235474


Schismic
[1, -8, -14, -10, 25, -15]   [[1, 0, 15, 25], [0, 1, -8, -14]]

bad   212.0930465   g   8.612526914   rms   2.859336356

1   1902.140161



[9, 10, -3, -35, 30, -5]   [[1, 4, 5, 2], [0, 9, 10, -3]]

bad   323.7195354   g   8.937416729   rms   4.052704060

1   -321.8581275



[4, 16, 9, -24, -3, 16]   [[1, 3, 8, 6], [0, 4, 16, 9]]

bad   300.2293184   g   9.336988664   rms   3.443812018

1   -425.9591136



[2, 8, -11, -48, 23, 8]   [[1, 1, 0, 6], [0, 2, 8, -11]]

bad   333.5422522   g   9.453300867   rms   3.732363180

1   348.3528923


Diaschismic
[2, -4, -16, -26, 31, -11]   [[2, 0, 11, 31], [0, 1, -2, -8]]

bad   342.7141199   g   9.469818377   rms   3.821630536

1/2   1903.737092


Octafifths
[8, 18, 11, -25, 5, 10]   [[1, 1, 1, 2], [0, 8, 18, 11]]

bad   227.7375065   g   10.50332216   rms   2.064339812

1   88.14540670



[5, -11, -12, 3, 33, -29]   [[1, 4, -3, -3], [0, 5, -11, -12]]

bad   316.6090581   g   10.83403515   rms   2.697384486

1   -580.4242148



[3, 17, -1, -50, 10, 20]   [[1, 1, -1, 3], [0, 3, 17, -1]]

bad   324.7554230   g   10.90855391   rms   2.729116326

1   234.4104087


Shrutar
[4, -8, 14, 55, -11, -22]   [[2, 1, 9, -2], [0, 2, -4, 7]]

bad   281.7169931   g   11.18841619   rms   2.250483424

1/2   652.8935173


Catakleismic
[6, 5, 22, 37, -18, -6]   [[1, 0, 1, -3], [0, 6, 5, 22]]

bad   209.7321406   g   11.41155044   rms   1.610555448

1   316.7238784


Hemiwuerschmidt
[16, 2, 5, 6, 37, -34]   [[1, 15, 4, 7], [0, 16, 2, 5]]

bad   120.8100402   g   11.74782291   rms   .8753631224

1   -1006.090063


Hemikleismic
[12, 10, -9, -49, 48, -12]   [[1, 0, 1, 4], [0, 12, 10, -9]]

bad   311.1962901   g   12.80971160   rms   1.896512488

1   158.7324720


Hemithird
[15, -2, -5, -6, 50, -38]   [[1, 4, 2, 2], [0, 15, -2, -5]]

bad   299.6293822   g   13.15572684   rms   1.731229740

1   -193.2841226



[12, -2, 20, 52, 2, -31]   [[2, 1, 5, 2], [0, 6, -1, 10]]

bad   327.4131763   g   13.66566083   rms   1.753213789

1/2   216.7129478



[0, 12, 24, 22, -38, 19]   [[12, 19, 0, -22], [0, 0, 1, 2]]

bad   283.6535726   g   13.76571634   rms   1.496892545

1/12   2784.052566



[13, -10, 6, 42, 27, -46]   [[1, 2, 2, 3], [0, 13, -10, 6]]

bad   331.7213164   g   14.05800468   rms   1.678518039

1   -38.46612668


Amt
[5, 13, -17, -76, 41, 9]   [[1, 3, 6, -2], [0, 5, 13, -17]]

bad   195.3007298   g   15.19489337   rms   .8458796028

1   -339.4147298



[2, 25, 13, -40, -15, 35]   [[1, 1, -5, -1], [0, 2, 25, 13]]

bad   137.7813896   g   15.34473078   rms   .5851564738

1   351.4712147


Ennealimmal
[18, 27, 18, -34, 22, 1]   [[9, 1, 1, 12], [0, 2, 3, 2]]

bad   37.51193854   g   16.95758830   rms   .1304491741

1/9   884.3341826

Message: 4988 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 04:40:04

Subject: Re: A 7-limit best list

From: genewardsmith

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


> I'm sorry. I just gave enough info for anyone to recognise and 
> distinguish them, and assumed you would have your software _generate_ 
> the more detailed list, using the cutoffs I gave.


I wanted the list to find the cutoffs, but it's all done now.

 I eagerly await your 

> next list. I fear it will still contain some temperaments that are of 
> no real interest to musicians.


We've got people who sniff suspiciously at anything remotely resembling an audible deviation from RI.

Message: 4989 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 16:48:41

Subject: Amended best list

From: genewardsmith

I had a problem merging lists (I neglected to consider that two
different temperaments can have the same complexity.) This led to the
only addition to my list, which I'm calling "supersharp" because of
its very very sharp fifth generator of 17/28, being left off.



Decimal
[4, 2, 2, -1, 8, -6]   [[2, 0, 3, 4], [0, 2, 1, 1]]

bad   184.7010517   g   2.777313996   rms   23.94525150

1/2   950.9775006


Dominant seventh
[1, 4, -2, -16, 6, 4]   [[1, 0, -4, 6], [0, 1, 4, -2]]

bad   197.3456024   g   3.128478105   rms   20.16328150

1   1902.225978


Diminished
[4, 4, 4, -2, 5, -3]   [[4, 0, 3, 5], [0, 1, 1, 1]]

bad   189.2082747   g   3.144366918   rms   19.13699259

1/4   1885.698207


Quintal
[0, 5, 0, -14, 0, 8]   [[5, 8, 0, 14], [0, 0, 1, 0]]

bad   171.2126805   g   3.290247187   rms   15.81535241

1/5   2789.386745


Augmented
[3, 0, 6, 14, -1, -7]   [[3, 0, 7, -1], [0, 1, 0, 2]]

bad   224.2088808   g   3.675273386   rms   16.59867843

1/3   1889.740309


Pajara
[2, -4, -4, 2, 12, -11]   [[2, 0, 11, 12], [0, 1, -2, -2]]

bad   169.1429833   g   3.938677761   rms   10.90317755

1/2   1908.814330


Supersharp
[2, 6, 6, -3, -4, 5]   [[2, 0, -5, -4], [0, 1, 3, 3]]

bad   292.6389492   g   3.938677761   rms   18.86388876

1/2   1928.512337


Hexadecimal
[1, -3, 5, 20, -5, -7]   [[1, 0, 7, -5], [0, 1, -3, 5]]

bad   296.7523121   g   3.995964429   rms   18.58450012

1   1873.109081


Tertiathirds
[4, -3, 2, 13, 8, -14]   [[1, 2, 2, 3], [0, 4, -3, 2]]

bad   225.9663420   g   4.305717081   rms   12.18857055

1   -125.4687958


Kleismic
[6, 5, 3, -7, 12, -6]   [[1, 0, 1, 2], [0, 6, 5, 3]]

bad   234.2754881   g   4.368916409   rms   12.27380956

1   316.6640534


Tripletone
[3, 0, -6, -14, 18, -7]   [[3, 0, 7, 18], [0, 1, 0, -2]]

bad   173.7904007   g   4.631825456   rms   8.100678834

1/3   1911.279336



[2, 8, 1, -20, 4, 8]   [[1, 0, -4, 2], [0, 2, 8, 1]]

bad   293.7346084   g   4.811111338   rms   12.69007837

1   947.2576878


Meantone
[1, 4, 10, 12, -13, 4]   [[1, 0, -4, -13], [0, 1, 4, 10]]

bad   103.8247475   g   5.322447240   rms   3.665035228

1   1896.647968


Injera
[2, 8, 8, -4, -7, 8]   [[2, 0, -8, -7], [0, 1, 4, 4]]

bad   320.3524287   g   5.343650829   rms   11.21894132

1/2   1893.651026


Double wide
[8, 6, 6, -3, 13, -9]   [[2, 1, 3, 4], [0, 4, 3, 3]]

bad   334.6430427   g   5.746952448   rms   10.13226624

1/2   325.6113679


Porcupine
[3, 5, -6, -28, 18, 1]   [[1, 2, 3, 2], [0, 3, 5, -6]]

bad   231.9225067   g   5.836211169   rms   6.808961862

1   -162.3778142


Superpythagorean
[1, 9, -2, -30, 6, 12]   [[1, 0, -12, 6], [0, 1, 9, -2]]

bad   246.9834642   g   6.207109365   rms   6.410458352

1   1910.384820



[5, 1, -7, -19, 25, -10]   [[1, 0, 2, 5], [0, 5, 1, -7]]

bad   347.0112248   g   6.305725722   rms   8.727168682

1   377.6398806


Neutral thirds
[2, -9, -4, 16, 12, -19]   [[1, 1, 5, 4], [0, 2, -9, -4]]

bad   265.2522216   g   6.517068879   rms   6.245315858

1   356.3080310


Flattone
[1, 4, -9, -32, 17, 4]   [[1, 0, -4, 17], [0, 1, 4, -9]]

bad   329.1049270   g   6.557956330   rms   7.652394368

1   1893.456080


Magic
[5, 1, 12, 25, -5, -10]   [[1, 0, 2, -1], [0, 5, 1, 12]]

bad   190.6152791   g   6.786228469   rms   4.139050792

1   380.5064474


Small diesic
[10, 9, 7, -9, 17, -9]   [[1, 9, 9, 8], [0, 10, 9, 7]]

bad   181.6005942   g   7.395689110   rms   3.320167332

1   -890.0485289


Semisixths (tiny diesic)
[7, 9, 13, 5, -1, -2]   [[1, 6, 8, 11], [0, 7, 9, 13]]

bad   278.2627568   g   7.420887650   rms   5.052931030

1   -756.3796144


Orwell
[7, -3, 8, 27, 7, -21]   [[1, 0, 3, 1], [0, 7, -3, 8]]

bad   142.6121910   g   7.421511799   rms   2.589237496

1   271.3263633


Miracle
[6, -7, -2, 15, 20, -25]   [[1, 1, 3, 3], [0, 6, -7, -2]]

bad   94.80434091   g   7.609147969   rms   1.637405196

1   116.5729472


Quartaminorthirds
[9, 5, -3, -21, 30, -13]   [[1, 1, 2, 3], [0, 9, 5, -3]]

bad   179.6938179   g   7.655669978   rms   3.065961726

1   77.70708740


Supermajor seconds
[3, 12, -1, -36, 10, 12]   [[1, 1, 0, 3], [0, 3, 12, -1]]

bad   214.5541544   g   7.742330569   rms   3.579262150

1   232.1235474


Schismic
[1, -8, -14, -10, 25, -15]   [[1, 0, 15, 25], [0, 1, -8, -14]]

bad   212.0930465   g   8.612526914   rms   2.859336356

1   1902.140161



[9, 10, -3, -35, 30, -5]   [[1, 4, 5, 2], [0, 9, 10, -3]]

bad   323.7195354   g   8.937416729   rms   4.052704060

1   -321.8581275



[4, 16, 9, -24, -3, 16]   [[1, 3, 8, 6], [0, 4, 16, 9]]

bad   300.2293184   g   9.336988664   rms   3.443812018

1   -425.9591136



[2, 8, -11, -48, 23, 8]   [[1, 1, 0, 6], [0, 2, 8, -11]]

bad   333.5422522   g   9.453300867   rms   3.732363180

1   348.3528923


Diaschismic
[2, -4, -16, -26, 31, -11]   [[2, 0, 11, 31], [0, 1, -2, -8]]

bad   342.7141199   g   9.469818377   rms   3.821630536

1/2   1903.737092


Octafifths
[8, 18, 11, -25, 5, 10]   [[1, 1, 1, 2], [0, 8, 18, 11]]

bad   227.7375065   g   10.50332216   rms   2.064339812

1   88.14540670



[5, -11, -12, 3, 33, -29]   [[1, 4, -3, -3], [0, 5, -11, -12]]

bad   316.6090581   g   10.83403515   rms   2.697384486

1   -580.4242148



[3, 17, -1, -50, 10, 20]   [[1, 1, -1, 3], [0, 3, 17, -1]]

bad   324.7554230   g   10.90855391   rms   2.729116326

1   234.4104087


Shrutar
[4, -8, 14, 55, -11, -22]   [[2, 1, 9, -2], [0, 2, -4, 7]]

bad   281.7169931   g   11.18841619   rms   2.250483424

1/2   652.8935173


Catakleismic
[6, 5, 22, 37, -18, -6]   [[1, 0, 1, -3], [0, 6, 5, 22]]

bad   209.7321406   g   11.41155044   rms   1.610555448

1   316.7238784


Hemiwuerschmidt
[16, 2, 5, 6, 37, -34]   [[1, 15, 4, 7], [0, 16, 2, 5]]

bad   120.8100402   g   11.74782291   rms   .8753631224

1   -1006.090063


Hemikleismic
[12, 10, -9, -49, 48, -12]   [[1, 0, 1, 4], [0, 12, 10, -9]]

bad   311.1962901   g   12.80971160   rms   1.896512488

1   158.7324720


Hemithird
[15, -2, -5, -6, 50, -38]   [[1, 4, 2, 2], [0, 15, -2, -5]]

bad   299.6293822   g   13.15572684   rms   1.731229740

1   -193.2841226



[12, -2, 20, 52, 2, -31]   [[2, 1, 5, 2], [0, 6, -1, 10]]

bad   327.4131763   g   13.66566083   rms   1.753213789

1/2   216.7129478



[0, 12, 24, 22, -38, 19]   [[12, 19, 0, -22], [0, 0, 1, 2]]

bad   283.6535726   g   13.76571634   rms   1.496892545

1/12   2784.052566



[13, -10, 6, 42, 27, -46]   [[1, 2, 2, 3], [0, 13, -10, 6]]

bad   331.7213164   g   14.05800468   rms   1.678518039

1   -38.46612668


Amt
[5, 13, -17, -76, 41, 9]   [[1, 3, 6, -2], [0, 5, 13, -17]]

bad   195.3007298   g   15.19489337   rms   .8458796028

1   -339.4147298



[2, 25, 13, -40, -15, 35]   [[1, 1, -5, -1], [0, 2, 25, 13]]

bad   137.7813896   g   15.34473078   rms   .5851564738

1   351.4712147


Ennealimmal
[18, 27, 18, -34, 22, 1]   [[9, 1, 1, 12], [0, 2, 3, 2]]

bad   37.51193854   g   16.95758830   rms   .1304491741

1/9   884.3341826

Message: 4990 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 12:13:57

Subject: Re: Amended best list

From: Carl Lumma

>I had a problem merging lists (I neglected to consider that two different 
>temperaments can have the same complexity.) This led to the only addition
>to my list, which I'm calling "supersharp" because of its very very sharp
>fifth generator of 17/28, being left off.


For someone who doesn't understand why everyone else thinks weighted
complexity is better, could we see the same thing with unweighted
complexity?

-Carl

Message: 4991 - Contents - Hide Contents

Date: Sun, 09 Jun 2002 12:22:39

Subject: Re: A 7-limit best list

From: Carl Lumma

>We've got people who sniff suspiciously at anything remotely resembling an
>audible deviation from RI. 


We do?  Then I argue they would have little interest in temperament.

-Carl

Message: 4992 - Contents - Hide Contents

Date: Mon, 10 Jun 2002 03:47:03

Subject: Re: A 7-limit best list

From: genewardsmith

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


>> We've got people who sniff suspiciously at anything remotely resembling an
>> audible deviation from RI. 


> We do?  Then I argue they would have little interest in temperament.


I did say audible. Over on the main list, they are sniffing at the scale I gave in 612-et, but logically, should they?

Message: 4993 - Contents - Hide Contents

Date: Mon, 10 Jun 2002 16:31:00

Subject: Re: Amended best list

From: genewardsmith

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


> For someone who doesn't understand why everyone else thinks weighted
> complexity is better, could we see the same thing with unweighted
> complexity?


It still has the property which screwed me up before--the same complexity as pajara. This time the value in question is sqrt(18) =
4.242640686.

Message: 4994 - Contents - Hide Contents

Date: Mon, 10 Jun 2002 16:32:59

Subject: Re: A 7-limit best list

From: genewardsmith

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


> Most probably not.  I haven't seen that thread yet, but it doeesn't sound
> like there's a reason for us to sniff about giving it in JI, either.


Just what I was planning--give some JI scales, and point out the
2401/2400 appromimations also.

Message: 4995 - Contents - Hide Contents

Date: Mon, 10 Jun 2002 10:52:41

Subject: Re: A 7-limit best list

From: Carl Lumma

>Just what I was planning--give some JI scales, and point out the
>2401/2400 appromimations also.


Cool.  Even I am wondering how such a small comma can add many
consonances in an 11-tone scale.  Actually, if you really want
to win friends and influence people, you'll have Maple draw a
picture of the block in the lattice.  Some of the naysayers might
not even know what you mean by "increases consonances", that one
can leave everything in JI and still ignore the wolf.

-Carl

Message: 4996 - Contents - Hide Contents

Date: Mon, 10 Jun 2002 20:27:33

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> Hi George,
> 
> In fact, (and I've been making gentle noises about this possibility 
for a
> some time now), I'm willing to throw away everything we agonised 
over with
> regard to rational complements and instead adopt a simple system 
that
> applies automatically to all ETs and rational tunings.
> 
> I propose that the complement of a|b is always b||a, except that the
> complement of |//| (natural) is /||\ and the complement of /|\ 

is /|\ if it

> represents the same number of steps and (|) if it represents a 
different
> number of steps.
> 
> ...


I agree that the biggest disadvantage of the rational complements is 
that they are not easy to remember.  And the second biggest 
disadvantage is that not all of the single-shaft symbols have 
rational complements.  However, I don't like the symbol arithmetic 
being completely different in the double-shaft symbols; it is counter-
intuitive, especially in the fact that more flags make a smaller 
alteration, e.g., /||~  is a smaller alteration than either /|| or 
||~, which occurs in your porposal for 111-ET.

After carefully considering your mirroring proposal, I am making a 
counter-proposal for the determination of apotome complements that 
also eliminates both of these biggest disadvantages of the rational 
complements.  This will look familiar, except that it has one added 
clause (to cover wavy flags):

<< For a symbol consisting of:
1) a left flag (or blank)
2) a single (or triple) stem, and
3) a right flag (or blank):
4) convert the single stem to a double (or triple to an X);
5) replace the left and right flags with their opposites according to 
the following:
 a) a straight flag is the opposite of a blank (and vice versa);
 b) a convex flag is the opposite of a concave flag (and vice versa);
 c) a wavy flag is its own opposite.

This preserves most of the symbol arithmetic without encountering 
either of the two disadvantages you gave for mirrored complements.

It also retains most of the advantages of your mirroring proposal.


>> Advantages:
>> 
>> Simple to remember -- check!

>> Covers all tunings.  -- check!

>> Flags are more strongly associated with particular primes because 

the flags don't change when the comma is complemented; and

>> No new flag types ever need to be introduced merely to handle 
complements.


For straight flags used alone -- check! (they just change sides; 
otherwise, used in combination, they just disappear)
For convex right flag used alone -- check! Observe that |) has as 
complement /||(, which is the virtual equivalent of ||), which may 
then be used as the complement -- we discussed this previously.

For wavy flags  -- check! (these completely retain their identities)

For convex left flag used in combination with straight or convex 
right flag -- check!  (these don't require double-shaft complements)

This covers the situation for most of the lower-numbered ET's, which 
should keep the simple things simple.

Convex flags are not used with concave or wavy flags (nor are concave 
flags generally used at all) until the notation starts getting more 
complicated, and I don't think that the complementation is going to 
result in too many new flags in those situations (we would have to 
try this to see).


>> Doesn't require /| and ||\ as a special case.  -- check!


Plus there are the following additional advantages:

Doesn't require /||\ to be an exception.

Retains most of the symbol arithmetic used in single-shaft symbols.  
The exceptions are with the left convex and left concave flags.  
After looking at several ET's in which these are used, I believe that 
problems with these can be easily avoided in most cases, especially 
in the lower-numbered ET's.

Let me know what you think about this (not-so-new) idea.

--George

Message: 4997 - Contents - Hide Contents

Date: Tue, 11 Jun 2002 00:13:17

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

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

> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> I agree that the biggest disadvantage of the rational complements is 
> that they are not easy to remember.  And the second biggest 
> disadvantage is that not all of the single-shaft symbols have 
> rational complements.  However, I don't like the symbol arithmetic 
> being completely different in the double-shaft symbols; it is 
counter-
> intuitive, especially in the fact that more flags make a smaller 
> alteration, e.g., /||~  is a smaller alteration than either /|| or 
> ||~, which occurs in your porposal for 111-ET.


Yes. More flags being a smaller alteration, is a serious problem with 
mirrored complements.


> After carefully considering your mirroring proposal, I am making a 
> counter-proposal for the determination of apotome complements that 
> also eliminates both of these biggest disadvantages of the rational 
> complements.  This will look familiar, except that it has one added 
> clause (to cover wavy flags):
> 
> << For a symbol consisting of:
> 1) a left flag (or blank)
> 2) a single (or triple) stem, and
> 3) a right flag (or blank):
> 4) convert the single stem to a double (or triple to an X);
> 5) replace the left and right flags with their opposites according 
to 
> the following:
>  a) a straight flag is the opposite of a blank (and vice versa);
>  b) a convex flag is the opposite of a concave flag (and vice 
versa);
>  c) a wavy flag is its own opposite.


Wavy being its own opposite isn't new either. I think I proposed that 
back when we were still deciding what the wavy's would mean.


> This preserves most of the symbol arithmetic without encountering 
> either of the two disadvantages you gave for mirrored complements.
> 
> It also retains most of the advantages of your mirroring proposal.
> 
>>> Advantages:
>>> 

>>> Simple to remember -- check!
> 
>>> Covers all tunings.  -- check!
> 
>>> Flags are more strongly associated with particular primes 
because 

> the flags don't change when the comma is complemented; and

>>> No new flag types ever need to be introduced merely to handle 
> complements.
> 

> For straight flags used alone -- check! (they just change sides; 
> otherwise, used in combination, they just disappear)
> For convex right flag used alone -- check! Observe that |) has as 
> complement /||(, which is the virtual equivalent of ||), which may 
> then be used as the complement -- we discussed this previously.
> 
> For wavy flags  -- check! (these completely retain their identities)
> 
> For convex left flag used in combination with straight or convex 
> right flag -- check!  (these don't require double-shaft complements)
> 
> This covers the situation for most of the lower-numbered ET's, which 
> should keep the simple things simple.
> 
> Convex flags are not used with concave or wavy flags (nor are 
concave 
> flags generally used at all) until the notation starts getting more 
> complicated, and I don't think that the complementation is going to 
> result in too many new flags in those situations (we would have to 
> try this to see).
> 

>>> Doesn't require /| and ||\ as a special case.  -- check!
> 

> Plus there are the following additional advantages:
> 
> Doesn't require /||\ to be an exception.
> 
> Retains most of the symbol arithmetic used in single-shaft symbols.  
> The exceptions are with the left convex and left concave flags.  
> After looking at several ET's in which these are used, I believe 
that 
> problems with these can be easily avoided in most cases, especially 
> in the lower-numbered ET's.
> 
> Let me know what you think about this (not-so-new) idea.


Now that we've exhaustively (exhaustingly?) considered the 
alternatives, I think it looks absolutely brilliant!!!!

If you have time, could you repost your latest proposals, including 
any whose first half-apotome we've agreed on, using these complements?

I've been working on what I call horizontal consistency in the first 
half-apotome. I believe it is more important than vertical 
consistency. Vertical consistency is between ETs where fifth-size is 
the same but number of steps per apotome in one is a multiple of the 
other, e.g. 48-ET and 96-ET. Horizontal consistency is between ETs 
that have the same number of steps per apotome, but have slightly 
different fifth sizes. ETs with same steps-per-apotome and adjacent 
fifth sizes, always differ by 7, e.g. 41-ET, 48-ET, 55-ET.

Here are the proposals that have come from that investigation so far. 
I've added complements as per the above.

To be notated as subsets of larger ETs:
2,3,4,5,6,7,8,9,10,11,13,14,15,16,18,20,21,23,25,28,30,33,35,40,47. 
1 step per apotome
12,19,26:  /||\
2 steps per apotome
17,24,31,38: /|\  /||\
45,52:  /|)  /||\    [13-comma]
3 steps per apotome
22,29:  /|  ||\  /||\
36:  |)  ||)  /||\
43,50,57,64: /|)  (|\  /||\    [13-commas]
4 steps per apotome
27:  /|  /|)  ||\  /||\      [13-comma]
34,41,(48?):  /|  /|\  ||\  /||\
(48?),55:  ~|)  /|\  ~||(  /||\
62:  |)  /|\  (|\  /||\     [13-commas]
69,76:  |)  ??  (|\  /||\     [13-comma]
5 steps per apotome
32:  )|  /|\  (|)  (||\  /||\
39,46,53:  /|  /|\  (|)  ||\  /||\
60:  /|  |)  ||)  ||\  /||\
67,74: ~|)  /|)  (|\  ~||(  /||\
81,88: )|)  /|)  (|\  (||(  /||\         [13-commas]
6 steps per apotome
37,44,51: )|  /|  /|)  ||\  (||\  /||\         [13-commas]
or
37,44,51: |)  )|)  /|)  (||(  ||)  /||\         [13-commas]
58: /|  |\  /|\  /||  ||\  /||\
or
58: /|  |)  /|\  ||)  ||\  /||\      [13-comma]
65,72,79:  /|  |)  /|\  ||)  ||\  /||\
86,93,100:  )|)  |)  )|\  (|\  (||(  /||\   [13-commas]
or
86,100:  )|(  |)  )|\  (|\  (||)  /||\   [13-commas]
93:  |(  |)  )|\  (|\  /||)  /||\   [13-commas]

Message: 4998 - Contents - Hide Contents

Date: Tue, 11 Jun 2002 23:18:53

Subject: Re: A twelve-note, 11-limit scale

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

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

>> This results from tempering a variety of Fokker blocks using the 
>> planar temperament defined by 126/125~176/175~1. I've used the 120-
>> et for the results; since I already called the 108-et the crazy 
>> uncle of the family, I don't know where to place 120.
> 

> i mention it in my paper, it's a pajara temperament.


Perhaps it would be better to say that 120-tET "supports" pajara 
temperament (rather than "is" a pajara temperament), since pajara in 
120-tET does not use 120-tET's best approximations to all the relevant 
rational intervals.

Message: 4999 - Contents - Hide Contents

Date: Tue, 11 Jun 2002 09:11:22

Subject: Help requested

From: kalleaho

Hi! 

What should I read in the Web and in the Lists to get a good 
understanding of the notation and terminology used in tuning-math? 

I understand what linear temperaments are but the notation used is 
not self-evident to me. I also have a basic understanding of 
periodicity blocks but hmm... wedges? commatic/chromatic unison 
vectors? 

Kalle

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