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Message: 8125

Date: Wed, 12 Nov 2003 21:58:19

Subject: Re: Definition of microtemperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...>
> > wrote:
> > > I would have said "would always be less than about 3 cents" 
> or "... 
> > > less than 3.5 cents" in order to include Miracle.  Or don't you 
> > > consider that a microtemperament, and if not, then what should 
we 
> > > call it?
> > 
> > I've always considered miracle to be a microtemperament at the 7-
> limit
> > (2.4 c) but not at the 9 or 11 limits (3.3 c).
> 
> I don't follow this.  The error of 4:5 in Miracle (with minimax 
> generator) is ~3.323c.

we were focusing on the 72-equal incarnation of miracle.

> If you're going to use 
> anything on the order of half the error of meantone as your cutoff, 
> then you should also extend this to half the error of 8:9 in 
meantone 
> for a 9 limit.

why? there's no analogy there. 1/4-comma meantone was not used for 
music where 8:9 is used as a consonance.

> The beating harmonics in a tempered 8:9 are much more difficult to 
> hear than for 2:3,

shouldn't that consideration lower the weight of 8:9 in the 
calculation, compensating this next point?

> hence that interval is more difficult to play in 
> tune with flexible-pitch instruments, hence the actual error for 
that 
> interval in a live performance is likely to be greater.

> > It's all pretty arbitrary, but I think we need to draw such a line
> > somewhere.
> 
> Yes.

noooooooooooooooooo! :)


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Message: 8126

Date: Wed, 12 Nov 2003 00:44:40

Subject: Re: Eponyms

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > so how about "mapping" instead of "val" with the implication 
> > (preferably stated along with n) that we are talking about ET.
> 
> I don't see the point. What about optimal et?

we're talking about how to optimally map primes to a given et, right?


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Message: 8127

Date: Wed, 12 Nov 2003 22:03:08

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> > Here's what you've given us so far...
> 
> I've given way, way way more than that. I can't force anyone to 
read 
> it.
> 
> > ...It appears that in the case of the "standard 3-val for the 5-
> limit",
> > n=3.  Is that why you called it a 3-val? 
> 
> > Where did 3 come from?
> 
> A division of the octave into three parts, or in other words, a 
> mapping of 2 to 3.

excuse me, but i think the answer to carl's question is "the complete 
5-limit otonal chord has *3* notes". right?


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Message: 8128

Date: Wed, 12 Nov 2003 01:17:48

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> > I don't see the point. What about optimal et?
> 
> we're talking about how to optimally map primes to a given et, 
right?

I don't count it as an et unless it has a mapping; anyway "optimal 
val" is shorter and sweeter.


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Message: 8129

Date: Wed, 12 Nov 2003 22:03:26

Subject: Re: 7-limit optimal et vals

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > what is the optimality criterion?
> 
> Minimax error in the 7-limit.

any differences if you use rms?


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Message: 8130

Date: Wed, 12 Nov 2003 01:23:42

Subject: Re: Eponyms

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > I don't see the point. What about optimal et?
> > 
> > we're talking about how to optimally map primes to a given et, 
> right?
> 
> I don't count it as an et unless it has a mapping; anyway "optimal 
> val" is shorter and sweeter.

ok.


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Message: 8131

Date: Wed, 12 Nov 2003 22:04:44

Subject: Re: Definition of microtemperament

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> 
> wrote:
> 
> > I have a particular 13-limit temperament in mind...
> 
> Which is?

... more useful than Miracle, in my opinion.

If you want to try it out in Scala, then be advised that, like 
Miracle, it comes in several sizes (of 17, 29, and 41 tones).  The 41-
tone size is a superset of all of the others, and the 17-tone version 
comes in several keys (none of which are a subset of the 29-tone 
version).  "Set notation 41e" or "sa41" misses one of the tones, so 
you should "set notation sahtt" to get all of them (and if you want 
to see conventional sharps and flats, then "set sagittal mixed".

Here are the file contents for a few of these:


! secor17htt1.scl
!
George Secor's 17-tone high-tolerance temperament subset #1 on C (5/4 
& 7/4 exact)
 17
!
30.08878
140.19633
207.15739
265.24719
347.35372
5/4
496.42131
554.51111
612.60091
703.57869
733.66747
843.77502
882.73506
7/4
1050.93241
1089.89245
2/1


! secor17htt2.scl
!
George Secor's 17-tone high-tolerance temperament subset #2 on Ev 
(5/4 & 7/4 exact)
 17
!
30.08878
116.17960
207.15739
237.24617
347.35372
5/4
496.42131
554.51111
612.60091
703.57869
733.66747
843.77502
910.73608
7/4
1050.93241
1089.89245
2/1


! secor17htt3.scl
!
George Secor's 17-tone high-tolerance temperament subset #3 on G (5/4 
& 7/4 exact)
 17
!
58.08980
116.17960
207.15739
237.24617
347.35372
5/4
472.40458
554.51111
593.47114
703.57869
733.66747
843.77502
910.73608
7/4
1050.93241
1089.89245
2/1


! secor29htt.scl
!
George Secor's 29-tone 13-limit high-tolerance temperament (5/4 & 7/4 
exact)
 29
!
58.08980
97.04984
140.19633
179.15637
207.15739
265.24719
296.73557
347.35372
5/4
414.31478
472.40458
496.42131
554.51111
593.47114
633.37025
679.56197
703.57869
761.66849
800.62853
843.77502
882.73506
910.73608
7/4
992.84261
1050.93241
1089.89245
1117.89347
1175.98327
2/1


! secor41htt.scl
!
George Secor's 13-limit high-tolerance temperament superset (5/4 & 
7/4 exact)
 41
!
30.08878
58.08980
97.04984
116.17960
140.19633
179.15637
207.15739
237.24617
265.24719
296.73557
323.33699
347.35372
5/4
414.31478
444.40355
472.40458
496.42131
526.51008
554.51111
593.47114
612.60091
636.61764
679.56197
703.57869
733.66747
761.66849
800.62853
819.75830
843.77502
882.73506
910.73608
940.82486
7/4
992.84261
1026.91568
1050.93241
1089.89245
1117.89347
1147.98225
1175.98327
2/1


Have fun analyzing these, and let me know if you think they sound 
like JI!

--George


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Message: 8132

Date: Wed, 12 Nov 2003 02:05:19

Subject: 7-limit optimal et vals

From: Gene Ward Smith

Here is a list of all of them which are not already standard vals, for
n from 1 to 1 100. No torsion issues arise. In some cases other vals
scored nearly as well.

<1 2 3 3]

<3 5 7 9]

<8 13 19 23]

<11 18 26 31]

<13 20 30 36]

<14 22 32 39]

<17 27 40 48]

<20 31 46 56]

<23 36 53 64]

<28 44 65 78]

<30 47 69 84]

<33 52 76 92]

<34 54 79 96]

<39 62 91 110]

<48 76 112 135]

<52 83 121 146]

<54 85 125 151]

<64 102 149 180]

<65 103 151 183]

<66 104 153 185]

<67 106 155 188]

<71 112 165 199]

<85 135 198 239]

<86 136 199 241]

<96 152 223 269]

<98 155 227 275]

<100 159 232 281]


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Message: 8133

Date: Wed, 12 Nov 2003 14:16:14

Subject: Re: Vals?

From: Carl Lumma

>> Here's what you've given us so far...
>
>I've given way, way way more than that. I can't force anyone to
>read it.

I've read everything you've ever posted to this list, much of it
more than once, and much of it I've saved locally.

-Carl


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Message: 8134

Date: Wed, 12 Nov 2003 03:18:51

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 
> > Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping
> > in exactly the same way? 
> 
> Go ahead and do so, however a val is not necessarily an et-mapping of
> any kind.

In this message
Yahoo groups: /tuning-math/message/7528 * [with cont.] 
you said the two could be identified (except for a question about
finiteness)?

Here is a possible Monz dictionary definition of an ET-prime-mapping
(improved from the one I already gave in the above message, that you
seem to have missed):
----------------------------------------------------------------------
ET-prime-mapping

A list of the (whole) numbers of steps of some equal temperament (ET)
(not necessarily octave based) used to approximate each prime number
(considered as a frequency ratio). An "n-limit" ET-prime-mapping
(where n is a whole number) only lists numbers of steps for primes no
greater than n.

The "standard" prime mapping for an ET is the one that gives the best
approximation for each prime, but note that this is not guaranteed to
give the best approximation for all ratios, and other mappings may be
more useful in some cases.

To find the number of steps approximating some ratio in some ET,
express the ratio as a prime-exponent-vector and multiply its elements
by the corresponding elements in the chosen ET-prime-mapping and sum
the products. This is called the dot product or inner product or
scalar product of the two vectors.

For example the standard 7-limit ET-prime-mapping for 12-EDO is [12 19
28 34]. These numbers can be calculated for any EDO as
Round(N*ln(p)/ln(2)) where N is the number of divisons per octave and
p is the prime number.

To find how many steps of 12-EDO approximate a 7/5 frequency ratio,
first express the ratio as a prime-exponent-vector.
7/5 = 2^0 * 3^0 * 5^-1 * 7^1 = [0 0 -1 1]
now find its dot product with the prime-mapping
[0 0 -1 1].[12 19 28 34] = 0*12 + 0*19 + -1*28 + 1*34 = 6
So 7/5 is approximated by 6 steps, a tritone.
----------------------------------------------------------------------
 
> And then a "p-limit ET-mapping" would be a
> > restricted one.
> > 
> > So they're exactly the same!!!!
> 
> I havn't seen your definition.

I suspect you just didn't recognise it as such, because it was in
plain English.

> > So why have we been calling them "vals" all this time? 
> 
> Because there wasn't a good word for "finitely generated homomorphic
> mapping from Q+ to Z" or "Z-linear combination of padic valuations"
> already in existence.
> 
> A mathworld
> > search on the term finds nothing. Did you invent the term? 
> 
> You bet. We needed a term for it, and there wasn't one.

_You_ might have needed a term for "finitely generated homomorphic
mapping from Q+ to Z" or "Z-linear combination of padic valuations",
but I don't think anyone else on this list did.

> Is it
> > merely an obscure synonym for "homomorphism", or "group homomorphism"?
> > The fact that it _is_ a group homomorphism is far from being its most
> > important characteristic as far as microtonality is concerned. The
> > fact that it maps ratios to steps of ETs is of far more interest.
> 
> A val *does not necessarily* map ratios to steps of an ET, but it *is*
> always a homomorphism. 

So what would you estimate is the percentage of the vals posted to
tuning-math that could not be read as mapping ratios to steps of an
ET. Please give some examples of these and explain what they _do_ mean
in tuning terms.

> If you insist, you could replace the term with
> "finitely generated homomorphic mapping from Q+ to Z", I suppose, but
> I imagine "a p-limit homomorphic mapping to the integers" or something
> like that would suit you better.

Those certainly don't suit me. By all means use the word "val" to
stand for this abstract mathematical category. But this is the
_tuning_ math list. We want names that indicate their meaning as
applied to _tuning_.

For example, in the application area of electrical theory we use
vectors to represent the magnitude and phase of sinusoidal voltages
and currents. But we don't just call them all vectors. We want names
that tell us what they _mean_. We call them "voltage phasors" and
"current phasors".

I assume that the val, strictly speaking, is the operation of taking
the dot product with a vector of step numbers, not the actual vector
of step numbers itself. That's ok. The term "mapping" is used
similarly ambiguously. It doesn't usually cause any misunderstanding.

> It most certainly would not have served me better. You do what you
> like, but please don't expect me to follow your lead.

So are you saying that you don't really care if only two or three
people on this list understand how the things you write about apply to
tuning?


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Message: 8135

Date: Wed, 12 Nov 2003 22:36:40

Subject: Re: Definition of microtemperament

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> wrote:
> > > > I would have said "would always be less than about 3 cents" 
or "... 
> > > > less than 3.5 cents" in order to include Miracle.  Or don't 
you 
> > > > consider that a microtemperament, and if not, then what 
should we 
> > > > call it?
> > > 
> > > I've always considered miracle to be a microtemperament at the 
7-limit
> > > (2.4 c) but not at the 9 or 11 limits (3.3 c).
> > 
> > I don't follow this.  The error of 4:5 in Miracle (with minimax 
> > generator) is ~3.323c.
> 
> we were focusing on the 72-equal incarnation of miracle.

This then *requires* moving the ~2.8c boundary to 3.0 cents, not just 
putting it there because it looks less arbitrary.  If Dave said there 
was a gap between ~2.8 and ~3.1 cents, and if it takes a special (non-
11-limit-optimal) version of Miracle to make the 7-limit cut, then I 
would say that the boundary is in the wrong place, and he should have 
left it at ~2.8 cents and allowed Miracle (in all of its 
incarnations) to fall into a different category (for which I'm still 
awaiting a name).

> > If you're going to use 
> > anything on the order of half the error of meantone as your 
cutoff, 
> > then you should also extend this to half the error of 8:9 in 
meantone 
> > for a 9 limit.
> 
> why? there's no analogy there. 1/4-comma meantone was not used for 
> music where 8:9 is used as a consonance.

No, it wasn't historically, but that doesn't mean that someone 
*couldn't* use an extended meantone temperament for 9-limit harmony.  
I will admit that there are better options for that purpose, but then 
that's the whole point -- this gives us a baseline against which we 
can compare those options.

> > The beating harmonics in a tempered 8:9 are much more difficult 
to 
> > hear than for 2:3,
> 
> shouldn't that consideration lower the weight of 8:9 in the 
> calculation, compensating this next point?
> 
> > hence that interval is more difficult to play in 
> > tune with flexible-pitch instruments, hence the actual error for 
that 
> > interval in a live performance is likely to be greater.

That was precisely my point.  More leeway can be allowed for some of 
these less-consonant consonances.

> > > It's all pretty arbitrary, but I think we need to draw such a 
line
> > > somewhere.
> > 
> > Yes.
> 
> noooooooooooooooooo! :)

Do you mean noooooooooooooooooo categories or noooooooooooooooooot 
arbitrary, or booooooooooooooooooth?  :-)

--George


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Message: 8136

Date: Wed, 12 Nov 2003 03:51:39

Subject: Re: Definition of microtemperament

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...>
wrote:
> I would have said "would always be less than about 3 cents" or "... 
> less than 3.5 cents" in order to include Miracle.  Or don't you 
> consider that a microtemperament, and if not, then what should we 
> call it?

I've always considered miracle to be a microtemperament at the 7-limit
(2.4 c) but not at the 9 or 11 limits (3.3 c).

I originally said "less than half the 5-limit error of 1/4-comma
meantone", i.e. less than 2.7 c.

I let it creep up already so a couple of temperaments with 2.8 c
errors could scrape in, and I went up to 3 for this definition just
because it seemed silly to be as precise as 2.8 c, so I definitely
wouldn't want it to creep _past_ 3 cents.

Gene would like the limit set at 1 c, although I haven't read why.
However I believe this definition caters for that, by allowing the ear
to arbitrate, and mentionaing the context dependence. In some contexts
a temperament with an error between 1 and 3 cents may not be a
microtemperament.

All I'm saying with the 3 cent thing is that there is no context in
which an error _greater_ than 3 cents would be considered a
microtemperament, ear or no ear.

It's all pretty arbitrary, but I think we need to draw such a line
somewhere.


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Message: 8137

Date: Wed, 12 Nov 2003 22:43:16

Subject: Re: Definition of microtemperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:

> > why? there's no analogy there. 1/4-comma meantone was not used 
for 
> > music where 8:9 is used as a consonance.
> 
> No, it wasn't historically, but that doesn't mean that someone 
> *couldn't* use an extended meantone temperament for 9-limit 
harmony.  

right, but then they'd be more likely to use something like 1/5-comma 
or 1/6-comma meantone.

> > > > It's all pretty arbitrary, but I think we need to draw such a 
> line
> > > > somewhere.
> > > 
> > > Yes.
> > 
> > noooooooooooooooooo! :)
> 
> Do you mean noooooooooooooooooo categories or noooooooooooooooooot 
> arbitrary, or booooooooooooooooooth?  :-)
> 
> --George

there's no way you could *hear* the point at which the line is drawn 
(nor should it necessarily be drawn according to minimax), so i'd 
prefer to use 'microtemperament' in a looser way -- if anyone cares 
to check on the 'microtemperedness' of a particular temperament, the 
exact numbers should be readily available. maybe 2.8 to 3.1 can be 
considered a 'gray zone', where *context* will determine whether the 
effect is one of microtemperament or not.


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Message: 8138

Date: Wed, 12 Nov 2003 05:26:52

Subject: Re: Vals?

From: Gene Ward Smith

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

> In this message
> Yahoo groups: /tuning-math/message/7528 * [with cont.] 
> you said the two could be identified (except for a question about
> finiteness)?

I understood you to mean any kind of prime mapping, whether it could
be called an et or not.

> Here is a possible Monz dictionary definition of an ET-prime-mapping
> (improved from the one I already gave in the above message, that you
> seem to have missed):
> ----------------------------------------------------------------------
> ET-prime-mapping
> 
> A list of the (whole) numbers of steps of some equal temperament (ET)
> (not necessarily octave based) used to approximate each prime number
> (considered as a frequency ratio). An "n-limit" ET-prime-mapping
> (where n is a whole number) only lists numbers of steps for primes no
> greater than n.

OK, but this isn't equivalent to val.


> So what would you estimate is the percentage of the vals posted to
> tuning-math that could not be read as mapping ratios to steps of an
> ET. 

Who knows? Maybe 50-50.

Please give some examples of these and explain what they _do_ mean
> in tuning terms.

<0 1 4 10| should be familiar from the meantone temperament. 

> For example, in the application area of electrical theory we use
> vectors to represent the magnitude and phase of sinusoidal voltages
> and currents.

Unless you use complex numbers, of course.

> I assume that the val, strictly speaking, is the operation of taking
> the dot product with a vector of step numbers, not the actual vector
> of step numbers itself. 

It's the mapping that generates, yes.

> So are you saying that you don't really care if only two or three
> people on this list understand how the things you write about apply to
> tuning?

I've explained what a val is numerous times. I can't insist you pay
attention to everything I say; these days you and George tend to lose
me, after all, which is fair enough.


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Message: 8139

Date: Wed, 12 Nov 2003 05:34:35

Subject: Gaps between Zeta function zeros and ets

From: Gene Ward Smith

I took a list of the first 10000 zeros of the Riemann zeta function,
rescaled so that they read in terms of equal divisions of the octave.
The top zero is then 1089.695, so we are looking at ets from 1 to 1089. 
The gaps between these zeros are on average 1/log2(n) in the vicinity
of n, so I took the gaps between successive zeros and multiplied by
log2 of the average of the two successive zeros. As I expected, for
the largest gaps these are centered around equal divisions of the
octave. The first fifty of these, in order, are as follows:

954, 1012, 311, 764, 422, 581, 270, 814, 742, 935, 718, 494, 882, 
1041, 525, 908, 571, 1065, 342, 836, 653, 851, 1084, 460, 354, 1075,
692, 1029, 684, 566, 624, 472, 711, 400, 863, 639, 988, 243, 997, 441,
643, 597, 373, 1046, 795, 449, 224, 513, 328, 966

Some tendency for the ets in question to be the kind approximating a
lot of primes (311) instead of a relatively few (411) seems to be in
evidence, but it isn't really clear what is going on.


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Message: 8140

Date: Wed, 12 Nov 2003 05:36:57

Subject: Re: Definition of microtemperament

From: Gene Ward Smith

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

> Gene would like the limit set at 1 c, although I haven't read why.

"Micro" to me means small enough that the error hardly matters.

> It's all pretty arbitrary, but I think we need to draw such a line
> somewhere.

There's always my magnitude scale, with lines differing by a factor of
two.


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Message: 8141

Date: Wed, 12 Nov 2003 07:40:16

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 
> > In this message
> > Yahoo groups: /tuning-math/message/7528 * [with cont.] 
> > you said the two could be identified (except for a question about
> > finiteness)?
> 
> I understood you to mean any kind of prime mapping, whether it could
> be called an et or not.

I wrote:
> > How is a val different from an ET-mapping? i.e. a list of the
> > numbers of steps approximating each prime in some ET.

So I'm rather surprised you didn't know I was talking about ETs?

But that's good, because now it looks like "val", as applied to
tuning, can be replaced by "prime-mapping", which is even simpler than
"ET-prime-mapping".

> <0 1 4 10| should be familiar from the meantone temperament. 

This looks like one row of the 7-limit prime-mapping for the meantone
linear temperament using a fifth as the generator, in particular the
row giving the mapping to fifth generators. Isn't it somewhat
incomplete without the other row that gives the mapping to octave
generators (periods)?

Why do we want to give the same name to something which in one case is
the complete mapping for an ET (a 1D temperament), and in the other
case only a part of the mapping for an LT (a 2D temperament)?

But assuming that there's a good reason, I'd simply call them
"prime-mappings" or "1D-prime mappings".

But I'd prefer to be more specific and call one an ET-mapping and the
other an LT-generator-mapping. I'd call the missing row the
LT-period-mapping. Together the LT-generator-mapping and the
LT-period-mapping make up the LT-mapping. The word "prime" can be
inserted before the word "mapping" whenever this is not clear from the
context.

> > So are you saying that you don't really care if only two or three
> > people on this list understand how the things you write about
apply > > to tuning?
> 
> I've explained what a val is numerous times. I can't insist you pay
> attention to everything I say; these days you and George tend to lose
> me, after all, which is fair enough.

If all your explanations were similar to this one
Definitions of tuning terms: val, (c) 2001 by ... * [with cont.]  (Wayb.)
I'm afraid it wouldn't have made any difference if I'd read them all.
But I can't hold it against anyone that they are not good at
explaining things.

I'm pretty sure I did read an early one, and said to myself, "I have
no idea what that means. I guess I need a bit more mathematical
background. I'll look into it later."

But it appears that little or no explanation would have been necessary
if you had simply called them prime mappings.

So is a val, as applied to tuning theory, simply a prime-mapping, or a
1D-prime-mapping?


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Message: 8142

Date: Thu, 13 Nov 2003 02:16:33

Subject: Re: Vals?

From: Dave Keenan

Actually, if you need a shorter term than "prime-mapping", it seems
like "mapping" would do. What other kinds of mappings do we use in
tuning-math?


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Message: 8144

Date: Thu, 13 Nov 2003 22:13:53

Subject: Re: Definition of microtemperament

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" 
<gdsecor@y...> 
> wrote:
> 
> > I have never thought of any tuning as anything less than 11-limit 
if 
> > it contains 11-limit intervals.
> 
> You mean 11-prime limit? Miracle is of course a temperament, and it 
> can be derived easily and naturally from the 7-limit lattice. You 
> simply temper our 225:224 and 2401:2400 (or 225:224 and 1029:1024).
> 
> You mean 11-odd limit? Well, meantone contained excellent 
> approximations to ratios of 7, but practically no one considered 
them 
> consonant historically. So i see no problem with considering 
miracle 
> a 7-limit temperament if someone uses it in a style where ratios of 
> 11 or their approximation are used as dissonances. In fact miracle 
is 
> one of the very best choices for a 7-limit temperament.
> 
> > Even from an historical perspective 
> > miracle has always been an 11-limit tuning.
> 
> Not true at all!! I think the 11-limit was an unexpected bonus, 
that 
> mostly Dave Keenan was keeping track of at the time.^

Okay, you win!

--George


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Message: 8145

Date: Thu, 13 Nov 2003 17:23:09

Subject: Re: Definition of microtemperament

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > Maximum error in cents
> > 
> > Magnitude 0: 0.25-0.5
> > Magnitude 1: 0.5-1.0
> > Magnitude 2: 1.0-2.0
> > Magnitude 3: 2.0-4.0
> > 
> > -log2(4 * error) is the formula.
> 
> Well this looks like a good system to me. But I'm afraid it will 
make
> too many people unhappy to try to align the definition of
> "microtemperament" with it.
> 
> > Miracle is a third magnitude temperament by this.
> 
> Hopefully this will make George happy.

Maybe and maybe not, because it doesn't take into account whether 
there might be better boundaries that would be suggested by the gaps 
indicated in a tabulated list of microtemperaments, which could also 
suggest that a factor of magnitude other than 2 might be more 
appropriate.  Then again, after doing this you might find that 
whatever boundaries and factor you chose would still be rather 
arbitrary and there would not be any reason to use anything different 
from what Gene gave above.

I'm not trying to give you guys a hard time about this, but just 
offering some constructive criticism.  And considering what I have to 
say below, it might be best to forget about boundaries altogether and 
express the magnitude to one decimal place (as with stellar magnitude 
in astronomy).

> Paul,
> 
> I wasn't considering the 72-EDO incarnation. When I gave 2.4 c and 
3.3
> c I was considering the minimax optima for 7-limit and 11-limit
> respectively (9-limit is the same as 11-limit).
> 
> George,
> 
> I think most of us find it quite natural to speak of "7-limit 
miracle"
> even though some may consider the "true" miracle temperament to be
> 11-limit. 

I have never thought of any tuning as anything less than 11-limit if 
it contains 11-limit intervals.  Even from an historical perspective 
miracle has always been an 11-limit tuning.

> We have enough trouble agreeing on names as it is. I'd hate
> to have to find different names for different lower-limit subsets 
(or
> whatever the right term is) of the same mapping.

My primary concern in engaging you in this discussion was to 
determine whether I should be calling my high-tolerance temperament 
(which I described here:)
Yahoo groups: /tuning-math/message/7574 * [with cont.] 
a microtemperament, or whether some other category-name would be more 
appropriate.  (See my comments on your definition, below.) 

> The most important thing is for Monz to get rid of the current false
> definition of microtemperament. 
> 
> Monz,
> 
> You could just change every ocurrence of "microtemperament" in it to
> "planar temperament" and change its name to "planartemp.htm" and
> there's your definition for planar temperament.
> 
> I'm hoping that the following will make everyone happy. I've changed
> "would always be less than 3 cents" to "would typically be less than
> 2.8 cents". I've also changed "JI scale" to "JI tuning" throughout.
> 
> I'm sure this definition could be improved, but can we just get
> something in place of that bad definition in Monz's dictionary?
> 
> --------------------------------------------------------------------
--
> Microtemperament
> 
> A microtemperament is a temperament where the consonances sound 
justly
> intoned to most listeners in ordinary musical use. The allowed 
errors
> in the approximated ratios are therefore somewhat context-dependent
> but would typically be less than 2.8 cents.

This language is very good in that it acknowledges that the term 
relates to one's (subjective) *perception* of a tuning rather than a 
hard-and-fast (objective) error limit, so that classification as a 
microtemperament for a particular tuning with approximations 
approaching or exceeding 2.8-cents is left open for debate.  (And 
given the subjectivity of this definition, I would agree with Aaron's 
recommendation that this be changed to 3 cents.)  The orders of 
magnitude that Gene gave could be useful for predicting the 
*probability* of a particular tuning being perceived as a 
microtemperament.

> A JI tuning might be microtempered to increase the number of 
available
> consonances or to regularise the scale for some purpose such as
> allowing more full-width continuous frets on a stringed instrument.
> Microtemperament may also be used to introduce deliberate slight
> mistunings to avoid phase-locking when a JI tuning is implemented 
on an
> electronic instrument.
> 
> A microtemperament may be equal, linear, planar or of any dimension
> less than that of the JI tuning being approximated.
> --------------------------------------------------------------------
--

Excellent!

--George


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Message: 8146

Date: Thu, 13 Nov 2003 23:24:11

Subject: Re: Definition of microtemperament

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > i changed it a couple of days ago when you proposed the
> > earlier version of the part i snipped here.  now it's as
> > per your latest definition:
> > 
> > Definitions of tuning terms: microtemperament,... * [with cont.]  (Wayb.)
> 
> Could you change this back to "always less than three cents"? 2.8 
> cents seems an absurd line to draw, and "usually" means it isn't even 
> a line.

This whole thread is hilarious. :-) I haven't had such a good laugh
from tuning-math in a long time, but I admit I've been taking myself
too seriously lately.

Back when it said "always", and it had 3 cents (because I thought, as
a lot of people apparently do) that 2 significant digits of cents was
a bit too precise, George said "if you mean 2.8 cents then say 2.8
cents or you'll just encourage further creepage" or words to that
effect. I thought he had a good point. But now that we've changed it
to "typically" (which I understand most people support) then even
George agrees 2.8 is too finicky. So "typically less than 3 cents" is
ok with me.

Paul, I assume you were merely arguing that "typically less than 2.8
cents" is as about as good as any other nearby number as a
just-noticeble-difference, and you wouldn't really mind if the
microtemperament definition was changed to "typically less than 3 cents".

Gene, I wanted an actual cutoff too - an "always" rather than a
"typically" - but it looks like we're outvoted. Or to put it another
way, I can live with a "typically" for the sake of consensus.


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Message: 8147

Date: Thu, 13 Nov 2003 05:41:58

Subject: Re: 7-limit optimal et vals

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
> > wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> > > wrote:
> > > > what is the optimality criterion?
> > > 
> > > Minimax error in the 7-limit.
> > 
> > any differences if you use rms?
> 
> and are you allowing the octaves to be tempered? i.e. Do they apply
> strictly to EDOs or to ET's generally?

EDOs only, but I didn't know you were calling non=integer mappings ets.


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Message: 8148

Date: Thu, 13 Nov 2003 18:51:27

Subject: Re: Vals?

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > > Where did 3 come from?
> > 
> > A division of the octave into three parts, or in other words, a 
> > mapping of 2 to 3.
> 
> excuse me, but i think the answer to carl's question is "the 
complete 
> 5-limit otonal chord has *3* notes". right?

Maybe I misunderstood the question.


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Message: 8149

Date: Thu, 13 Nov 2003 23:30:01

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > I think we have quite complementary skills. You come up with the the
> > math tools and methods and I may _eventually_ be able to understand
> > them enough to put them into terms that others on this list can more
> > easily understand. But I don't think you should worry too much if my
> > explanations or recasting of terminology misses some of the more
> > subtle points as far as the pure mathematician is concerned, at 
> least
> > on a first pass.
> 
> Sounds reasonable, but I don't think you should worry to much if I 
> want to make precise mathematical definitions for things, or make the 
> definitions the way they are for reasons not immediately apparent to 
> you.

It's a deal. :-)


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