Tuning-Math Digests messages 5800 - 5824

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Message: 5800

Date: Sun, 29 Dec 2002 01:46:41

Subject: Re: Temperament notation

From: Dave Keenan

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

> Whatever you do, I suggest you keep in mind that notating linear
temperaments is just as important as notating ets, if not more so.

Good point.

> Note that what we have now is a linear temperament notation adapted
for use as a 12-et notation, and not the other way around.

Good point.

Thanks to the efforts of you and Graham Breed and others on this list
we now have agreed lists of the most important 5-limit and 7-limit
linear temperaments which we can attempt to notate. 

We also have Graham's lists of LTs for higher limits up to 21, 
Automatically generated temperaments *
maybe you could check that his algorithm hasn't missed any important
ones, starting with 9-limit.

We should try to develop rules for notating them which do not depend
on predetermined ET notations. Ideally the notation would depend only
on the LT's mapping from gens and periods to primes, and not on any
particular value for the generator.

The problem is that we can't specify a notation that will work for
unlimited length chains of an LT while only allowing one or two
symbols against each note. And what's more, for even moderate length
chains we will typically need to use symbols for primes that do not
appear in the LT's map. For example with a chain of more than 21 notes
in 5-limit meantone, we wish to have a single-symbol alternative to
double-sharps and double-flats, e.g. Fx may be written G^ so that in
pitch order the letters can remain monotonic. This ^ symbol must
relate to a prime greater than 5. In this case either 7 or 11 will do.
For longer chains we may need to use symbols for 13-commas.

So even though it might only be used for 5-limit, the notation ends up
being for a particular 7, 11 or 13 limit mapping of meantone, when in
fact the range of generator sizes that are of interest for 5-limit
meantone, may encompass more than one 7 or 11 or 13-limit mapping. How
do we choose which one to use?


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Message: 5801

Date: Sun, 29 Dec 2002 20:16:48

Subject: Re: Temperament notation

From: Carl Lumma

>>I don't know how you expect to do that. The most obvious approach 
>>seems to me to pick nominals for a MOS with 26 steps or fewer,
>>pick the generator which has the lowest height in the correct
>>p-limit, and use something like sharps and flats. That is
>>completely at odds with what you are doing. If I understand how
>>Graham's decimal notation works, it would be a generalization of
>>that.
> 
>Yes, it is completely at odds. Yes, it is a generalisation of 
>multi-sharp/flat meantone notation and Graham's decimal notation
>for miracle temperament, and the 4, 7 and 8 natural notations for
>kleismic described in my "Chain of minor thirds" article.

I believe that the simplest way to notate 'diatonic' music is to
put the transposition in the fingers and have the scale degrees
make sense on paper.  Handing the mind scale degrees is
indispensible pre-processing for working with 'diatonic' music.
The generalized keyboard would reduce the number of fingerings for
each scale -- the musician could learn twelve tunings, reading
from 'diatonic' notation in each case, with the same amount of
effort needed to learn to read standard notation on the piano.

But George's thoughts go a long way with me... could it be that
for a strict performer, who had to cover lots of tunings,
'transpositionally invarient' notation is the way to go?

>By the way, although there are 26 letters in our alphabet, there
>are good reasons, relating to human cognition, why one should aim
>to have between 5 and 9 naturals if such a proper MOS exists for
>the temperament, and otherwise keep it as close to 9 as possible,

Dave's right, Gene.  Where did you get 26 from?

>more than 12 is probably useless. I think that even 4 would be
>better than 13 or more, if such a choice were available.

I'd hate to compare like this.  4 is seriously too small.  If we're
allowed to write music that allows the listener to subset melodies,
then 13 would be far better.  If we're forced to write tone rows
then probably 13 would be worse.

>But as you say, this is not what George and I are trying to do.
>When one learns one such temperament-specific notation there is
>almost nothing one can carry over to an unrelated temperament,
>particularly if it involves a different number of nominals.

I disagree (see above).  I imagine that once the fingerings are
learned, the mind could transform them to scale degrees and back
fairly easily, and that much of the ability to extract scale degree
motion from one 'diatonic' notation would work on other 'diatonic'
notations (it's quite graphic, after all).  Again, I'd take very
seriously any input from George on this matter.

>Nor is there anything much one can use from one's knowledge of
>conventional notation.

I would think that learning some new fingerings would be easier
than keeping track of all the bizare accidental motion if you
force decatonic music (say) onto 7 nominals.

>We want notations where C:G is always the temperament's
>approximation of a 1:3 (if it has one) and C:E* is always it's
>approximation of a 1:5, where * stands for a single saggital
>symbol or none at all, and so on up the primes.

Easy!  You just use 7-et and slap on as many accidentals as needed.

>But we want more than this. We want it so the notation for any
>reasonable-sized MOS in the temperament can be arranged in pitch
>order with monotonic letters, e.g. one is never forced to have
>any C* higher in pitch than any D*,

?  You mean 7 monotonic letters?

If you're forcing 7 nominals, you'll have to give up the idea that
accidentals represent chromatic uvs.

>and one is not forced to use more than one symbol with any letter.

Well, you can always get that by just adding more flags.  Not sure
if such flag profusion is any better or worse that stacking a
single flag... in standard notation, double-sharp has a dedicated
flag, while double-flat does not... IOW, flags are independent of
accidentals.

-Carl


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Message: 5803

Date: Sun, 29 Dec 2002 23:29:01

Subject: Re: A common notation for JI and ETs

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "M. Schulter" <mschulter@m...> wrote:
> Hello, everyone, and here are two "tests" of the JI symbols, or
> possibly rather of my imperfect understanding of them as a beginner.
> 
> There are some ratios which I'm not sure how best to express -- an
> exercise which could help in seeing how best to use the present
> symbols or add new ones -- but I've chosen two examples where the
> symbols on hand seem sufficient, provided that I've used and
> interpreted them correctly.
> 
> First, here's a 12-note JI system with a 7-note Pythagorean chain at
> F-B plus some others ratios for the accidentals. As this example
> illustrates, I often am looking to write 13:14 as a chromatic
> semitone, or apotome at 2048:2187 plus 28672:28431 (~14.613 cents),
> with the 17' comma ~|( at 4096:4131 (~14.730 cents) as a neat
> solution, as also noted in a recent post from George. 
> 
> >                              deg217    deg494
> > C#  2048:2187    ~113.685c      21       47
> > vs. 14/13        ~128.298c      23       53
> > 28431:28672       ~14.613c       2        6
> > 17' comma ~|(     ~14.730c       3        6
> 
>          14/13    7/6         21/16      21/13   7/4
>         C~|||(   E!!!)         F!)      G~|||(  B!!!)
>      C         D        E    F       G        A        B     C
>     1/1       9/8     81/64 4/3     3/2     27/16   243/128 2/1   

Hi Margo,

I don't read multishaft sagittal very well, particularly in ASCII, so
permit me to rewrite it in single-shaft (dual symbol).

          14/13    7/6         21/16      21/13   7/4
          C#~|(    Eb!)         F!)       G#~|(   Bb!)
      C         D        E    F       G        A        B     C
     1/1       9/8     81/64 4/3     3/2     27/16   243/128 2/1   

Yes, thats quite correct.

C#~|( and G#~|( could also be notated Db(|( and Ab(|( but of course
these involve a larger comma (larger by a Pythagorean comma) and so
there seems little point in doing so. In multishaft those would be
D~!!( and A~!!(

> Here's an example of a diatonic scale in a 17-note JI tuning I use,
> showing the 351:352 and 891:896 symbols -- very intuitive for me,
> since 351:352 is very close to half of 891:896.
> 
>     B\!!/    C|(     D)|(   E\!!/    F     G|(   A)|(  B\!!/
>      1/1    44/39    14/11   4/3    3/2   22/13  21/11  2/1

Yes, quite correct. It does work out nicely doesn't it.
 
> Of course, the precise sagittal symbols wouldn't always be necessary:
> from a user's viewpoint, this is simply a "justly tempered" diatonic
> scale with some fifths pure and others wide by around a 351:352 or
> about 5 cents.
> 
> Anyway, there might be two points to this post: I find these symbols
> useful, and also intuitive, especially the 351:352 and 891:896
> symbols. Of course, I realize that they're not valid for certain ET's,
> but if I'm using them, it suggests a precise kind of JI notation
> bringing out the "rational mapping apart from an ET" style.
> 
> By the way, speaking of ET's and standard symbol sets, I should offer a
> bit of reassurance that when notating in 72-ET, I would write
> 
>               4A/|  4B\!!/
>               4G    4F
>               4E/|  4F
>               4C    3B\!!/
> 
> rather than
> 
>               4A)|( 4B\!!/
>               4G    4F
>               4E)|( 4F
>               4C    3B\!!/

Right.


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Message: 5804

Date: Sun, 29 Dec 2002 06:18:56

Subject: Re: Temperament notation

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" 
<d.keenan@u...> wrote:
> 
> > We should try to develop rules for notating them which do not 
depend
> > on predetermined ET notations. Ideally the notation would depend 
only
> > on the LT's mapping from gens and periods to primes, and not on 
any
> > particular value for the generator.
> 
> I don't know how you expect to do that. The most obvious approach 
seems to me to pick nominals for a MOS with 26 steps or fewer, pick 
the generator which has the lowest height in the correct p-limit, and 
use something like sharps and flats. That is completely at odds with 
what you are doing. If I understand how Graham's decimal notation 
works, it would be a generalization of that.

Yes, it is completely at odds. Yes, it is a generalisation of 
multi-sharp/flat meantone notation and Graham's decimal notation for 
miracle temperament, and the 4, 7 and 8 natural notations for kleismic 
described in my "Chain of minor thirds" article. 

I have nothing against these temperament-specific notations. Very 
early in the "Common notation ..." thread I said the same, and one 
could certainly use sagittal symbols consistently for this purpose.

By the way, although there are 26 letters in our alphabet, there are 
good reasons, relating to human cognition, why one should aim to have 
between 5 and 9 naturals if such a proper MOS exists for the 
temperament, and otherwise keep it as close to 9 as possible, more 
than 12 is probably useless. I think that even 4 would be better than 
13 or more, if such a choice were available.

But as you say, this is not what George and I are trying to do. When 
one learns one such temperament-specific notation there is almost 
nothing one can carry over to an unrelated temperament, particularly 
if it involves a different number of nominals. Nor is there anything 
much one can use from one's knowledge of conventional notation.

George and I are attempting to design an _evolution_ for those who are 
unlikely to be interested in such a _revolution_, and we'd love your 
help.

We want notations where C:G is always the temperament's approximation 
of a 1:3 (if it has one) and C:E* is always it's approximation of 
a 1:5, where * stands for a single saggital symbol or none at all, and 
so on up the primes. But we want more than this. We want it so the 
notation for any reasonable-sized MOS in the temperament can be 
arranged in pitch order with monotonic letters, e.g. one is never 
forced to have any C* higher in pitch than any D*, and one is not 
forced to use more than one symbol with any letter.

As one extends the chains of generators in any temperament, there 
eventually comes a proper MOS which is so close to an equal 
temperament that there is little point in adding more notes. There 
seems no disagreement that this happens at 72 for miracle, but for 
meantone it is unclear to me whether it is 31 or 50. It seems to be 50 
in the 5-limit optimised case, and 31 for 7 and 11 limit.

For schismic is it 41 or 53?
For kleismic is it 53 or 72?
etc.


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Message: 5806

Date: Mon, 30 Dec 2002 23:10:12

Subject: Re: A common notation for JI and ETs

From: M. Schulter

Hello, Dave and everyone, and thank you for your response to my
examples of JI notation.

As I've read the draft of the article for Xenharmonikon 18, and
followed some of the discussions about reviewing and possibly adding
to the list of commas and dieses, I've noticed that the available
symbols cover many of my favorite ratios, but leave a few questions.

The following JI system, which I came up with earlier this year, is
based in part on an arithmetic or subharmonic series a la Kathleen
Schlesinger of 28-27-26-24-23-22-21, with the ratio of 22:28 or 11:14
divided into whole-tone steps of 39:44 and 242:273. The first ratio in
this division is wider than 8:9 by 351:352, the second by 363:364.

The tuning, as it happens, is generally similar to the 17-note
triaphonic system of John Chalmers. He and Erv Wilson, it turns out,
got the idea of combining Schlesinger's _harmoniai_ or arithmetic
divisions with a tetrachord (3:4) structure some years before I did.

The general philosophy of my Just Octachord Tuning (JOT-17) -- with an
"octachord" or eight inclusive steps for each 3:4 tetrachord -- is
that of a JI system offering many of the features of a 17-note
well-temperament, but with some quirks making it "a bit different"
than a closed circle.

Like a 17-note well-temperament, JOT-17 has at each position some kind
of diatonic "thirdtone" (1deg17) ranging in size from 88:91 to 21:22;
each whole-tone (3deg17) is divided into three such steps. Minor
thirds (4deg17) range from Pythagorean to septimal, with major thirds
(6deg17) having a similar range, and neutral thirds (5deg17) from
52:63 to 21:26, etc.

However, the presence of some pure septimal ratios and sonorities such
as 12:14:18:21 or 14:18:21:24 is associated with two fifths (10deg17)
wide by a full 63:64 (~27.264 cents) and 729:736 (~16.544 cents). Thus
a sagittal notation, at least, might seek to modify a usual 17-ET
notation for these special intervals in the interest of "least
astonishment," as well as to reflect the availability of such
sonorities as a just 16:21:24:28 in one position.

With JOT-17 we might therefore ask two questions: how _can_ we
precisely notate the diverse intervals in an approach showing
modifications of Pythagorean tuning (following the sagittal convention
of disregarding small schismas such as 10647:10648)?; and how _should_
we notate such a system to best effect, at once following many of the
norms of a 17-ET notation and duly representing divergences of
interest to performers?

Here is a Scala file for JOT-17, presented for simplicity in an
arrangement where the "1/1" is B\!!/ or Bb, the lowest note of the
first 3:4 tetrachord or "octachord":

! jot17a.scl
!
Just octachord tuning -- 4:3-9:8-4:3 division, 17 steps (7 + 3 + 7), Bb-Bb
 17
!
 28/27
 14/13
 44/39
 7/6
 28/23
 28/22
 4/3
 112/81
 56/39
 3/2
 14/9
 21/13
 22/13
 7/4
 42/23
 21/11
 2/1

Here's a Scala "show scale" data file:

|
Just octachord tuning -- 4:3-9:8-4:3 division, 17 steps (7 + 3 + 7), Bb-Bb
  0:          1/1            0.000000 unison, perfect prime
  1:         28/27           62.96093 1/3-tone   
  2:         14/13           128.2983 2/3-tone
  3:         44/39           208.8353
  4:          7/6            266.8710 septimal minor third
  5:         28/23           340.5516
  6:         14/11           417.5081 undecimal diminished fourth
  7:          4/3            498.0452 perfect fourth
  8:        112/81           561.0061
  9:         56/39           626.3435
 10:          3/2            701.9553 perfect fifth
 11:         14/9            764.9162 septimal minor sixth
 12:         21/13           830.2536
 13:         22/13           910.7907
 14:          7/4            968.8264 harmonic seventh
 15:         42/23           1042.507
 16:         21/11           1119.463
 17:          2/1            1200.000 octave

In my 17-ET notation, I'd really like to consider the 1/1 as B\!!/,
placing the 21:32 between the steps A!!!) and E\!!/ -- however, for
this try, why don't I instead consider the 1/1 as C, to simplify a
certain point I'll explain.

Here are the steps I can readily notate, and some on which I'm unsure:

  1/1            C           C
 28/27           D!!!)       Db!)
 14/13           C~|||(      C#~|(
 44/39           D|(         D|(
  7/6            E!!!)       Eb!)
 28/23           This is Pythagorean D/||\ + 452709:458752 (~22.957c)
 14/11           E)|(        E)|(
  4/3            F           F
112/81           G!!!(       Gb!)
 56/39           F~|||(      F#~|(
  3/2            G           G
 14/9            A!!!)       Ab!)
 21/13           G~|||(      G#~|(       
 22/13           A|(         A|(
  7/4            B!!!)       Bb!)
 42/23           This is Pythagorean A/||\ + 452709:458752 (~22.957c)
 21/11           B)|(        B)|(
  2/1            C           C 

Interestingly, the 23:28 or 23:42 is very close to a Pythagorean
augmented second or sixth plus a Pythagorean comma -- so if there's a
sign for a Pythagorean comma, that might do (within ~0.5 cents of the
actual ratio).

Now for the complication. When I took B\!!/ for the 1/1, I realized
than using 17-ET conventions, A/||\ would be a thirdtone higher, here
a just 27:28 -- but in Pythagorean, this is only a Pythagorean comma,
so the modifications become unclear to me as a beginner.

In practice, I would guess that keeping track of the 351:352
adjustments and the like would be unnecessary, much as in a 17-tone
well-temperament. However, three fifths do seem to invite some
explicit sagittal modifications. Taking 1/1 as B\!!/ or Bb, these
would be in conventional notation F#-C# (69:104, wide by 207:208 or
~8.343 cents); G#-D# (243:368, wide by 729:736 or ~16.544 cents); and
Ab-Eb (21:32, wide by 63:64 or ~27.264 cents). 

Anyway, the problem of notating the 23:28 as a modification of a
Pythagorean augmented second might be another reason for a Pythagorean
comma sign, which as I recall has been proposed by adding at least one
flag to one of the basic symbols.

Most appreciatively,

Margo
mschulter@xxxxx.xxx


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Message: 5808

Date: Wed, 01 Jan 2003 20:37:15

Subject: Re: A common notation for JI and ETs

From: David C Keenan

In tuning-math@xxxxxxxxxxx.xxxx "M. Schulter" <mschulter@m...> wrote:

>The following JI system, which I came up with earlier this year, is
>based in part on an arithmetic or subharmonic series a la Kathleen
>Schlesinger of 28-27-26-24-23-22-21, with the ratio of 22:28 or 11:14
>divided into whole-tone steps of 39:44 and 242:273. The first ratio in
>this division is wider than 8:9 by 351:352, the second by 363:364.

Oh dear! You're really testing us aren't you? _SUB_harmonic series, ratios 
of 23, and tempering (albeit by ratios).

>Here is a Scala file for JOT-17, presented for simplicity in an
>arrangement where the "1/1" is B\!!/ or Bb, the lowest note of the
>first 3:4 tetrachord or "octachord":
>
>! jot17a.scl
>!
>Just octachord tuning -- 4:3-9:8-4:3 division, 17 steps (7 + 3 + 7), Bb-Bb
>  17
>!
>  28/27
>  14/13
>  44/39
>  7/6
>  28/23
>  28/22
>  4/3
>  112/81
>  56/39
>  3/2
>  14/9
>  21/13
>  22/13
>  7/4
>  42/23
>  21/11
>  2/1

...

>In my 17-ET notation, I'd really like to consider the 1/1 as B\!!/,
>placing the 21:32 between the steps A!!!) and E\!!/ -- however, for
>this try, why don't I instead consider the 1/1 as C, to simplify a
>certain point I'll explain.
>
>Here are the steps I can readily notate, and some on which I'm unsure:
>
>   1/1            C           C
>  28/27           D!!!)       Db!)
>  14/13           C~|||(      C#~|(
>  44/39           D|(         D|(
>   7/6            E!!!)       Eb!)
>  28/23           This is Pythagorean D/||\ + 452709:458752 (~22.957c)
>  14/11           E)|(        E)|(
>   4/3            F           F
>112/81           G!!!(       Gb!)
>  56/39           F~|||(      F#~|(
>   3/2            G           G
>  14/9            A!!!)       Ab!)
>  21/13           G~|||(      G#~|(
>  22/13           A|(         A|(
>   7/4            B!!!)       Bb!)
>  42/23           This is Pythagorean A/||\ + 452709:458752 (~22.957c)
>  21/11           B)|(        B)|(
>   2/1            C           C
>
>Interestingly, the 23:28 or 23:42 is very close to a Pythagorean
>augmented second or sixth plus a Pythagorean comma -- so if there's a
>sign for a Pythagorean comma, that might do (within ~0.5 cents of the
>actual ratio).

The above notation is quite correct.

You _could_ use a Pythagorean comma symbol for 28/23 and 42/23, but I don't 
think we've agreed on that symbol, and some of the candidates are rather 
complicated 3-flaggers, and in any case I have a much simpler suggestion. 
Use the 5-comma symbol /| .

28/23           D/|||      D#/|
42/23           A/|||      A#/|

You will find that this does not imply any actual ratios of 5 in this 
tuning and happens to be consistent with 212-ET, which models it rather well.

Proposal
212-ET: |(  )|(  ~|(  /|  |)  (|  (|(  //|  /|\  (/|  (|)

I'm not saying that /| is always appropriate to represent a 7:23 comma, but 
in this tuning I do not believe these two pitches are justly intoned 
relative to any combination of other pitches in the tuning (except each 
other) and therefore I feel that the 1.5 cent notational schisma it 
involves is insignificant. We can just as easily decide to read an Archytas 
comma symbol as a 7:23-comma, as we can a Pythagorean comma symbol.

>Now for the complication. When I took B\!!/ for the 1/1, I realized
>than using 17-ET conventions, A/||\ would be a thirdtone higher, here
>a just 27:28 -- but in Pythagorean, this is only a Pythagorean comma,
>so the modifications become unclear to me as a beginner.

There is no way, in the rational sagittal notation, to notate a 27:28 above 
a Bb, as a variety of A#. Nor do I think there ever should be. That's 
equivalent to wanting to notate a 4:7 above C as a variety of Gx.

With 1/1 as Bb, 28/27 would be Cb!) or C !!!) and 112/81 would be Fb!) or 
F!!!).

You could invoke the 212-ET notation and call them A#(|( and D#(|( but this 
seems wrong to me because it actually makes use of the slight difference 
between the 212-ET fifth and the just fifth, which we are otherwise ignoring.

>In practice, I would guess that keeping track of the 351:352
>adjustments and the like would be unnecessary, much as in a 17-tone
>well-temperament. However, three fifths do seem to invite some
>explicit sagittal modifications. Taking 1/1 as B\!!/ or Bb, these
>would be in conventional notation F#-C# (69:104, wide by 207:208 or
>~8.343 cents); G#-D# (243:368, wide by 729:736 or ~16.544 cents); and
>Ab-Eb (21:32, wide by 63:64 or ~27.264 cents).

What's wrong with 1/1 as C and then the wolves are
G#~|( to D#/| 8 cents wide,
A#/|  to Gb!) 17 cents wide,
Bb!)  to F    27 cents wide.

Surely you would want A#-Gb to be a wolf?

I'm afraid I don't understand the advantage of the Bb 1/1.

We also have 5c wide fifths
G    to D|(
A|(  to E)|(
B)|( to F#~|(

>Anyway, the problem of notating the 23:28 as a modification of a
>Pythagorean augmented second might be another reason for a Pythagorean
>comma sign, which as I recall has been proposed by adding at least one
>flag to one of the basic symbols.

Yes one proposal is to add a very tiny straight right flag (or some other 
tiny graphical addition) to the 5-comma symbol /| . Using Pythagorean 
rather than Archtus certainly would make the distinction between the 5 cent 
wide and the 8 cent wide fifths, which are not distinguished in a 212-ET 
mapping.

There's another angle to this which I won't go into since you seem happy 
with the way it is. That is: How would you notate it if you did not want to 
have C-something being a higher pitch than D-something etc.; what I refer 
to as having monotonic letters?

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5809

Date: Wed, 01 Jan 2003 21:25:27

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 01:21 PM 23/12/2002 -0800, George Secor wrote:
> > The 7:11 comma that is relevant to Peppermint is 891:896 which I
>don't
> > believe we have considered before in regard to the sagittal notation.
>The
> > appropriate symbol for it would be )|(, however it vanishes in
>Peppermint
> > (and 121-ET).
>
>The symbol )|( is not valid as 891:896 in either 217 or 494, but that
>shouldn't stop anyone from using it for JI,  unless we can figure out
>something else.  Well, here's something else:  The (19'-17)-5' comma
>~)|' would come within 0.3 cents and would be valid in both 217 and 494
>(plus 224, 270, 282, 311, 342, 364, 388, 400, 525, and 612, to name
>more than a few).  The only thing I can say against it is that it seems
>rather contrived and not at all intuitive, but it works in more places
>than I would have expected.

I'd prefer to go with )|( as the 7:11 comma since it only involves a 0.55 
cent schisma. I feel that a 3 flag symbol for something under 10 cents 
could not be justified when a 2 flagger is within 0.98 cents.

It seems 891:896 )|( should be called the 7:11 comma while the comma 
represented by (| is called the 7:11'-comma.

Are there any ETs in which we should now prefer )|( over some other symbol 
given that it now has such a low prime-limit or low product complexity?

>They are all 7-related.  In a 13-limit heptad (8:9:10:11:12:13:14) it
>is 7 that introduces scale impropriety; e.g., the fifth 5:7 is smaller
>than the fourth 7:10.  Replace 14 with 15 in the heptad and I believe
>the scale is proper.  So it would not be surprising that someone might
>want to respell the intervals involving 7 -- 4:7 as a sixth, 5:7 as a
>fourth, 6:7 as a second, 7:9 as a fourth, 11:14 as a third, and 13:14
>as an altered unison.
>
>So we would want to notate the following ratios of 7 using these
>commas:
>
>                              deg217   deg494
>                              ------   ------
>A# 32768:59049  ~1019.550c     185      420
>vs. 7/4          ~968.826c     175      399
>57344:59049       ~50.724c      10       21
>(apotome complement of 27:28 - this could be called the 7' comma)
>11:19 comma (|~   ~49.895c       9       21
>But a new symbol /|)` would represent it exactly
>(if the flags are added up separately ­ 5+7+5' comma)

I really don't think it is necessary or desirable to notate this 7'-comma. 
It is larger than the standard 7-comma and it involves a longer chain of 
fifths than _any_ other comma we've ever used.

I think we should only accept the need for a _larger_ alternative comma for 
some prime (or ratio of primes) if it involves a _shorter_ chain of fifths.

>Expressed another way:

I don't see the following quote as expressing the above quote another way. 
It is a completely different 7-comma. With this comma a 4:7 above C would 
be a kind of A, not A#.

A  16:27
vs. 4:7

>F  3:4           ~498.045c      90      205
>vs. 9/7          ~435.084c      79      179
>27:28             ~62.961c      11       26
>symbol  )||                     12       26
>But a new symbol (|\' would represent it exactly

It is very large, and the absolute value of its power of 3 is still larger 
than that of the standard 7 comma, although only by 1. I'm not convinced 
there's any need for it.

>F#  512:729      ~611.730c     111      252
>vs. 7/5          ~582.512c     105      240
>3584:3645         ~29.218c       6       12
>This is the 5:7' comma, or 7+5' comma, or 7'-5 comma
>A new symbol  |)` would represent it exactly

This contains 3^6 while the standard 5:7-comma has 3^-6 so I think 
there  could be some demand for this one. I think the proposed symbol is 
good, being only 2 flags, however I'd like it even better if we could come 
up with some way that the 5'-comma (ordinary schisma) could be notated as a 
modification of the shaft rather than as a flag, or if the two flags were 
not on the same side.

But in any case, it seems we should avoid using it if possible because of 
its containing that very unfamiliar flag. It's kind of strange if we should 
need to use this obscurte new flag as low as the 7-limit. You should leave 
it out of the XH18 paper.

>E  64:81        ~407.820c      74      168
>vs. 14/11        ~417.508c      75      172
>891:896            ~9.688c       1        4
>5:7+19 comma )|(   ~9.136c       2        3

Agreed.

>C#  2048:2187    ~113.685c      21       47
>vs. 14/13        ~128.298c      23       53
>28431:28672       ~14.613c       2        6
>17' comma ~|(     ~14.730c       3        6

Agreed. I though we already had that one. I believe we called this the 7:13 
comma while (|( is the 7:13' comma.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5815

Date: Fri, 3 Jan 2003 14:17:59

Subject: Re: Minimax generator

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

Minimax generators can also be calculated with Scala by the
"calculate/minimax" command. It shows the least squares optimum
at the same time, so you don't need to enter everything twice
if you want that also.

The next version still to come will have a new dialog to support 
the easy calculation of equal beating temperaments, in the Tools
menu.

Manuel


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Message: 5818

Date: Fri, 03 Jan 2003 08:32:44

Subject: Re: Fwd: Re: A common notation for JI and ETss

From: David C Keenan

George Secor:

>Filling out the rational complementation for a complete apotome this
>would be:
>
>212a:  |(  )|(  ~|(  /|  |)  (|  (|(  //|  /|\  (/|  (|)  ~||  ~||(  )
>||~  ||)  ||\  (||(  ||~)  /||)  /||\  (DK)
>
>I have only one question: Since the 17th harmonic is so far off in
>212-ET as to be almost midway between tones (and inconsistent
>besides), whereas the 23rd is almost exact, would it be more
>appropriate to substitute the 23 comma for the 17' comma symbol?
>That would give:
>
>212b:  |(  )|(  |~  /|  |)  (|  (|(  //|  /|\  (/|  (|)  ~||  ~||(  )
>||~  ||)  ||\  ~||)  ||~)  /||)  /||\  (GS)
>
>But if you still prefer 212a for the standard set, then at least
>Margo could use 212b as a modification, since 23 is present in her
>tuning.

23 is present in the tuning, but not in those pitches that might be notated 
with the 3deg212 symbol. Margo and I both chose ~|( because of its 
interpretation as a 7:13 comma, not a 17 comma. So whatever we might decide 
for 3deg212, ~|( seems like the right symbol for Margo's tuning.

In determining what is best for 3deg212 I agree that 17 commas should be 
avoided because of the inconsistency and inaccuracy, but should the primary 
interpretation of ~|( be the 17' comma or the 7:13 comma? The only 
popularity stats we have, say that ignoring powers of 2 and 3, 17/1 is 
twice as popular as 13/7, so 17' should be the primary interpretation. 
However these same stats say that 13/7 is slightly more popular than 23/1, 
so perhaps we should use ~|( for that reason. 212-ET is at least 
1,3,9,17-consistent.

Is there any other advantage conferred by using |~ instead of ~|( ?
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5820

Date: Fri, 03 Jan 2003 08:44:29

Subject: Re: Fwd: Re: Temperament notationn

From: David C Keenan

George Secor:

> > It depends on what method you're using to optimize.  The 5-limit
> > minimax generator is a 1/4-comma fifth, which would also favor 31.
> > In order to get the slow-beating minor third of 50-ET you also get
>a
> > slow-beating major third, plus a faster-beating fifth that is not
> > acceptable to some.  So I always thought that 31 was the clear
>choice
> > for meantone.

Me too. But some prefer RMS error, and some minimax beating, both of which 
favour something closer to 50-ET in the 5-limit.

Perhaps what we should do is give upper and lower bounds on the size of 
generator for which a particular ET notation is valid. Perhaps we 
should  propose standard notations that fully cover the spectrum of 
generator sizes from say 50 cents to 600 cents. This would be for single 
chain (octave period) temperaments. We'd have to repeat this for 2, 3, 4, 5 
chain.

On second thoughts maybe the most popular temperaments would be enough.

> >
> > > For schismic is it 41 or 53?
> >
> > How about 94?
>
>Now that I've more time to look at this, I would say definitely 94,
>for two reasons:
>
>1) If you're including the 7th harmonic, then you might as well take
>this to the 11 limit (since the minimax generator for both is the
>same).  The 11th harmonic occurs in the series of fifths in 41, 53,
>and 94 in the +23 position, but in 41 it is also closer -- in the -18
>position, which is not typical for the schismic family of
>temperaments.  So I eliminate 41 as my choice.
>
>2) The 7 and 11-limit minimax generator is ~702.193c (7:9 being
>exact) giving a maximum error of ~4.331c (for 5:7 and 5:9).  The 53-
>ET fifth is ~701.887c (max. error ~12.681c), but the 94-ET fifth is
>much closer to the ideal: ~702.178c (max. error ~4.722c), and the
>same may be said for the 13 and 15-limit minimax generator
>(~702.109c, 13:14 being exact).  So I eliminate 53, leaving 94 as my
>choice.

OK. 94 sounds good.

> > > For kleismic is it 53 or 72?
> >
> > I'll have to taken a better look at this one.
>
>I've done that and I conclude that, unless you are sticking with a 5
>limit, the choice is clearly 72 over 53.  It is best to put the
>figures in a table to show this:
>
>Generator     Size       Max. error   Exact
>
>5-minimax   ~316.993c     ~1.351c     2:3
>53-ET       ~316.981c     ~1.408c
>72-ET       ~316.667c     ~2.980c
>125-ET      ~316.800c     ~2.314c
>
>7,9-minimax ~316.765c     ~2.732c     4:7
>53-ET       ~316.981c     ~6.167c
>72-ET       ~316.667c     ~3.910c
>125-ET      ~316.800c     ~3.088c
>
>11-minimax  ~316.745c     ~2.976c     9:11
>53-ET       ~316.981c    ~12.681c
>72-ET       ~316.667c     ~3.910c
>125-ET      ~316.800c     ~4.892c
>
>I threw 125 in there also, since it does slightly better than 72 at
>the 7 and 9 limit (and also at the 13 and 15 limit, which has the
>same minimax generator as for the 7 and 9 limit).  But since the 11
>limit is the highest you can go while keeping the max error under 4
>cents, that's the limit I would use, and 72 has the advantage.
>
>Something else I noticed about the choice of 72 as the notation for
>both the Miracle and kleismic temperaments:  the progression of
>sagittal symbols for a 72-ET panchromatic scale (one passing through
>all the tones) is the same as that for a sequence of tones differing
>by the generating interval in both temperaments.  To illustrate:
>
>72-ET:     C  C\!  C!)   C\!/   B|)  B/|   B
>Miracle:   C  B\!  Bb!)  A\!/   G|)  F#/|  F
>kleismic:  C  A\!  F#!)  Eb\!/  B|)  G#/|  F
>
>The pattern then repeats.  I believe that this is a useful property
>that provides a further justification for basing the kleismic
>notation on 72.

OK. 72 is good.

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5823

Date: Fri, 03 Jan 2003 02:56:09

Subject: Re: Temperament notation

From: Carl Lumma

Hi George,

>Let me offer an example from my viewpoint that may shed some
>light on some of the issues involved.  Carl, you seem to be
>looking at the problem of notation from a keyboardist's
>viewpoint (from your mention of fingerings), but one's
>understanding and framework for a notation must be broader than
>and/or independent of that.

Everything I said should apply to any instrument.  I used
keyboarding as a 'worst-case' example.

>But even if we stick to a keyboard application for the moment,
>let me give an example of how a "transpositionally invariant"
>notation (if I understand the term correctly) could pose more
>problems than it would solve.

That was a bad choice of terminology.  The issue that Gene and
I are raising is: should notation be based on the melodic scale
being used, or should they be based on the 7-tone meantone
diatonic scale musicians are already familiar with?

The issue sort-of assumes the notation will be used to write
what I've been calling "diatonic music" -- music that takes
melodies and primary harmonies from the same small scale, as
90% of Western music does.

>Let's consider the Miracle tuning notated using Graham's decimal 
>notation, which I presume you would consider transpositionally 
>invariant.

Let's nix that term, but yes, Graham's decimal notation is what
I'd advocate for Miracle.

>But what value is the decimal notation to me, and what incentive
>would I have to learn it without a decimal keyboard?

() It gives you an invaluable tool for understanding the music.

() The re-learning won't be as bad as you fear on a decimal
keyboard.

() Since your meantone generalized keyboard is at root a planar
hexagonal tiling, many mappings exist to make it more 'decimal'.
It may be that none reflect the secor:octave cycles in the
correct way (as far as the distance of the keys from the player,
etc.), but there should be a way to get the ten nominals of the
decimal scale under the fingers.

>Now give me the same decimal keyboard with Partch's 11-limit JI 
>mapped onto it (observe that this was the reason that I originally 
>came up with the layout).  Again, I should do just fine with 72-ET 
>sagittal notation, assuming that I am proficient with the
>keyboard.

You seem to be saying that it's easier to learn to find pitches
on a keyboard than it is to learn to find pitches in a notation...
For me, it's the opposite.

>If I decide to use the Miracle temperament instead of 11-limit JI
>for Partch's music, is there any advantage in using the decimal
>notation with 10 nominals for this purpose over a 7-nominal
>sagittal notation?

Let's ask it this way: take the well-tempered clavier and re-
write it with 6 nominals.  Is that only a slight disadvantage?

Yes, my way means more work for the person interested in learning
multiple scales.  But:

() If you use a generalized keyboard, you could learn 12 scales
with the same amount of work as people spend on learning the
diatonic scale on the piano, excluding the work it takes to
find the pitches in a tuning from blobs on paper... which I seem
to think is easier than you do...

>We're all familiar with a 7-nominal / heptatonic / diatonic
>notation, and that's what Dave and I have been building on to
>produce what I call a "generalized" notation -- one that is
>semantically independent of any particular tonal geometry or
>division of the octave.  It may not be the best notation for
>everything, but it will do a lot of different things and will
>do many of them extremely well.

Definitely a worthwhile project.

-Carl


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Message: 5824

Date: Fri, 03 Jan 2003 23:55:54

Subject: Re: Temperament notation

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> 
> Would someone explain "minimax" generator (I understand rms generator)
> 
> Thanks

Frankly I think "minimax" is a silly term. I prefer to call it the
max-absolute (MA) generator. Obviously we're trying to minimise the
error measure in both cases, but we don't say "miniRMS". In one case
we minimise (I'd prefer to say "optimise") the Root of the Mean of the
Squares of the errors and in the other it is the Maximum of the
Absolute-values of the errors. RMS and MA.


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