A template for temperaments

Here's a sort of form to fill in for these. One question is whether
error/complexity/badness should be added, and if so, using what measures.

family name:
period:
generator:

5-limit

name:
comma:
mapping:
poptimal generator:
TOP period: mapping:
TM basis:
MOS:

7-limit

name:
wedgie:
mapping:
poptimal generator(s):
TOP period: generator:
TM basis:
MOS:

Higher limits are the same

please don't stop! keep them coming.

-monz

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

> Here's a sort of form to fill in for these. One question is whether
> error/complexity/badness should be added, and if so, using what
measures.
>
> family name:
> period:
> generator:
>
> 5-limit
>
> name:
> comma:
> mapping:
> poptimal generator:
> TOP period: mapping:
> TM basis:
> MOS:
>
> 7-limit
>
> name:
> wedgie:
> mapping:
> poptimal generator(s):
> TOP period: generator:
> TM basis:
> MOS:
>
> Higher limits are the same

monz wrote:

> please don't stop! keep them coming.

Well, I don't have convenient archives of things like poptimal generators and TM basis, so I'll let someone else fill those in. I'll start with a couple of 5-limit temperaments.

Names: mavila, pelogic
(from original name "meta-mavila", given by Erv Wilson)
Comma: 135;128 [-7, 3, 1>
Mapping: [<1, 2, 1|, <0, -1, 3|]
Generator/period: fourth, octave
Generator (TOP): 521.52 cents, period = 1206.55 cents
Generator (Erv Wilson's golden horagram #30): 527.15 cents
ET's: 7, 9, 16, 23
Typical MOS: 7, 9, 16

Names: kleismic, hanson
Comma: 15625;15552 [-6, -5, 6>
Mapping: [<1, 0, 1|, <0, 6, 5|]
Generator/period: minor third, octave
Generator (TOP): 317.34 cents, period = 1200.29 cents
Generator (Erv Wilson's golden horagram #9): 317.17 cents
ET's: 15, 19, 23, 30, 34, 38, 49, 53, 57, 68, 72, ......
Typical MOS: 15, 19, 34, 53, 87

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Names: kleismic, hanson
> Comma: 15625;15552 [-6, -5, 6>
> Mapping: [<1, 0, 1|, <0, 6, 5|]
> Generator/period: minor third, octave
> Generator (TOP): 317.34 cents, period = 1200.29 cents
> Generator (Erv Wilson's golden horagram #9): 317.17 cents
> ET's: 15, 19, 23, 30, 34, 38, 49, 53, 57, 68, 72, ......
> Typical MOS: 15, 19, 34, 53, 87

Doesn't "minor thirds" (as in "the minor thirds temperament" or "the
chain of minor thirds") count as a name for this? It's what we
called it for a long time, before "kleismic" gained currency.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Names: mavila, pelogic
> (from original name "meta-mavila", given by Erv Wilson)
> Comma: 135;128 [-7, 3, 1>
> Mapping: [<1, 2, 1|, <0, -1, 3|]
> Generator/period: fourth, octave
> Generator (TOP): 521.52 cents, period = 1206.55 cents
> Generator (Erv Wilson's golden horagram #30): 527.15 cents
> ET's: 7, 9, 16, 23
> Typical MOS: 7, 9, 16
>
> Names: kleismic, hanson
> Comma: 15625;15552 [-6, -5, 6>
> Mapping: [<1, 0, 1|, <0, 6, 5|]
> Generator/period: minor third, octave
> Generator (TOP): 317.34 cents, period = 1200.29 cents
> Generator (Erv Wilson's golden horagram #9): 317.17 cents
> ET's: 15, 19, 23, 30, 34, 38, 49, 53, 57, 68, 72, ......
> Typical MOS: 15, 19, 34, 53, 87

There is no need for separate "ET's:" and "Typical MOS:" fields.
They ought to be the same list of numbers, i.e. the denominators of
the convergents and semiconvergents of the gen/period ratio
multiplied by the number of periods per octave.

We probably should start with the smallest greater than 4. There
isn't much point in including any beyond a point where acceptable
changes in the generator would lead to different numbers.

It's conventional to place those which are only semi-convergent in
parenthesis. Then we know they are not very good ETs (in relative
terms) for the temperament and as MOS are improper (L/s > 2).

e.g. Using
http://dkeenan.com/Music/MyhillCalculator.xls
(54kB Excel spreadsheet)

TOP 5-limit Pelogic
Typical ET/MOS/DE cardinalities: (5) 7 (9 16 23)

TOP 5-limit Kleismic
Typical ET/MOS/DE cardinalities: (7 11) 15 19 34 53

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

> We probably should start with the smallest greater than 4. There
> isn't much point in including any beyond a point where acceptable
> changes in the generator would lead to different numbers.

"Acceptable changes" is vague; I've been considering what covers the
whole poptimal range. Moreover, five is too small a number for the
more complex temperaments. What's the point of five notes of schismic?

> It's conventional to place those which are only semi-convergent in
> parenthesis. Then we know they are not very good ETs (in relative
> terms) for the temperament and as MOS are improper (L/s > 2).

This assumes we have a single, precisely defined generator in mind,
but probably we don't.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> > We probably should start with the smallest greater than 4.
There
> > isn't much point in including any beyond a point where
acceptable
> > changes in the generator would lead to different numbers.
>
> "Acceptable changes" is vague; I've been considering what covers
the
> whole poptimal range.

I know it's vague. It would be nice to nail it down. But p-optimal
doesn't include TOP does it? It's equally-weighted for all p isn't
it? And in any case, isn't it possible that in some cases the p-
optimals might all be very close to each other, and yet we might
find much greater variations in generator-size acceptable?

I was thinking more of something like the range of generator sizes
over which the TOP error is no more than than double its minimum
value.

> Moreover, five is too small a number for the
> more complex temperaments. What's the point of five notes of
schismic?

A possible set of nominals.

> > It's conventional to place those which are only semi-convergent
in
> > parenthesis. Then we know they are not very good ETs (in
relative
> > terms) for the temperament and as MOS are improper (L/s > 2).
>
> This assumes we have a single, precisely defined generator in mind,
> but probably we don't.

Well yeah. I assumed everyone was gaga about TOP.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> monz wrote:
>
> > please don't stop! keep them coming.
>
> Well, I don't have convenient archives of things like poptimal
> generators and TM basis, so I'll let someone else fill those in. I'll
> start with a couple of 5-limit temperaments.
>
> Names: mavila, pelogic
> (from original name "meta-mavila", given by Erv Wilson)
> Comma: 135;128 [-7, 3, 1>
> Mapping: [<1, 2, 1|, <0, -1, 3|]
> Generator/period: fourth, octave
> Generator (Erv Wilson's golden horagram #30): 527.15 cents
Poptimal generator: 10/23
TOP period: 1206.548 generator: 521.520
> ET's: 7, 9, 16, 23
> Typical MOS: 7, 9, 16
>
> Names: kleismic, hanson
> Comma: 15625;15552 [-6, -5, 6>
> Mapping: [<1, 0, 1|, <0, 6, 5|]
> Generator/period: minor third, octave
Poptimal generator: 65/246
> Generator (TOP): 317.34 cents, period = 1200.29 cents
> Generator (Erv Wilson's golden horagram #9): 317.17 cents
> ET's: 15, 19, 23, 30, 34, 38, 49, 53, 57, 68, 72, ......
MOS: 7, 11, 15, 19, 34, 53, 87, 140

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

> I know it's vague. It would be nice to nail it down. But p-optimal
> doesn't include TOP does it?

Nope; it's strictly a pure-octaves system. However pure octaves make
sorting this stuff out less of a headache.

It's equally-weighted for all p isn't
> it?

It weights all the odd limit consonances of the appropriate odd limit
the same, in 9 or above that means 3 is weighted more.

And in any case, isn't it possible that in some cases the p-
> optimals might all be very close to each other, and yet we might
> find much greater variations in generator-size acceptable?

Oh, absolutely. The poptimal tuning could consist of a single number
not obtainable in any equal temperament. Normally it does not.

> I was thinking more of something like the range of generator sizes
> over which the TOP error is no more than than double its minimum
> value.

There's a thought; should we try that?

> > > It's conventional to place those which are only semi-convergent
> in
> > > parenthesis. Then we know they are not very good ETs (in
> relative
> > > terms) for the temperament and as MOS are improper (L/s > 2).
> >
> > This assumes we have a single, precisely defined generator in mind,
> > but probably we don't.
>
> Well yeah. I assumed everyone was gaga about TOP.

Should we assume TOP tuning for the generators? That would give us MOS
right off the bat, from the convergents and semiconvergents of the
ratio of the generators.

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

> It's conventional to place those which are only semi-convergent in
> parenthesis. Then we know they are not very good ETs (in relative
> terms) for the temperament and as MOS are improper (L/s > 2).

Hi Dave,

Why is this so? Is 81 not good ET for meantone? Why is that so and
how does it follow from the impropriety of 81-tone MOS?

Kalle

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > I was thinking more of something like the range of generator
sizes
> > over which the TOP error is no more than than double its minimum
> > value.
>
> There's a thought; should we try that?

It would be interesting, if it's easy for you to crunch the numbers.
This might also allow us to resolve questions about which two ETs we
should choose for the two-ET method of naming. Namely those two,
among the semiconvergents, that fall nearest to these double-error
generators. It would be a good test of this double-error business to
see if it agrees with people's existing knowledge and intuitions
about extreme ETs for temperaments.

We could start by looking at those listed in Paul's draft paper.

> Should we assume TOP tuning for the generators? That would give us
MOS
> right off the bat, from the convergents and semiconvergents of the
> ratio of the generators.

It's OK by me, but I'd want some kind of check, like a comparison of
TOP generators with ordinary unweighted minimax (for 7-prime-limit
TOP you'd need to compare both 7-odd-limit and 9-odd-limit minimax)
giving both TOP error and minimax error for both TOP and minimax
optimum generators, so we can see when they differ wildly.

The only such I've found so far is 5-limit dominant, but I've only
looked at the 5-limit ones in Paul's paper so far.

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > It's conventional to place those which are only semi-convergent
in
> > parenthesis. Then we know they are not very good ETs (in
relative
> > terms) for the temperament and as MOS are improper (L/s > 2).
>
> Hi Dave,
>
>
> Why is this so? Is 81 not good ET for meantone? Why is that so and
> how does it follow from the impropriety of 81-tone MOS?
>
> Kalle

Hi Kalle,

Notice I said "in relative terms". What this means is, if you take
the error (e.g. in cents) and divide it by the size of the ET step
(e.g. in cents), you'll find that 81 isn't as good as 31. But in
terms of absolute cents, 81 is better.

How does that follow from impropriety? I'd say instead that both MOS-
impropriety and lower ET relative-accuracy follow from whether the
cardinality relates to the denominator of a convergent or only a
semiconvergent rational approximation of the irrational ratio of
generator/period. Why? This is something I don't really understand,
but have just accepted.

You can examine the calculation of both convergents and
semiconvergents by looking at the formulae in this spreadsheet, but
maybe Gene knows of a good exposition on them, on the web.
http://dkeenan.com/Music/MyhillCalculator.xls

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

> > Why is this so? Is 81 not good ET for meantone? Why is that so
and
> > how does it follow from the impropriety of 81-tone MOS?

> Notice I said "in relative terms". What this means is, if you take
> the error (e.g. in cents) and divide it by the size of the ET step
> (e.g. in cents), you'll find that 81 isn't as good as 31. But in
> terms of absolute cents, 81 is better.

Oh, I have done that a lot! Did you know that 12 rules the 5-limit
meantone ETs if you take the TOP error divided by ET step? 5-limit
minimax divided by ET step gives 19.

> How does that follow from impropriety? I'd say instead that both
MOS-
> impropriety and lower ET relative-accuracy follow from whether the
> cardinality relates to the denominator of a convergent or only a
> semiconvergent rational approximation of the irrational ratio of
> generator/period. Why? This is something I don't really understand,
> but have just accepted.

This is interesting.

> You can examine the calculation of both convergents and
> semiconvergents by looking at the formulae in this spreadsheet, but
> maybe Gene knows of a good exposition on them, on the web.
> http://dkeenan.com/Music/MyhillCalculator.xls

Funny, just yesterday I learned about continued fractions and
convergents. But I'm not really sure what a semiconvergent is. Is it
just a rational approximation that is not in the series of
convergents?

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> Oh, I have done that a lot! Did you know that 12 rules the 5-limit
> meantone ETs if you take the TOP error divided by ET step? 5-limit
> minimax divided by ET step gives 19.

Correction:

...5-limit max error, not minimax...

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> Funny, just yesterday I learned about continued fractions and
> convergents. But I'm not really sure what a semiconvergent is. Is it
> just a rational approximation that is not in the series of
> convergents?

No, it's what you get by filling in the gaps between the convergents
by mediants with the lower convergent. For instance, convergents to
log2(3/2) are 1/2, 3/5, 7/12, 24/41, 31/53... For p1/q1 and p2/q2, the
mediant is (p1+p2)/(q1+q2); for example the mediant of 7/12 and 24/41
is 31/53, so there is no gap to fill in between 41 and 53. On the
other hand, the mediant of 1/2 and 3/5 is 4/7, which is a
semiconvergent; we fill in the gap once since the mediant of 3/5 and
4/7 is 7/12. Between 7/12 and 24/41 we fill in twice, since the
mediant of 7/12 and 3/5 is 10/17, the mediant of 7/12 and 10/17 is
17/29, and the mediant of 7/12 and 17/29 is 24/41. Both 10/17 and
17/29 therefore are semiconvergents; and 17 and 29 define Pythagorean MOS.

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

> No, it's what you get by filling in the gaps between the convergents
> by mediants with the lower convergent. For instance, convergents to
> log2(3/2) are 1/2, 3/5, 7/12, 24/41, 31/53... For p1/q1 and p2/q2,
the
> mediant is (p1+p2)/(q1+q2); for example the mediant of 7/12 and
24/41
> is 31/53, so there is no gap to fill in between 41 and 53. On the
> other hand, the mediant of 1/2 and 3/5 is 4/7, which is a
> semiconvergent; we fill in the gap once since the mediant of 3/5 and
> 4/7 is 7/12. Between 7/12 and 24/41 we fill in twice, since the
> mediant of 7/12 and 3/5 is 10/17, the mediant of 7/12 and 10/17 is
> 17/29, and the mediant of 7/12 and 17/29 is 24/41. Both 10/17 and
> 17/29 therefore are semiconvergents; and 17 and 29 define
Pythagorean MOS.

Aha! Thanks, Gene.