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Message: 4500 - Contents - Hide Contents

Date: Wed, 03 Apr 2002 20:43:42

Subject: Re: A common notation for JI and ETs

From: gdsecor

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

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

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

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

>>>> Which is one reason we have both 26:27 ; |, and 1024:1053 { } 
> as 

>>>> 13-commas. Are you saying we shouldn't have 1024:1053 at all?
>>> 

>>> so the rule is that every comma and its 2187:2048 complement 
has a
>>> unique symbol?
>> 

>> No, not every comma. So far George and I have only agreed on the 
>> desirability of apotome complements of those commas which are 
close 
> to 

>> the half-apotome, say those between 1/3 and 2/3 apotome. So far 
> that's 

>> only the 11 and 13 commas (dieses).
> 

> but these 'sizes' won't come out anything like that in many, if not 
> most, equal temperaments. right?


Not really.  When notating those strange and difficult ET's that I 
must assume you have in mind (inasmuch as the twelve ET's under 100 
that I have tried so far behave quite nicely, thank you*), you can 
pick and choose the most appropriate symbols to use for the various 
degrees of alteration.  There are at present (in the 217-ET-based 
notation) 13 different symbols less than half an apotome in size from 
which to select.

Only the two largest of these have dedicated apotome-complement 
symbols.  These four symbols taken as a group are the four varieties 
of "semisharps" or "semiflats" representing ratios of 11 and 13, and 
their distinctive appearance would make it relatively easy to 
translate them at sight from JI to ET notation -- a primary reason 
for providing the dedicated symbols.

By the way, the approximate 1/3 to 2/3 apotome range given above has 
turned out to be narrower: more like 4/10 to 6/10 apotome.

--George

*The twelve ET's are 22, 27, 34, 41, 43, 46, 50, 53, 58, 72, 94, and 
96.  I didn't bother to count the more trivial ones, such as 12, 17, 
19, and 31.

Message: 4504 - Contents - Hide Contents

Date: Thu, 04 Apr 2002 07:43:08

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

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

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

>> -- Dave Keenan
> 

> who's anneliz and does she like to annelize things as much as you 
> do? :)

Tee hee.


She's Anne Elizabeth and she's my sister and no she certainly does 
not analyze. If you saw her fingers flying on her violin fingerboard 
during some of the Irish jigs she plays, you'd know that if she 
thought for a millisecond about what she was actually doing, the whole 
thing would just explode into a shower of wood, catgut and horsehair. 
:-)

Message: 4505 - Contents - Hide Contents

Date: Thu, 04 Apr 2002 21:44:29

Subject: Re: A common notation for JI and ETs

From: David C Keenan

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

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> With the 29 factor we have passed the point (in both 217 and 311) 
> where all of the ratios within the harmonic limit are a unique 

number 
> of degrees, so some bridging is inevitable.  However, what's most 
> important is that the number of degrees for the 29 comma is 
> consistent with this flag in both 217 (6 deg) and 311 (9 deg).


I had mistakenly thought that 311-ET was 31-limit-unique until Graham
corrected me. What's the smallest ET that has a 31-limit-unique mapping,
even if it's not consistent? Could it be 624-ET. How about a 35-limit
unique mapping? Never mind.


> I prefer to consider the xL flag as 715:729 with the additional 
> meaning of 45056:45927.  
...
> Or am I just splitting hairs, inasmuch as conflating 4095:4096 would 
> allow us to look at it either way (or both ways at once)?


Yes. Splitting hairs. All I'm saying is, if someone only needs an 11-limit
notation then they shouldn't have to know anything about xL also being
715:729 (13-limit).


> As I mentioned above, in the process of counting sharp-vs.-flat 
> occurrences for 23 in my JI heptatonic scales, I also counted the 
> number of sharp-vs.-flat occurrences for 17 (3#, 3b), 19 (4#, 5b), 

25 
> (5#, 3b), and 29 (no#, 4b).  I conclude that there has to be a 
> provision for spelling any interval in at least two different ways, 
> which is a compelling reason for providing a complete set of symbols 
> for whatever division we settle on for the JI notation.


Oh dear, I think I've agreed with certain things you proposed which have
turned out to be the beginning of a slippery slope that I don't want to go
down. One was to accept certain schismas vanishing in order to minimise the
number of different flags making up the symbols. The other was to accept
the need for more than one comma for certain primes.

Now you say we need two commas for _every_ prime. Why stop there? Shouldn't
we provide for anyone who wants to notate the prime intervals from C as
varieties of any of the following:
5	E,Fb
7	A#,Bb
11	E#,F,F#,Gb
13	G#,Ab,A
17	C#,Db
19	D#,Eb
23	F#,Gb
29	A#,Bb,Cb
31	B,Cb,C

These all involve commas smaller than 3/5 apotome. There are _only_ 23 of
them. I am of course being facetious. I'm more inclined to go back to a
strict single comma per prime. In all the lower ETs where either an
11-comma or a 13-comma is a semisharp or semiflat, its apotome complement
is the same number of steps, so there isn't any need for both.

In rational tunings, I don't see why we have to cater for all the base
notes (powers of 3 to which the prime commas are applied) remaining within
a single diatonic scale. Actually it isn't even possible, unless you allow
commas much larger than 3/5 apotome. I see no problem with requiring a
chromatic scale for the base notes, or worse, a few enharmonics such as Ab
and G# together.

Remember I'm happy to use up to 3 symbols against a note (for rational
tunings and very large ETs) rather than have too many new symbols for
people to learn.


> In addition, being able to notate all of the degrees would ensure 
> that no matter how much modulation is done in JI, at least one would 
> never run out of symbols.  If this were not done, then we would lose 
> one of the principal advantages of mapping the JI notation to a 
> specific division.
...
> So the question now becomes:  Are we left with any good reason for 
> basing the JI notation on 311 instead of 217?


From your point of view, I would say that you are better off with 217-ET.

However I do not wish to base a JI (rational) notation on _any_ temperament
that has errors larger than 0.5 c. For me, 217-ET and 311-ET were merely a
way of looking for schismas that might be notationally usable (less than
0.5 c), and of checking that things were working sensibly, and it was nice
to actually be able to notate those ETs themselves. But I'm taking Johnny
Reinhard at his word when he says (or implies) that nothing less than
1200-ET is good enough as an ET-based JI notation.

What is the first ET above 1200 that is 35-limit consistent? 35-limit unique?

But I am more interested in a 31-limit (or 35-limit) temperament (not equal
or linear or even planar, but maybe 6D or 7D or 8D) with as many
sub-half-cent schismas as possible, vanishing. Hence the challenge which
no-one's taken up yet, except in part.

I don't share your obsession with packing all the necessary information
into a single symbol. Some folks may well be willing to notate their JI
piece by mapping to the nearest degree of 217-ET, and they may well be glad
of the possibility to do it with only one accidental per note, but that
won't be everyone's cup of tea. Given that it may involve errors of up to 3
cents, it must be a conscious decision, not something forced upon us by the
notation.

However, it seems that most would be willing to wear it if a 0.5 c error
were forced upon them by the notation. The only objections to this will be
"philosophical" ones, not ones that anyone can hear (except in something
like Lamonte Young's Dream House).

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

Message: 4506 - Contents - Hide Contents

Date: Thu, 4 Apr 2002 12:50 +01

Subject: Re: A common notation for JI and ETs

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a8fkev+5808@xxxxxxx.xxx>
gdsecor wrote:


> Something else in favor of 217: I noted earlier that it is 7 times 
> 31.  If you make instruments for 31-ET (or train string players to 
> play 31-ET), then your JI can be reckoned in alterations of +/- 1 to 
> 3 increments of 217-ET (i.e., multiples of ~5.5 cents), to a maximum 
> of 16.6 cents, which is not an unreasonable amount of intonation 
> adjustment for instruments of flexible pitch.  This would be a decent 
> practical alternative to alterations reckoned relative to 12-ET, 
> whether for (too-coarse) 72-ET or Johnny Reinhard's (too-small) one-
> cent increments.  Plus you get 31-ET in the bargain.  And I should 
> mention that a series of 41 fifths in 217 (kept within an octave) 
> brings you only one degree away from your starting point, giving you 
> a very close approximation of 41-ET.


31-ET for the 31-limit, great!

In fact, a temperament with a period of 1/31 octaves is already my top 
inconsistent 21-limit temperament:

1/3, 16.0 cent generator

basis:
(0.032258064516129031, 0.013300385023233607)

mapping by period and generator:
[(31, 0), (49, 0), (72, 0), (87, 0), (107, 0), (115, -1), (127, -1), (132, 
-1)]

mapping by steps:
[(62, 31), (98, 49), (144, 72), (174, 87), (214, 107), (229, 115), (253, 
127), (263, 132)]

highest interval width: 1
complexity measure: 31  (62 for smallest MOS)
highest error: 0.009287  (11.145 cents)

Without a history of usage of such temperaments, it's really impossible to 
say if that 11 cent error is good or bad.  It's lower in absolute terms 
than 12-equal in the 5-limit, which many put up with.  But if the 21-limit 
is taken to mean the difference between 22:21 (80.5 cents) and 21:20 
(84.5) cents is significant, 11 cents is huge.  Of course, any 
inconsistent temperament will remove such distinctions.  Here are the 
21-limit equivalences for this particular temperament:

15:14 =~ 16:15
10:9 =~ 19:17 =~ 9:8
12:11 =~ 11:10
22:19 =~ 15:13
13:11 =~ 19:16
16:13 =~ 21:17
9:7 =~ 14:11
21:16 =~ 17:13
17:16 =~ 19:18
15:11 =~ 11:8 =~ 26:19
22:21 =~ 21:20
20:19 =~ 18:17


The online scripts and results won't handle the 31-limit, but I can run it 
locally (it's a trivial change).  The temperament George describes is 
roughly


>>> temper.Temperament(31, 217, temper.primes)


1/8, 5.3 cent generator

basis:
(0.032258064516129031, 0.0043924816210988982)

mapping by period and generator:
[(31, 0), (49, 1), (72, 0), (87, 0), (107, 2), (115, -2), (127, -2), (132, 
-2),
(140, 2), (151, -3), (154, -3)]

mapping by steps:
[(217, 31), (344, 49), (504, 72), (609, 87), (751, 107), (803, 115), (887, 
127),
 (922, 132), (982, 140), (1054, 151), (1075, 154)]

highest interval width: 6
complexity measure: 186  (217 for smallest MOS)
highest error: 0.002255  (2.706 cents)


217-equal is only accurate to 3.2 cents, so there is some kind of 
improvement here.  The complexity may be too high for the tradeoff to be 
worthwhile.


An interesting alternative is


>>> temper.Temperament(62,217,temper.primes)


4/9, 16.7 cent generator

basis:
(0.032258064516129031, 0.013944526171363311)

mapping by period and generator:
[(31, 0), (50, -2), (72, 0), (87, 0), (109, -4), (116, -3), (128, -3), 
(133, -3)
, (142, -4), (151, -1), (154, -1)]

mapping by steps:
[(217, 62), (344, 98), (504, 144), (609, 174), (751, 214), (803, 229), 
(887, 253
), (922, 263), (982, 280), (1054, 301), (1075, 307)]

highest interval width: 6
complexity measure: 186  (217 for smallest MOS)
highest error: 0.002263  (2.715 cents)


If you're prepared to forego ratios involving 27, the complexity is only 
4*26, compared to 5*26 for the other version.  One disadvantage is that 
the 3:1 has a more complex approximation.  But that may not be important, 
as you can modulate by as many 31-equal fifths as you like, and they're 
good enough in another context.  Again, there isn't enough (any!) history 
of tempered 31-limit music to pronounce on the importance of these 
differences.


                  Graham

Message: 4509 - Contents - Hide Contents

Date: Fri, 5 Apr 2002 10:12 +01

Subject: Re: A common notation for JI and ETs

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <3.0.6.32.20020404214429.00b1bb90@xx.xxx.xx>
David C Keenan wrote:


> What is the first ET above 1200 that is 35-limit consistent? 35-limit 
> unique?


Above 1200?  1600.  The only ones below that are 311 and 388.  If I can 
remember how I did uniqueness for the 11-limit, I'll try a search on that 
as well.

Anyway, the simplest linear temperament from consistent ETs is

236/699, 405.2 cent generator

basis:
(1.0, 0.3376257079318315)

mapping by period and generator:
[(1, 0), (8, -19), (-23, 75), (18, -45), (46, -126), (1, 8), (-29, 98), 
(11, -20
), (42, -111), (16, -33), (61, -166)]

mapping by steps:
[(388, 311), (615, 493), (901, 722), (1089, 873), (1342, 1076), (1436, 
1151), (1
586, 1271), (1648, 1321), (1755, 1407), (1885, 1511), (1922, 1541)]

highest interval width: 316
complexity measure: 316  (388 for smallest MOS)
highest error: 0.001078  (1.293 cents)

The next two are unique

41/497, 24.7 cent generator

basis:
(0.25, 0.020624732047905801)

mapping by period and generator:
[(4, 0), (7, -8), (13, -45), (1, 124), (6, 95), (20, -63), (18, -20), (8, 
109),
(11, 86), (21, -19), (10, 119)]

mapping by steps:
[(1600, 388), (2536, 615), (3715, 901), (4492, 1089), (5535, 1342), (5921, 
1436)
, (6540, 1586), (6797, 1648), (7238, 1755), (7773, 1885), (7927, 1922)]

highest interval width: 214
complexity measure: 856  (1212 for smallest MOS)
highest error: 0.000284  (0.341 cents)
unique


467/1911, 293.3 cent generator

basis:
(1.0, 0.24437509095621543)

mapping by period and generator:
[(1, 0), (27, -104), (-38, 165), (-49, 212), (56, -215), (45, -169), (-79, 
340),
 (-61, 267), (49, -182), (53, -197), (79, -303)]

mapping by steps:
[(1600, 311), (2536, 493), (3715, 722), (4492, 873), (5535, 1076), (5921, 
1151),
 (6540, 1271), (6797, 1321), (7238, 1407), (7773, 1511), (7927, 1541)]

highest interval width: 696
complexity measure: 696  (978 for smallest MOS)
highest error: 0.000298  (0.358 cents)
unique


which isn't enough to prove that 1600-equal is unique.


                            Graham

Message: 4510 - Contents - Hide Contents

Date: Fri, 05 Apr 2002 15:06:51

Subject: Re: A common notation for JI and ETs

From: gdsecor

I double-checked the following and found it to be in error:

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

> 
> To further complicate things in 311, I also noticed in my notes 
that, 
> while I favored using 19 as an E-flat (where 1/1 is C), there were 
> almost as many instances where a heptatonic scale called for a D-
> sharp (using the 19-comma 19456:19683, 3^9:2^14*19, ~20.082 
cents).  
> This usage turns out to be *inconsistent* in 311 (but consistent in 
> 217), a problem that I didn't expect to find.  While I compared the 
> inconsistency of a few ratios of 23 in 217 with driving a car 
> slightly onto the shoulder, the 311 problem (affecting all ratios 
of 
> 19) is more like attempting to drive in the less-traveled direction 
> on route 19 and finding yourself in the wrong lane moving against 
> traffic. So I am beginning to have serious doubts about going to 
311.


The 19-comma 19456:19683 is 2^10*19:3^9 (~20.082 cents), and its 
usage is consistent in *both* 217 and 311.

I'm sorry about the misinformation, but I'm very delighted that I was 
wrong about this.

--George

Message: 4511 - Contents - Hide Contents

Date: Fri, 5 Apr 2002 17:34:50

Subject: Re: A common notation for JI and ETs

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

Dave wrote:

>I had mistakenly thought that 311-ET was 31-limit-unique until Graham
>corrected me. What's the smallest ET that has a 31-limit-unique mapping,
>even if it's not consistent? Could it be 624-ET.


Yes, 624 to 633-tET are all 35-limit unique.
1600-tET is 37-limit consistent and 55-limit unique.

Manuel

Message: 4513 - Contents - Hide Contents

Date: Fri, 05 Apr 2002 21:50:41

Subject: Re: A common notation for JI and ETs

From: gdsecor

Note:  I posted this more than a couple of hours ago, but it still 
hasn't shown up, so, after having made a correction, I am trying 
again.  Please disregard the duplicate, as I suspect that it will 
eventually show up.  --gs

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

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

>> With the 29 factor we have passed the point (in both 217 and 311) 
>> where all of the ratios within the harmonic limit are a unique 
number 
>> of degrees, so some bridging is inevitable.  However, what's most 
>> important is that the number of degrees for the 29 comma is 
>> consistent with this flag in both 217 (6 deg) and 311 (9 deg).
> 

> I had mistakenly thought that 311-ET was 31-limit-unique until 
Graham
> corrected me. ...


>> ...  I conclude that there has to be a 
>> provision for spelling any interval in at least two different 
ways, 
>> which is a compelling reason for providing a complete set of 
symbols 
>> for whatever division we settle on for the JI notation.
> 

> Oh dear, I think I've agreed with certain things you proposed which 
have
> turned out to be the beginning of a slippery slope that I don't 

want to go

> down. One was to accept certain schismas vanishing in order to 
minimise the
> number of different flags making up the symbols. The other was to 
accept
> the need for more than one comma for certain primes.
> 
> Now you say we need two commas for _every_ prime. Why stop there? 
Shouldn't
> we provide for anyone who wants to notate the prime intervals from 
C as
> varieties of any of the following:
> 5	E,Fb
> 7	A#,Bb
> 11	E#,F,F#,Gb
> 13	G#,Ab,A
> 17	C#,Db
> 19	D#,Eb
> 23	F#,Gb
> 29	A#,Bb,Cb
> 31	B,Cb,C
> 
> These all involve commas smaller than 3/5 apotome. There are _only_ 
23 of
> them. I am of course being facetious. I'm more inclined to go back 
to a
> strict single comma per prime. In all the lower ETs where either an
> 11-comma or a 13-comma is a semisharp or semiflat, its apotome 
complement
> is the same number of steps, so there isn't any need for both.


That is correct if you are indicating "semisharp" and "semiflat" to 
be exactly half an apotome in those ET's (or is this what you really 
meant to say?).  In 39, 46, and 53-ET this is not the case, and you 
either need to have both 11-commas for the two different kinds of 
semisharps/semiflats, or else have the 11 comma for one and the 13 
comma for the other (although these would then be faux complements).

I think that I have probably misunderstood what you were trying to 
say and that my going on about this is sort of pointless, considering 
that we are getting the notation for the secondary 11 and 13 commas 
as a freebie with the 4095:4096 schisma.  I think that your point is 
that each pair of 11 and 13 commas consists of apotome-complements, 
so that the larger of each could be notated as an apotome minus the 
smaller; hence we really have only one comma for each prime in the 
semantics of the notation; but for convenience in handling the 
semisharps & semiflats we have symbols that make it appear as if 
there were two.

Anyway, in pointing out the need for being able to respell ratios at 
will, my primary objective was to indicate why we needed to notate 
all of the degrees in the ET, not to open up a can of worms whereby 
we need to have multiple commas per prime.  One apiece should do.


>> ... So the question now becomes:  Are we left with any good 
reason for 
>> basing the JI notation on 311 instead of 217?
> 

> From your point of view, I would say that you are better off with 
217-ET.


This amounts, then, to a 19-limit-unique-&-consistent, polyphonic-
readable sagittal notation with non-unique capability up to the 35-
odd limit.  That sounds like something that fulfills (and in some 
ways exceeds) our original objective (as I understood it).


> However I do not wish to base a JI (rational) notation on _any_ 
temperament
> that has errors larger than 0.5 c. For me, 217-ET and 311-ET were 
merely a
> way of looking for schismas that might be notationally usable (less 
than
> 0.5 c), and of checking that things were working sensibly, and it 
was nice
> to actually be able to notate those ETs themselves. But I'm taking 
Johnny
> Reinhard at his word when he says (or implies) that nothing less 
than
> 1200-ET is good enough as an ET-based JI notation.
> 
> What is the first ET above 1200 that is 35-limit consistent? 35-
limit unique?


Graham suggests 1600-ET in his message #3947.  It looks like a good 
choice, inasmuch as:

1) It is 37-limit consistent;

2) The largest error for any 37-limit consonance is ~0.36441 cents 
(for 19:25);

3) It conflates all three of my schismas: 4095:4096, 3519:3520, and 
20735:20736 (but not the 31-schisma that I tried, 59024:59049, which 
was also unusable in 311);

4) And I strongly suspect that it will be found to be at least 37-
limit unique, inasmuch as the largest superparticular ratios that are 
not unique are 54:55 and 55:56 (both 42deg1600).


> But I am more interested in a 31-limit (or 35-limit) temperament 
(not equal
> or linear or even planar, but maybe 6D or 7D or 8D) with as many
> sub-half-cent schismas as possible, vanishing. Hence the challenge 
which
> no-one's taken up yet, except in part.


Well, good luck on that one!


> I don't share your obsession with packing all the necessary 
information
> into a single symbol. Some folks may well be willing to notate 
their JI
> piece by mapping to the nearest degree of 217-ET, and they may well 
be glad
> of the possibility to do it with only one accidental per note, but 
that
> won't be everyone's cup of tea. Given that it may involve errors of 

up to 3

> cents, it must be a conscious decision, not something forced upon 

us by the

> notation.


You're talking about two separate issues here:

1) One altering symbol per note:

Since we seem to have concluded that the sagittal JI notation is 
going with 217, we can now determine what each ET notation is going 
to look like.  This means that I can now go ahead and present my Sims 
vs. sagittal comparison on the main tuning list without having to 
issue a caveat that the 72-ET sagittal notation might not be the 
final version.  In that comparison I will show an instance in which 
double symbols (Sims or otherwise) could present some confusion.  I 
will also make the point that, once they are learned, single symbols 
can be read more quickly than double ones (particularly in chords), 
since there is less to read.

2) Mapping to 217-ET:

There is a question that needs to be asked: are we notating JI or are 
we notating 217-ET?  I understood that we were notating JI (mapped 
onto 217 for convenience in understanding some of the size 
relationships among the various ratios), which makes discussion about 
3-cent errors a bit irrelevant.

Now one may also want to make use of the 217 mapping as a convenience 
in conceptualizing a way of arriving at the approximate pitches 
represented by those ratios (which are in turn represented by symbols 
that correlate with a 217 mapping), and without any fine-tuning (by 
ear) you would be entitled to contemplate 3-cent errors.  (Come to 
think of it, anyone coming within 3 cents -- less than the Miracle 
tuning minimax deviation -- is doing pretty well by almost anybody's 
standard.)  But this is more of a matter of how the composer is going 
to treat the notation:

1) Either sticking with a specific set of ratios (in which case the 
ratios and symbols could, for reference, be listed in a table 
alongside each other, with cents values, if that helps), in which 
case the 217 mapping (and error thereof) would have little or no 
relevance;

2) Or else freely employing whatever intervals are permitted by the 
notation, with little regard to keeping track of ratios, in which 
case it could very well turn into (at best) a 217-ET performance.

Putting this another way:  If I write a piece for 13-limit JI using 
just 12 tones per octave and map the tones (consistently) into 12, 
specifying the ratios that I want for each position, would you be 
entitled to claim that I would be getting errors close to 50 cents 
for some of the tones if I used the 12-ET notation?

I believe the problem is more of a matter of how the composer and 
performer understand the notation than with the notation itself.

--George

Message: 4514 - Contents - Hide Contents

Date: Fri, 05 Apr 2002 14:17:58

Subject: Fwd: Re: deriving classical major/minor scales

From: Carl Lumma

Gene, don't know if you saw this on the main list, or for
that matter, if it showed up.  Am re-posting here.


>Date: Sun, 31 Mar 2002 17:24:45 -0800
>To: tuning@xxxxxxxxxxx.xxx
>From: Carl Lumma <carl@xxxxx.xxx>
>Subject: Re: deriving classical major/minor scales
>

>> The periodicity block business, in suitable generality, ends
>> up saying that scales tend to be convex, which seems reasonable.
>
>Gene,
>

>Will this suitable generality be covered in your paper?  Or
>would you care to explain it on these lists?
>
>-Carl

Message: 4515 - Contents - Hide Contents

Date: Sat, 06 Apr 2002 21:13:08

Subject: Re: Blocks and convexity

From: genewardsmith

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


> i'd like to better understand the structure of the balls of various 
> diameters, given the tenney norm, and extensions to octave-equivalent 
> lattices . . . can you help?


You keep bringing up the Tenney norm in this context; it isn't priviledged. Fokker blocks have paralleopiped balls and are of
L-infinity type; the Tenney norm is of L1 type and would have
cross
polytope (generalized octahedron) type balls. Other norms are
possible, and given a convex set you can cook up the corresponding
norm, and vice-versa.

Let's look at an example. Using the sxymmetrical lattice norm on
octave classes in the 7-limit, so that

||3^a 5^b 7^c|| = sqrt(a^2+b^2+c^2+ab+ac+bc)

we find that the hexany 1-15/14-5/4-10/7-3/2-12/7 is contained in a
spherical (L2) ball centered at [1/2, 1/3, -1/3]. Is it a block? We
set up the five equations in four unknowns [a,b,c,d] which define the
val, if there is one, such that the nth degree is given value n;
solving this system of equations gives the unique solution 
{a=6,b=10,c=14,d=17}, so that the val in question exists and is h6;
hence the hexany is a block. The L2 norm is not the only norm which
would define the hexany as a block; one can always use the convec hull
to generate a ball--in this case we would get an L1 norm in that way.

Message: 4516 - Contents - Hide Contents

Date: Sat, 06 Apr 2002 22:04:18

Subject: Re: Blocks and convexity

From: genewardsmith

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


> we find that the hexany 1-15/14-5/4-10/7-3/2-12/7 is contained in a spherical (L2) ball centered at [1/2, 1/3, -1/3]. 


Sorry, this should be [1/2,1/2,-1/2].

Message: 4517 - Contents - Hide Contents

Date: Sat, 06 Apr 2002 23:15:51

Subject: Re: New Member-Question about ratios

From: dkeenanuqnetau

--- In tuning-math@y..., Alexandros Papadopoulos <alexmoog@o...> 
wrote:

> Hello , I am Alex Papadopoulos from Greece.
> 
> My mathematics knowledge is pretty low , so I wonder if this list is 
> accepting amateur-ish mathematics questions related to tunings.
> If not please tell me , so I will not bother you.

Hi Alex,


We accept any mathematics questions related to tunings.


> If yes , my question is:
> 
> In Helmholtz's book "the sensations of tone" in page 15 he states 
,that 
> since we know the ratios of the C ,E and G notes , we can find
> the rest of the C scale notes , by associating this C triad with the 
F 
> and G triad.
> So I thought that since G to B is a Major 3rd , I can add 5/4(Major 
3rd) 
> to G which has a ratio of 3/2.
> The B ratio is 15/8 , which is not what I find.
> 
> I must have something very wrong in my mind!


[If you are reading this on Yahoo's web interface you will need to 
choose Message Index and then Expand Messages to see the following 
correctly formatted. Or simply choose Reply.]

What is it that you find?

15/8 is the correct ratio for B in a _justly_intoned_ C major scale 
whose lattice follows. 

  A---E---B
 / \ / \ / \ 
F---C---G---D

  5/3-5/4-15/8
  / \ / \ / \ 
4/3-1/1-3/2-9/8

Of course this is not the same as its tuning in equal temperament (the 
most common tuning today). There all the notes are mapped to the 
nearest power of the twelfth root of two. In the case of B this is 
2^(11/12) ~= 1.888, whereas 15/8 is 1.875.

-- Dave Keenan

Message: 4518 - Contents - Hide Contents

Date: Sat, 06 Apr 2002 02:23:31

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:

>> In all the lower ETs where either an
>> 11-comma or a 13-comma is a semisharp or semiflat, its apotome 
> complement

>> is the same number of steps, so there isn't any need for both.
> 

> That is correct if you are indicating "semisharp" and "semiflat" to 
> be exactly half an apotome in those ET's (or is this what you really 
> meant to say?).


I was being sloppy. I didn't check the facts before I wrote. Sorry.


> In 39, 46, and 53-ET this is not the case, and you 
> either need to have both 11-commas for the two different kinds of 
> semisharps/semiflats, or else have the 11 comma for one and the 13 
> comma for the other (although these would then be faux complements).


Thanks. I agree that 46 and 53 are the first where this is a problem. 
39-ET is 1,3,13-inconsistent and one can choose a number of steps for 
1024:1053 that is the apotome complement of 32:33, but this is not 
possible for 46 and 53. So I guess you're assuming 26:27 as _the_ 13 
comma. Or are you assumng 704:729 as _the_ 11 comma?


> I think that I have probably misunderstood what you were trying to 
> say and that my going on about this is sort of pointless, 
considering 
> that we are getting the notation for the secondary 11 and 13 commas 
> as a freebie with the 4095:4096 schisma.  I think that your point is 
> that each pair of 11 and 13 commas consists of apotome-complements, 
> so that the larger of each could be notated as an apotome minus the 
> smaller; hence we really have only one comma for each prime in the 
> semantics of the notation; but for convenience in handling the 
> semisharps & semiflats we have symbols that make it appear as if 
> there were two.


That wasn't my point. As I say, I was just careless. But it's a good 
point. Thanks for making it.


> Anyway, in pointing out the need for being able to respell ratios at 
> will, my primary objective was to indicate why we needed to notate 
> all of the degrees in the ET, not to open up a can of worms whereby 
> we need to have multiple commas per prime.  One apiece should do.


Ok. Good. But hopefully we can have less than one apiece, with the 
other sub-half-cent schismas you found.


>>> ... So the question now becomes:  Are we left with any good 
> reason for 

>>> basing the JI notation on 311 instead of 217?
>> 

>> From your point of view, I would say that you are better off with 
> 217-ET.
> 

> This amounts, then, to a 19-limit-unique-&-consistent, polyphonic-
> readable sagittal notation with non-unique capability up to the 35-
> odd limit.  That sounds like something that fulfills (and in some 
> ways exceeds) our original objective (as I understood it).


Sure. But I don't understand what 217-ET or 311-ET have to do with it. 
217-ET just happens to be the highest ET that you can notate with it. 
The definitions of the symbols must be based on the commas, not the 
degrees of 217-ET.

What do you mean by "polyphonic-readable"? As opposed to what?


>> However I do not wish to base a JI (rational) notation on _any_ 
> temperament

>> that has errors larger than 0.5 c. For me, 217-ET and 311-ET were 
> merely a

>> way of looking for schismas that might be notationally usable 
(less 
> than

>> 0.5 c), and of checking that things were working sensibly, and it 
> was nice

>> to actually be able to notate those ETs themselves. But I'm taking 
> Johnny

>> Reinhard at his word when he says (or implies) that nothing less 
> than

>> 1200-ET is good enough as an ET-based JI notation.
>> 
>> What is the first ET above 1200 that is 35-limit consistent? 35-
> limit unique?
> 

> Graham suggests 1600-ET in his message #3947.  It looks like a good 
> choice, inasmuch as:
> 
> 1) It is 37-limit consistent;
> 
> 2) The largest error for any 37-limit consonance is ~0.36441 cents 
> (for 19:25);
> 
> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and 
> 20735:20736 (but not the 31-schisma that I tried, 59024:59049, which 
> was also unusable in 311);
> 
> 4) And I strongly suspect that it will be found to be at least 37-
> limit unique, inasmuch as the largest superparticular ratios that 
are 
> not unique are 54:55 and 55:56 (both 42deg1600).
> 

>> But I am more interested in a 31-limit (or 35-limit) temperament 
> (not equal

>> or linear or even planar, but maybe 6D or 7D or 8D) with as many
>> sub-half-cent schismas as possible, vanishing. Hence the challenge 
> which

>> no-one's taken up yet, except in part.
> 

> Well, good luck on that one!


You have in effect found one of these, with your 3 schismas. I'd just 
like to be sure that there isn't a whole 'nother larger set of 
31-limit schismas that give an even better compression of the number 
of flags, without exceeding the 0.5 cent.

But I guess I'll just assume 1600-ET. In other words, any schisma that 
vanishes in 1600-ET is acceptable to be built into the notation, but 
certainly not all the schismas that vanish in 217-ET (or 311-ET or 
388-ET).

So in that sense only, I could say that the notation is "based on" 
1600-ET, but there is certainly no desire to notate every degree of 
1600-ET using a single accidental per note, or even a single 
accidental in addition to a sharp or flat. In fact there is no desire 
to notate 1600-ET at all, and it is fine that 217-ET is the highest ET 
that can be so notated.


>> I don't share your obsession with packing all the necessary 
> information

>> into a single symbol. Some folks may well be willing to notate 
> their JI

>> piece by mapping to the nearest degree of 217-ET, and they may 
well 
> be glad

>> of the possibility to do it with only one accidental per note, but 
> that

>> won't be everyone's cup of tea. Given that it may involve errors 
of 

> up to 3

>> cents, it must be a conscious decision, not something forced upon 

> us by the
>> notation.
> 

> You're talking about two separate issues here:
> 
> 1) One altering symbol per note:
> 
> Since we seem to have concluded that the sagittal JI notation is 
> going with 217, we can now determine what each ET notation is going 
> to look like.  This means that I can now go ahead and present my 
Sims 
> vs. sagittal comparison on the main tuning list without having to 
> issue a caveat that the 72-ET sagittal notation might not be the 
> final version.


Agreed. I'm pretty sure it's only the concave flags whose meaning 
could still change. 


>  In that comparison I will show an instance in which 
> double symbols (Sims or otherwise) could present some confusion.  I 
> will also make the point that, once they are learned, single symbols 
> can be read more quickly than double ones (particularly in chords), 
> since there is less to read.
> 
> 2) Mapping to 217-ET:
> 
> There is a question that needs to be asked: are we notating JI or 
are 
> we notating 217-ET?  I understood that we were notating JI (mapped 
> onto 217 for convenience in understanding some of the size 
> relationships among the various ratios), which makes discussion 
about 
> 3-cent errors a bit irrelevant.


OK. Good. So I wish you'd stop talking about it being "based on" or 
"going with" 217-ET, or any other ET with larger than 0.5 cent errors.


> Now one may also want to make use of the 217 mapping as a 
convenience 
> in conceptualizing a way of arriving at the approximate pitches 
> represented by those ratios (which are in turn represented by 
symbols 
> that correlate with a 217 mapping), and without any fine-tuning (by 
> ear) you would be entitled to contemplate 3-cent errors.  (Come to 
> think of it, anyone coming within 3 cents -- less than the Miracle 
> tuning minimax deviation -- is doing pretty well by almost anybody's 
> standard.)  But this is more of a matter of how the composer is 
going 
> to treat the notation:
> 
> 1) Either sticking with a specific set of ratios (in which case the 
> ratios and symbols could, for reference, be listed in a table 
> alongside each other, with cents values, if that helps), in which 
> case the 217 mapping (and error thereof) would have little or no 
> relevance;


Yes. This is more fundamental to the notation.


> 2) Or else freely employing whatever intervals are permitted by the 
> notation, with little regard to keeping track of ratios, in which 
> case it could very well turn into (at best) a 217-ET performance.


Yes, so 217-ET is just one ET that could be used in this way. The 
notation is not based on it. It just happens to be the highest one 
that is fully notatable with single symbols.


> Putting this another way:  If I write a piece for 13-limit JI using 
> just 12 tones per octave and map the tones (consistently) into 12, 
> specifying the ratios that I want for each position, would you be 
> entitled to claim that I would be getting errors close to 50 cents 
> for some of the tones if I used the 12-ET notation?

No.

> I believe the problem is more of a matter of how the composer and 
> performer understand the notation than with the notation itself.


Yes. I'm glad we understand each other. Showing how the notation maps 
to 217-ET is no different from showing how it maps to 22-ET. The 
_definition_ of the symbols is in terms of the commas and schismas.

Message: 4519 - Contents - Hide Contents

Date: Sat, 06 Apr 2002 07:28:18

Subject: Blocks and convexity

From: genewardsmith

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


>> Will this suitable generality be covered in your paper?  Or
>> would you care to explain it on these lists?


The metric on a vector space defined from a norm

d(x,y) = ||x-y||

has
the property (not shared by metrics in general) of convexity--the open
and closed balls defined by the metric are convex. On the other hand,
given a convex closed set S containing the origin in its interior, we
can define a norm such that S is a unit ball. If we define a block as
a set of octave-equivalence lattice points which are epimorphic and
convex, from the above we can see this ends up equivalent to a
definition in terms of normed vector spaces, where a block is an
epimorphic set of lattice points of minimal diameter. The Fokker block
is simply then the special case of a linearlly transformed L-infinity
norm. I don't know if any of this needs to be expounded.

Message: 4521 - Contents - Hide Contents

Date: Sun, 07 Apr 2002 01:49:56

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

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

> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, and 
> 20735:20736 (but not the 31-schisma that I tried, 59024:59049, which 
> was also unusable in 311);


59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test anyway, 
being 0.73 c, however it might tempt me if it could be combined with 
other suitable schismas, as per my challenge.

Message: 4522 - Contents - Hide Contents

Date: Sun, 07 Apr 2002 21:12:52

Subject: Re: A common notation for JI and ETs

From: David C Keenan

In case anyone cares; there's a 37-schisma that vanishes in 1600-ET (and
388-ET) that lets us make a symbol for a 37-comma, with no new flags.

The 37 comma is 999:1024 (3^3*37 : 2^10) 42.8 cents, so a 1:37 from C is a
lowered Eb.

The schisma is 570236193:570425344 (3^12*29*37 : 2^25*17) 0.57 cents, which
means the 37 arrowhead combines the 17-flag and the 29-flag. Only trouble
is, these are presently both left flags, concave and convex.

I believe we have a rule that says that at each prime limit, the symbols
should be as simple as possible and that no higher prime should be allowed
to "reach down" and cause us to change the way we do the flags for a lower
limit. 

However, the only reason so far, to make the 17 flag a left flag is to make
a symbol for the large 23 comma by combining the 17 flag with the 11-5
flag. It isn't essential to have a symbol for the large 23 comma, so the 17
flag and (small) 23 flag (both concave) could swap sides. 8 steps of 217-ET
could still be notated as either the 25 symbol (sL+sL) or 7 flag + 23 flag
(vL+xR).

1:37 from C is very close to halfway between D and Eb, a pythagorean limma
apart, so there is a good argument for needing the large 37 comma of
(36:37) 47.4 cents as well. There is a 1600-ET schisma that gives us the
large 37 comma without new flags, provided we're willing to combine 3 of
them, 19 flag + 23 flag + 7 flag (cL+vL+xR), which actually do combine ok.
The schisma is 6992:6993 (2^4*19*23 : 3^3*7*37) 0.25 cents.

This is probably all pretty silly, catering for 37, and we should probably
just forget it and keep the large 23 comma symbol, but here's a pass at a
full set of 37-limit symbols anyway.

[If you're reading this on the yahoogroups website you will need to 
choose Reply or Message Index, Expand Messages, to see the following
symbols rendered correctly.]

5-comma 80:81

 /|
/ |
  |    \ /
  |
  |

7-comma 63:64
   _
  | \
  | |
  |    L P
  |
  |

11-comma 32:33

 /|\
/ | \
  |    v ^
  |
  |

13-comma 1024:1053
   _
 /| \
/ | |
  |    { }    flags based on vanishing of schisma 4095:4096
  |
  |

17-comma 2176:2187

  |
  |\_
  |    j f
  |
  |

19-comma 512:513
 _
(_)
  |
  |    o *
  |
  |

23-comma 729:736

  |
_/|
  |    w m
  |
  |

29-comma 256:261
 _
/ |
| |
  |    q d    flag based on vanishing of schisma 20735:20736
  |
  |

31-comma 243:248
 _
(_)\
  | \
  |    y h    flags based on vanishing of schisma 253935:253952
  |
  |

37-comma 999:1024
 _
/ |
| |\_
  |           flags based on vanishing of schisma 570236193:570425344
  |
  |

We also have optional symbols for larger 11, 13 and 37 commas.

11'-comma 704:729
 _ _
/ | \
| | |
  |    [ ]    flags based on vanishing of schisma 5103:5104
  |
  |

13'-comma 26:27
 _
/ |\
| | \
  |    ; |    flags based on vanishing of schisma 20735:20736
  |
  |

37'-comma 36:37
 _ _
(_) \
_/| |
  |          flags based on vanishing of schisma 6992:6993
  |
  |

If we really still wanted a symbol for the large 23 comma I guess we could
still combine the 17 and 11-5 flags like this:

23'-comma 16384:16767

  |
  |\_
  | \    W M    flags based on vanishing of schisma 3519:3520
  |
  |

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

Message: 4523 - Contents - Hide Contents

Date: Sun, 07 Apr 2002 03:20:13

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

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

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

>> 3) It conflates all three of my schismas: 4095:4096, 3519:3520, 
and 
>> 20735:20736 (but not the 31-schisma that I tried, 59024:59049, 
which 
>> was also unusable in 311);
> 

> 59024:59049 (2^4*7*17*31:3^10) doesn't pass the Reinhard test 
anyway, 
> being 0.73 c, however it might tempt me if it could be combined with 
> other suitable schismas, as per my challenge.


We can forget about that 31-schisma. What's wrong with 253935:253952
(3^5*5*11*19 : 2^13*31) 0.12 cents. Consistent with 311-ET 388-ET 
1600-ET, but not 217-ET.

31 comma = (11 comma - 5 comma) + 19 comma

Since (11 comma - 5 comma) is a single flag and 19 comma is a single 
flag (or blob) then this 31 comma can be represented by a pair of 
flags. The fact that it doesn't work in 217-ET doesn't matter because 
the notation is not "based on" 217-ET and the 31 comma is not needed 
in order to notate 217-ET.

Message: 4524 - Contents - Hide Contents

Date: Sun, 7 Apr 2002 01:28:36

Subject: Re: Blocks and convexity

From: Pierre Lamothe

Perhaps what follows is sufficiently correlated to your interest in this thread.
 
I failed until date to generalize something that seemed very simple in 3D for I believed that the
parrallelopiped was the elementary shape of periodicity blocks in which thefundamental domain
around the unison (corresponding to a set of steps) was decomposable. I waswrong.

I found the generalization way using vectorial drawing. I don't want to useneither words nor math
formalism to explain that now. I leave you to discover that by intuition with a simple 4D example.

I used animated images. The image fd4.swf shows the decomposition with 4 periodicity blocks of
the 4D fundamental domain associated with 4 steps. For comparison only, theimage fd3.swf shows
the decomposition with 3 periodicity blocks of the 3D fundamental domain having only 3 steps.

(I already showed similar hexagonal shapes decomposed with 3 parallelograms.) 

I guess that Gene will see immediately the link with norms and unit balls.
 
I could show later how matrices are nicely correlated with that. Particularly, I have now a new tool,
the TS-matrix, which is a variant of the S-matrix. (S means srutis - TS means steps and srutis)
permitting to look anew, geometrically, at the correlation between steps and unison vectors.


Don't forget to press "enter" or use "play" to start the animation.
 
http://www.aei.ca/~plamothe/tuning/fd4.swf - Type Ok * [with cont.]  (Wayb.)
http://www.aei.ca/~plamothe/tuning/fd3.swf - Type Ok * [with cont.]  (Wayb.)

Pierre


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