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.

--- 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.

It's been discussed before; I liked error less than a cent as the

cutoff.

--- 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

--- 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.

--- 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.

--- 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.

> 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.

--- 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

--- 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

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

wrote:

> > 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.

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.

Miracle is a third magnitude temperament by this.

> The term microtemperament has a long history of referring to

> temperaments with errors less than half that of 1/4-comma meantone.

I didn't know that. I do recall wafso-just.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>

wrote:

I have a particular 13-

> limit temperament in mind...

Which is?

--- 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! :)

--- 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

--- 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

--- 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.

--- 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 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.

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.

----------------------------------------------------------------------

--- 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

--- 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

Thanks Monz. I just needed to hit the reeload button.

I hope it's ok with everyone else.

> and i already had in the Dictionary a definition of

> "planar temperament" from Graham:

>

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

It's fine. You can leave it how it is, as far as I'm concerned.

> > 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'. Based on

psychoacoustic experiments for the JND, you can safely change

'2.8 cents' to 3 cents. I suggest making the change to 3 cents.

This is also the error threshold in the default tuning for the tonal

plexus.

http://www.members.aol.com/pitchcolor/instruments/plexus.html

Aaron

--- 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

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

wrote:

> 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.

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.

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

wrote:

> 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.

I would change "typically be less than 2.8 cents" to "at minimum be

less than three cents". I also wonder, if we adopt this definition,

what we would call something like ennealimmal or octoid.

>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

>I would change "typically be less than 2.8 cents" to "at minimum be

>less than three cents". I also wonder, if we adopt this definition,

>what we would call something like ennealimmal or octoid.

Howabout "typically less than 2 cents" (the error of a 12-equal

fifth)? Since this is meant to be applied by musicians, .1 cent

resolution should not be offerred.

-Carl

--- 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.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>

wrote:

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).

Works for me.

> I have never thought of any tuning as anything less than 11-limit

if

> it contains 11-limit intervals.

Miracle still makes a lot of sense if you don't use the 11-limit. Is

there a rule we need the 7 and 11 limits in meantone, simply because

they are there?

Even from an historical perspective

> miracle has always been an 11-limit tuning.

You don't need to use 11-limit harmony in miracle.

--- 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.

--- 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.

--- 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

--- 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

--- 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.

--- 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. 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.

Which, in fact, I have already done, which was why I was using 175

and not 72.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>

wrote:

I know you've already conceded, but I thought I'd mention this anyway.

By sacrificing the ratios of 9 and the ratios of 11 one can find a

different optimum generator that treats the 7-limit better (whether

you favour minimax or rms). This is at least one thing we mean by

7-limit miracle.

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

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

wrote:

> 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".

of course i wouldn't mind.

--- 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.

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

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