Yahoo Tuning Groups Ultimate Backup tuning-math Definition of microtemperament

Monz,

I just noticed that your definition of microtemperament is quite
wrong. Sorry I didn't pick this up sooner.

http://sonic-arts.org/dict/microtemp.htm

What you've actually defined here is "planar temperament".

Here's my definition of "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 always be less than 3 cents.

A JI scale 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 scale is implemented on an
electronic instrument.

A microtemperament may be equal, linear, planar or of any dimension
less than that of the JI scale being approximated.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗George D. Secor <gdsecor@yahoo.com>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > Monz,
> >
> > I just noticed that your definition of microtemperament is quite
> > wrong. Sorry I didn't pick this up sooner.
> >
> > http://sonic-arts.org/dict/microtemp.htm
> >
> > What you've actually defined here is "planar temperament".
> >
> > Here's my definition of "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 always be less than 3 cents.
>
> 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?
>
> --George

dave has said that miracle is a microtemperament in the 7-limit but
not in the 9-limit or 11-limit.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "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.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "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.

Exactly what I said in my definition. It's nice that we agree on
something. :-)

> > 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.

I can't find this by searching the archive. I tried all kinds of
things. Please explain or give a URL.

The term microtemperament has a long history of referring to
temperaments with errors less than half that of 1/4-comma meantone. So
the magnitude scale could go down by factors of two using the syntonic
comma as the basic unit. But it would be better to "carve nature at
its joints" if possible. That is, look at the minimax errors of large
numbers of the best temperaments and see where the gaps are, near to
these binary fractions of the comma. There seems to be one such gap
between about 2.8 c and 3.1 c for linear temperaments up to 15-limit.

Here's a definition from Feb 2000.
/tuning/topicId_8589.html#8589

And here's the first use (Feb 1999) of the earlier term "wafso-just"
that "microtemperament" replaced.
/tuning/topicId_1012.html#1012

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "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.

Anyway, what should I call a temperament in which all of the
consonances are within 3.33 cents and all of the 7-limit consonances
are within half of that amount, which I believe sounds like "JI
brought to life" (i.e., with the "stagnation" or "deadness"
eliminated with a small amount of tempering)? I have a particular 13-
limit temperament in mind, and I could send you a recording of a live
1975 performance on the Scalatron in that temperament, so you could
judge whether it sounds like JI or not. Or I could make an mp3 file
from the recording available, if anyone else wants to hear it.

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

I think you're comparing apples and oranges here. The max error of 9-
limit 1/4-comma meantone is twice that of 5-limit meantone, simply
because 8:9 will have twice the error of 2:3. 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. Otherwise, when you evaluate 9-limit (or higher)
temperaments against your standard, you are really using a 1/4-the-
error-of meantone standard.

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

> 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.

Then I guess that (without knowing exactly how much 3 cents has been
exceeded) you'll have to listen to my recording and then decide. But
whatever you decide, I think that you would definitely agree that the
accuracy is a magnitude or two better than meantone (and even
noticeably better than 72-ET).

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

Yes.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> ...
> The term microtemperament has a long history of referring to
> temperaments with errors less than half that of 1/4-comma meantone.
So
> the magnitude scale could go down by factors of two using the
syntonic
> comma as the basic unit. But it would be better to "carve nature at
> its joints" if possible. That is, look at the minimax errors of
large
> numbers of the best temperaments and see where the gaps are, near to
> these binary fractions of the comma. There seems to be one such gap
> between about 2.8 c and 3.1 c for linear temperaments up to 15-
limit.
>
> Here's a definition from Feb 2000.
> /tuning/topicId_8589.html#8589

If you want to draw a boundary at ~2.8 cents, then draw it there, not
at 3.0 cents, because you're inviting others to want the boundary to
creep upward.

BTW, what is the next (larger-error) gap? I am inclined to think
that a factor of two is too large for establishing magnitudes of
error.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "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! :)

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "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

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > > --- In tuning-math@yahoogroups.com, "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

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "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.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "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.

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. 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.

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 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.
----------------------------------------------------------------------

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> 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?

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:

http://sonic-arts.org/dict/microtemp.htm

and i already had in the Dictionary a definition of
"planar temperament" from Graham:

http://sonic-arts.org/dict/planartemp.htm

since i've already changed the definition of "microtemperament"
according to your revisions, can you post the old definition
which i should revamp into "planar temperament"?

... the old HTML is still in the source code of
"microtemperament" in a comment, in case you need it.

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗gooseplex <cfaah@eiu.edu>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "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:)
/tuning-math/message/7574
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

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
>
> > > http://sonic-arts.org/dict/microtemp.htm
>
>
> As a general definition, '2.8 cents' seems a bit finicky to me,
> especially with the qualifier 'depending on context'.

aaron, 'depending on context' was a qualifier inserted so that we
wouldn't seem so finicky as to insist on an exact figure of 2.8
cents! how is it that you are seeing this so differenly?

> Based on
> psychoacoustic experiments for the JND, you can safely change
> '2.8 cents' to 3 cents.

there are of course many JNDs, all of which are a function of
absolute frequency. which do you have in mind? of course johnny
reinhard on the tuning list can distinguish *melodic* changes of 1
cent (and most of us can distinguish *harmonic* changes of 1 cent
from a simple ratio), and he claims to be able to accurately
conceive, in his aural imagination, any interval to an accuracy of 1
cent.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "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.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I don't see how you can draw the line at 3 cents, though. You
*can*
> >hear the difference between that and JI pretty clearly.
>
> It depends on the instrument.
>
> -Carl

And it depends on the interval(s) involved. Error would be most
evident with the most consonant intervals, particularly fourths and
fifths. The language in question says "typically less than 3 cents",
which leaves quite a bit open to interpretation. I would expect that
the most consonant intervals would be quite a bit "less than 3
cents", while more dissonant "consonances" might even be acceptable
if allowed to stray a little over 3 cents. As soon as you are
considering 9-limit consonances you have automatically restricted the
allowable error for 2:3 and 3:4 to half of what will be allowed for
8:9 (assuming that we are dealing with regular temperaments).

--George

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "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

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "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:
> >
> > http://sonic-arts.org/dict/microtemp.htm
>
> 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.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

🔗monz <monz@attglobal.net>

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> You mean 11-odd limit? Well, meantone contained
> excellent approximations to ratios of 7, but practically
> no one considered them consonant historically.

that's not true, and you know it: meantone gave good
approximations to a 4:5:7 triad in its "augmented-6th"
chord, which was used a *lot* in the "common-practice"
era.

true, no-one at the time analyzed these chords as
consonant 4:5:7 chords, but in meantone, that's what
they were, and they were perfectly acceptable in
both theory and practice.

i wrote something to the main tuning list fairly recently
(perhaps this past spring?) that went into pretty good
detail about augmented-6th chords and meantone versions
of them. with your prodigious memory, you'll probably
recall them ...

-monz

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > You mean 11-odd limit? Well, meantone contained
> > excellent approximations to ratios of 7, but practically
> > no one considered them consonant historically.
>
>
> that's not true, and you know it: meantone gave good
> approximations to a 4:5:7 triad in its "augmented-6th"
> chord, which was used a *lot* in the "common-practice"
> era.

but not as a consonance -- so what i was saying is true.

> true, no-one at the time analyzed these chords as
> consonant 4:5:7 chords,

but huygens *did* find these ratios in augmented sixth chord.

> but in meantone, that's what
> they were, and they were perfectly acceptable in
> both theory and practice.

i didn't say they were unacceptable -- plenty of not-so-easy-to-ratio-
analyse sonorities were acceptable as dissonances as well -- just not
considered consonant, that is, it was not used as a chord to resolve
a dissonant chord to, but rather it was used as a chord that would
resolve *to* a consonant chord.

🔗monz <monz@attglobal.net>

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi paul,
> >
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > You mean 11-odd limit? Well, meantone contained
> > > excellent approximations to ratios of 7, but practically
> > > no one considered them consonant historically.
> >
> >
> > that's not true, and you know it: meantone gave good
> > approximations to a 4:5:7 triad in its "augmented-6th"
> > chord, which was used a *lot* in the "common-practice"
> > era.
>
> but not as a consonance -- so what i was saying is true.

OK, you're right.

> > true, no-one at the time analyzed these chords as
> > consonant 4:5:7 chords,
>
> but huygens *did* find these ratios in augmented sixth chord.

oops ... i *knew* you'd catch me on that one!

> > but in meantone, that's what
> > they were, and they were perfectly acceptable in
> > both theory and practice.
>
> i didn't say they were unacceptable -- plenty of
> not-so-easy-to-ratio-analyse sonorities were acceptable
> as dissonances as well -- just not considered consonant,
> that is, it was not used as a chord to resolve a dissonant
> chord to, but rather it was used as a chord that would
> resolve *to* a consonant chord.

OK, now i understand perfectly what you were saying.

the augmented-6th chord was always used as a "dissonant"
chord which had to resolve, as you say.

-monz

🔗Paul Erlich <perlich@aya.yale.edu>

but the appearance of this chord so easily in the meantone series
certainly makes one wonder what direction western music might have
progressed in had the movement for closure at 12 notes not won out.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > > hi paul,
> > >
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > > wrote:
> > >
> > > > You mean 11-odd limit? Well, meantone contained
> > > > excellent approximations to ratios of 7, but practically
> > > > no one considered them consonant historically.
> > >
> > >
> > > that's not true, and you know it: meantone gave good
> > > approximations to a 4:5:7 triad in its "augmented-6th"
> > > chord, which was used a *lot* in the "common-practice"
> > > era.
> >
> > but not as a consonance -- so what i was saying is true.
>
>
> OK, you're right.
>
>
> > > true, no-one at the time analyzed these chords as
> > > consonant 4:5:7 chords,
> >
> > but huygens *did* find these ratios in augmented sixth chord.
>
>
>
> oops ... i *knew* you'd catch me on that one!
>
>
> > > but in meantone, that's what
> > > they were, and they were perfectly acceptable in
> > > both theory and practice.
> >
> > i didn't say they were unacceptable -- plenty of
> > not-so-easy-to-ratio-analyse sonorities were acceptable
> > as dissonances as well -- just not considered consonant,
> > that is, it was not used as a chord to resolve a dissonant
> > chord to, but rather it was used as a chord that would
> > resolve *to* a consonant chord.
>
>
> OK, now i understand perfectly what you were saying.
>
> the augmented-6th chord was always used as a "dissonant"
> chord which had to resolve, as you say.
>
>
>
> -monz