Comments about Fokker's misfit metric

I have been reading Fokker's papers "Les mathematiques et la musique"
(1947) and the more recent "On the expansion of the musician's realm of
harmony" (1967).

In both papers, Fokkers uses a metric to evaluate the misfit of the
tempered scales to the just intonnation scales. He basically uses the
cuadratic mean of the Cent distances between the just intonation and
tempered values corresponding to the basic intervals (the octave, the
fifth, the third, the seveth, and so on).

Using this metric he clearly concludes, in summary, that the 31 equally
tempered is the option to improve the traditionally tempered scale.

Now it occurs to me that the metric used weights too much the significance
of the upper intervals, like the seventh, the eleventh and so for. I would
say that it could make more sense, specially if one is thinking of doing
tonal music, to really create a metric that will weigth more the most
basic intervals. Something like

metric = SQRT ( (0.5*distance_of_the_fifth)**2 +
(0.35*distance_of_the_third)**2 + (0.15*distance_of_the_seventh)**2 )

This would me also more consistent with the energy absorved by each of the
higher harmonics (my guess is that depending on the instrument the seventh
harmonic, i.e the first apperance of the interval 7/4 would be getting
less than 10% of the energy).

If you do use this metric just looking to the three intervals indicated,
the result of Fokker still holds and the 31 equally tempered scale is
still the first that produces a considerable reduction in the metric or
distance value. But my guess is that if the same procedure is used with
more harmonics then the result could vary significantly depending on the
weight factors selected.

At any rate, this seemed a more logical approach.

Comments would be wellcome.

Carlos

Carlos wrote:

> I have been reading Fokker's papers "Les mathematiques et la musique" > (1947) and the more recent "On the expansion of the musician's realm of > harmony" (1967).

Those are online at

http://www.xs4all.nl/~huygensf/doc/mm4.html
http://www.xs4all.nl/~huygensf/doc/realm.html

Sorry for the digression, but Paul! look!

The first paper clearly associates a periodicity block with 31-equal. Here's a quote:

"""
Ces trois vecteurs nous d�finissent dans le r�seau un parall�l�pip�de, une base de p�riodicit�. Seuls les notes int�rieures � ce parall�l�pip�de seront ind�pendantes.
Leur nombre est d�fini par son volume, qui, par les m�thodes de la g�om�trie analytique, se trouve � l'aide du d�terminant form� avec les coordonn�es des ar�tes:

| 4 -1 0 |
| 2 2 -1 | = 31
| 1 0 3 |

Cette m�thode nous fournit le temp�rament �gal de trente-et-un cinqui�mes de ton, tel qu'il a �t� calcul� par Christiaan Huygens.
"""

The last sentence translates as "This method provides us with the equal temperament of thirty-one fifth tones, as was calculated by Christiaan Huygens."

> Now it occurs to me that the metric used weights too much the significance
> of the upper intervals, like the seventh, the eleventh and so for. I would
> say that it could make more sense, specially if one is thinking of doing
> tonal music, to really create a metric that will weigth more the most
> basic intervals. Something like

If you're thinking of tonal music, it makes sense to restrict the search to meantones. That is, temperaments where 81:80 is tempered out. These include 12, 19, 31, 43, 50 and 55 note equal temperaments.

> If you do use this metric just looking to the three intervals indicated, > the result of Fokker still holds and the 31 equally tempered scale is > still the first that produces a considerable reduction in the metric or > distance value. But my guess is that if the same procedure is used with > more harmonics then the result could vary significantly depending on the > weight factors selected.

I'm all for treating intervals equally. If you care about it, it should be as much in tune as possible. I also find that more complex intervals have to be better tuned to be comprehensible. But that's all a matter of personal preference.

However, I also optimize on complete odd limits, whereas Fokker is here only using the prime numbers. When you hit the 9-limit, 3 is automatically weighted higher than 5 and 7.

Although 31-equal doesn't have such a good fifth, it still compares well with 41 in the 11-limit. I haven't checked the detail of Fokker's method, but probably it's the prime bias showing through here. 31-equal's approximation to 11:8 is 9.4 cents out, but 41-equal is only 4.8 cents out. In both cases, these are worse than the approximations to 3, 5 and 7. It's easy to see how 31 comes out looking a lot better than 41.

But for the 11-limit as a whole, the worst error in 31-equal is 11.1 cents, for 9:5. The worst error for 41-equal is 10.6 cents for 11:10 which is only slightly better. So including these second-order intervals, even when 9 is included as well, 31-equal is roughly as good as 41-equal, when you take the difference in complexity into account. Perhaps somebody could repeat Fokkers calculations with all 11-limit intervals to see what the result would be.

Graham

hi carlos,

one problem i have with fokker's metric is that it doesn't consider
the major sixth or minor third. major triads with a given absolute
error of the fifth and a given absolute error of the third can have
two possible errors for the minor third. when one of these is near
zero, the triad sounds considerably purer. the effect seems at least
as important for the minor triad.

around here we usually compute misfits using all the intervals within
a certain odd limit (following partch), rather than all the harmonics
or all the primes up to some limit. many weighting schemes have been
considered, but equal weighting doesn't seem too objectionable to
anyone, i think . . .

you might be interested to read my papers,
http://www-math.cudenver.edu/~jstarret/22ALL.pdf
http://sonic-arts.org/td/erlich/intropblock1.htm

-paul

Graham, thanks for your comments.

Ok, if I understand you and Paul allright you suggest to include many more
intervals in the metric no just the fundamental prime intervals of Fokker.
I am not sure what the physical sense of that is. As Paul points out, in
the case of the minor third, you certainly would like to keep it in the
metric as it is a nice consonant ratio (for our ears today).

At any rate, following your suggestion I have computed the misfit
(unweigheted misfit that is, based on a simple cuadratic mean) for all non
repetitive ratios up to the 11th limit. This means that when an interval
has a complementary to the octave, I have only retained one of them (like
the fifth and not the fourth and so on).

Also I have enforced that the octave should be absolutely just. Fokker
actually discusses this aspects and he indicate that he sees no reason to
treat the octave any different. This is not my case I enforce the octave
to be exact and all the other intervals to be approximate.

The intervals I have considered then are

Just fifth 3/2
Just mayor third 5/4
Just minor third 6/5
Harmonic seventh 7/4
Subminor fifth 7/5
Subminor third 7/6
Supersecond 8/7
Major tone 9/8
Super major third 9/7
Acute minor seventh 9/5
Trumpet interval 11/10
The 11th harmonic 11/8
Meshaqah quartertones 11/6
Unamed_1 11/9
Unamed_2 11/7

The names are those of Ellis. Interestingly I could not find a name for the
two last ones.

I have then computed the misfit in four cases:

- looking at the 5th, the major 3rd and the harmonic 7th
- looking at all the above + the minor 3rd
- looking at all the above + 7 and 9 limit ratios
- looking at all the above plus all the 11 limit ratios

The results still show that the 31 equally tempered scales is a very good
choice in all cases. However, when one looks to more and more ratios
(like in the fourth case above) the option of 41 and 53 become more
attractive.

I have created two (small !) files. One is a pdf (12.4 K) with a table that
shows the results. The other one, is a plot of the same table. I am
sending them in separate mails, after this one.

My conclusion out of this is that 31 ET is a very nice compromise of
accuracy and compexity.

Comments wellcome

Carlos

You say,
>
> However, I also optimize on complete odd limits, whereas Fokker is here
> only using the prime numbers. When you hit the 9-limit, 3 is
> automatically weighted higher than 5 and 7.
>

I undertstand that you like to include in the metric function then
intervals like 9/8 an others. Is that right?.

Paul points out that the would rather include the minor thirds. I agree, I
thought about that myself. In fact, wouldn't it be making more sense to
"a priori" select those intervals for which you want a good fit because
one plans to use them the most and the create a metric, maybe through a
weighting vector or so, that prioritizes them.

Here goes the table

Carlos

On Wednesday 13 August 2003 17:38, Carlos wrote:
> Graham, thanks for your comments.
>
> Ok, if I understand you and Paul allright you suggest to include many
> more intervals in the metric no just the fundamental prime intervals of
> Fokker. I am not sure what the physical sense of that is. As Paul points
> out, in the case of the minor third, you certainly would like to keep it
> in the metric as it is a nice consonant ratio (for our ears today).
>
> At any rate, following your suggestion I have computed the misfit
> (unweigheted misfit that is, based on a simple cuadratic mean) for all
> non repetitive ratios up to the 11th limit. This means that when an
> interval has a complementary to the octave, I have only retained one of
> them (like the fifth and not the fourth and so on).
>
> Also I have enforced that the octave should be absolutely just. Fokker
> actually discusses this aspects and he indicate that he sees no reason
> to treat the octave any different. This is not my case I enforce the
> octave to be exact and all the other intervals to be approximate.
>
> The intervals I have considered then are
>
> Just fifth 3/2
> Just mayor third 5/4
> Just minor third 6/5
> Harmonic seventh 7/4
> Subminor fifth 7/5
> Subminor third 7/6
> Supersecond 8/7
> Major tone 9/8
> Super major third 9/7
> Acute minor seventh 9/5
> Trumpet interval 11/10
> The 11th harmonic 11/8
> Meshaqah quartertones 11/6
> Unamed_1 11/9
> Unamed_2 11/7
>
> The names are those of Ellis. Interestingly I could not find a name for
> the two last ones.
>
> I have then computed the misfit in four cases:
>
> - looking at the 5th, the major 3rd and the harmonic 7th
> - looking at all the above + the minor 3rd
> - looking at all the above + 7 and 9 limit ratios
> - looking at all the above plus all the 11 limit ratios
>
> The results still show that the 31 equally tempered scales is a very
> good choice in all cases. However, when one looks to more and more
> ratios (like in the fourth case above) the option of 41 and 53 become
> more attractive.
>
> I have created two (small !) files. One is a pdf (12.4 K) with a table
> that shows the results. The other one, is a plot of the same table. I am
> sending them in separate mails, after this one.
>
> My conclusion out of this is that 31 ET is a very nice compromise of
> accuracy and compexity.
>
> Comments wellcome
>
> Carlos
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> You say,
>
> > However, I also optimize on complete odd limits, whereas Fokker is
> > here only using the prime numbers. When you hit the 9-limit, 3 is
> > automatically weighted higher than 5 and 7.
>
> I undertstand that you like to include in the metric function then
> intervals like 9/8 an others. Is that right?.
>
>
>
> Paul points out that the would rather include the minor thirds. I agree,
> I thought about that myself. In fact, wouldn't it be making more sense
> to "a priori" select those intervals for which you want a good fit
> because one plans to use them the most and the create a metric, maybe
> through a weighting vector or so, that prioritizes them.

The plot is in postcrip format

Carlos

On Wednesday 13 August 2003 17:38, Carlos wrote:
> Graham, thanks for your comments.
>
> Ok, if I understand you and Paul allright you suggest to include many
> more intervals in the metric no just the fundamental prime intervals of
> Fokker. I am not sure what the physical sense of that is. As Paul points
> out, in the case of the minor third, you certainly would like to keep it
> in the metric as it is a nice consonant ratio (for our ears today).
>
> At any rate, following your suggestion I have computed the misfit
> (unweigheted misfit that is, based on a simple cuadratic mean) for all
> non repetitive ratios up to the 11th limit. This means that when an
> interval has a complementary to the octave, I have only retained one of
> them (like the fifth and not the fourth and so on).
>
> Also I have enforced that the octave should be absolutely just. Fokker
> actually discusses this aspects and he indicate that he sees no reason
> to treat the octave any different. This is not my case I enforce the
> octave to be exact and all the other intervals to be approximate.
>
> The intervals I have considered then are
>
> Just fifth 3/2
> Just mayor third 5/4
> Just minor third 6/5
> Harmonic seventh 7/4
> Subminor fifth 7/5
> Subminor third 7/6
> Supersecond 8/7
> Major tone 9/8
> Super major third 9/7
> Acute minor seventh 9/5
> Trumpet interval 11/10
> The 11th harmonic 11/8
> Meshaqah quartertones 11/6
> Unamed_1 11/9
> Unamed_2 11/7
>
> The names are those of Ellis. Interestingly I could not find a name for
> the two last ones.
>
> I have then computed the misfit in four cases:
>
> - looking at the 5th, the major 3rd and the harmonic 7th
> - looking at all the above + the minor 3rd
> - looking at all the above + 7 and 9 limit ratios
> - looking at all the above plus all the 11 limit ratios
>
> The results still show that the 31 equally tempered scales is a very
> good choice in all cases. However, when one looks to more and more
> ratios (like in the fourth case above) the option of 41 and 53 become
> more attractive.
>
> I have created two (small !) files. One is a pdf (12.4 K) with a table
> that shows the results. The other one, is a plot of the same table. I am
> sending them in separate mails, after this one.
>
> My conclusion out of this is that 31 ET is a very nice compromise of
> accuracy and compexity.
>
> Comments wellcome
>
> Carlos
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
> You say,
>
> > However, I also optimize on complete odd limits, whereas Fokker is
> > here only using the prime numbers. When you hit the 9-limit, 3 is
> > automatically weighted higher than 5 and 7.
>
> I undertstand that you like to include in the metric function then
> intervals like 9/8 an others. Is that right?.
>
>
>
> Paul points out that the would rather include the minor thirds. I agree,
> I thought about that myself. In fact, wouldn't it be making more sense
> to "a priori" select those intervals for which you want a good fit
> because one plans to use them the most and the create a metric, maybe
> through a weighting vector or so, that prioritizes them.

hi Carlos,

i haven't really followed this thread, but when i saw
this i thought i could help illustrate something ...

> From: Carlos [mailto:garciasuarez@ya.com]
> Sent: Wednesday, August 13, 2003 8:38 AM
> To: tuning-math@yahoogroups.com
> Subject: Re: [tuning-math] Re: Comments about Fokker's misfit metric
>
>
> Graham, thanks for your comments.
>
> Ok, if I understand you and Paul allright you suggest
> to include many more intervals in the metric no[t] just
> the fundamental prime intervals of Fokker. I am not
> sure what the physical sense of that is. As Paul points
> out, in the case of the minor third, you certainly would
> like to keep it in the metric as it is a nice consonant
> ratio (for our ears today).

the point of using the minor-3rd as well as the major-3rd
and 5th (in the 5-limit) is that *all* other intervals
can be calculated as various combinations of those three.

in the 7-limit, in addition to the primary 3:2, 5:4, and
7:4, you would also need 6:5, 7:6, and 7:5.

it can be demonstrated on a lattice as follows:

(use "Expand Messages" mode if viewing on the stupid
Yahoo web interface)

5-limit
-------

primary:

5:4
/
/
/
/
1:1 ------- 3:2

primary and secondary:

5:4
/ \
/ \
/ \
/ \
1:1 ------- 3:2

7-limit
-------

primary:

5:4
/
/
/ 7:4
/.'
1:1 ------- 3:2

primary and secondary:

5:4
/| \
/ | \
/ 7:4 \
/.' '. \
1:1 ------- 3:2

it can be seen easily on the lattices that
adding lines to illustrate the secondary intervals
"completes" or "closes" the structure.

-monz

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> Graham, thanks for your comments.
>
> Ok, if I understand you and Paul allright you suggest to include
many more
> intervals in the metric no just the fundamental prime intervals of
Fokker.
> I am not sure what the physical sense of that is. As Paul points
out, in
> the case of the minor third, you certainly would like to keep it in
the
> metric as it is a nice consonant ratio (for our ears today).
>
> At any rate, following your suggestion I have computed the misfit
> (unweigheted misfit that is, based on a simple cuadratic mean) for
all non
> repetitive ratios up to the 11th limit. This means that when an
interval
> has a complementary to the octave, I have only retained one of them
(like
> the fifth and not the fourth and so on).
>
> Also I have enforced that the octave should be absolutely just.
Fokker
> actually discusses this aspects and he indicate that he sees no
reason to
> treat the octave any different. This is not my case I enforce the
octave
> to be exact and all the other intervals to be approximate.
>
> The intervals I have considered then are
>
> Just fifth 3/2
> Just mayor third 5/4
> Just minor third 6/5
> Harmonic seventh 7/4
> Subminor fifth 7/5
> Subminor third 7/6
> Supersecond 8/7
> Major tone 9/8
> Super major third 9/7
> Acute minor seventh 9/5
> Trumpet interval 11/10
> The 11th harmonic 11/8
> Meshaqah quartertones 11/6
> Unamed_1 11/9
> Unamed_2 11/7
>
> The names are those of Ellis. Interestingly I could not find a name
for the
> two last ones.
>
> I have then computed the misfit in four cases:
>
> - looking at the 5th, the major 3rd and the harmonic 7th
> - looking at all the above + the minor 3rd
> - looking at all the above + 7 and 9 limit ratios
> - looking at all the above plus all the 11 limit ratios
>
> The results still show that the 31 equally tempered scales is a
very good
> choice in all cases. However, when one looks to more and more
ratios
> (like in the fourth case above) the option of 41 and 53 become more
> attractive.

actually, 53-equal suffers from an additional problem with this type
of calculation, whether fokker's original or modified as you have
above. the problem is "inconsistency". if you consider each 11-limit
interval's best approximation in 53, you will think that you can
approximate 4:7 with 43 steps, 4:11 with 77 steps, and 7:11 with 35
steps. unfortunately, if you try to approximate a 4:7:11 triad in 53-
equal, you'll run into the problem that 43 + 35 does not equal 77
steps. thus you'll have to use a worse approximation for at least one
of these intervals than your original analysis suggested you'd need.

> My conclusion out of this is that 31 ET is a very nice compromise
of
> accuracy and compexity.

certainly. it's also a meantone which is hugely important for the
performance of western music. you'll find that 72-equal is much more
accurate than 31 for most of the intervals you've looked at, and
indeed 72 has gotten a huge amount of discussion on the tuning list
for this very reason. the advantage of 72 is that it's a multiple of
12 so can be taught as an extension of existing technique to modern
instrumentalists, allowing one to incorporate these higher ratios
into performable contemporary compositions (as, for example, joseph
pehrson has done). the disadvantage is that it's not a meantone, so
one can't use it to render the western repertoire c. 1480-1780 in a
more authentic and pleasing tuning. for this, a true revival of 31
would be a welcome development.

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:

> I undertstand that you like to include in the metric function then
> intervals like 9/8 an others. Is that right?.

i do, yes.

--- In tuning-math@yahoogroups.com, <monz@a...> wrote:
> hi Carlos,
>
>
> i haven't really followed this thread, but when i saw
> this i thought i could help illustrate something ...
>
>
> > From: Carlos [mailto:garciasuarez@y...]
> > Sent: Wednesday, August 13, 2003 8:38 AM
> > To: tuning-math@yahoogroups.com
> > Subject: Re: [tuning-math] Re: Comments about Fokker's misfit
metric
> >
> >
> > Graham, thanks for your comments.
> >
> > Ok, if I understand you and Paul allright you suggest
> > to include many more intervals in the metric no[t] just
> > the fundamental prime intervals of Fokker. I am not
> > sure what the physical sense of that is. As Paul points
> > out, in the case of the minor third, you certainly would
> > like to keep it in the metric as it is a nice consonant
> > ratio (for our ears today).
>
>
>
> the point of using the minor-3rd as well as the major-3rd
> and 5th (in the 5-limit) is that *all* other intervals
> can be calculated as various combinations of those three.

i don't follow.

hi paul,

> From: Paul Erlich [mailto:perlich@aya.yale.edu]
> Sent: Wednesday, August 13, 2003 11:52 AM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Comments about Fokker's misfit metric
>
>
> --- In tuning-math@yahoogroups.com, <monz@a...> wrote:
> > hi Carlos,
> >
> >
> > i haven't really followed this thread, but when i saw
> > this i thought i could help illustrate something ...
> >
> >
> > > From: Carlos [mailto:garciasuarez@y...]
> > > Sent: Wednesday, August 13, 2003 8:38 AM
> > > To: tuning-math@yahoogroups.com
> > > Subject: Re: [tuning-math] Re: Comments about Fokker's misfit
> metric
> > >
> > >
> > > Graham, thanks for your comments.
> > >
> > > Ok, if I understand you and Paul allright you suggest
> > > to include many more intervals in the metric no[t] just
> > > the fundamental prime intervals of Fokker. I am not
> > > sure what the physical sense of that is. As Paul points
> > > out, in the case of the minor third, you certainly would
> > > like to keep it in the metric as it is a nice consonant
> > > ratio (for our ears today).
> >
> >
> >
> > the point of using the minor-3rd as well as the major-3rd
> > and 5th (in the 5-limit) is that *all* other intervals
> > can be calculated as various combinations of those three.
>
> i don't follow.

OK, well, as i said, i wasn't really following the thread,
so maybe my comment has nothing to do with anything.

i was just pointing out that whatever the metric is
measuring, if it's meant to be used over the whole
tuning system, it only needs those 3 intervals in the
5-limit, or those 6 in the 7-limit, etc., to cover
the metric for any interval in the system.

-monz

Carlos wrote:

>Also I have enforced that the octave should be absolutely just. Fokker >actually discusses this aspects and he indicate that he sees no reason to >treat the octave any different. This is not my case I enforce the octave >to be exact and all the other intervals to be approximate.
> >
If the octaves weren't just, not only would you have to consider complements, like both 3:2 and 4:3, but also larger intervals like 3:1, 5:2, 7:3 and whatever. So it's a lot easier not to bother.

>Unamed_1 11/9
>Unamed_2 11/7
>
>The names are those of Ellis. Interestingly I could not find a name for the >two last ones.
> >
11/9 is a neutral third.

>The results still show that the 31 equally tempered scales is a very good >choice in all cases. However, when one looks to more and more ratios >(like in the fourth case above) the option of 41 and 53 become more >attractive.
>
So it's 26.5 for 31 against 20.07 for 41. Fokker deficiency is this number multiplied 2**((n-12)/12) for n notes to to the octave. Even by his method, the 70.8 for 31 is so close to 71.0 for 41 as to be negligible. With your figures I get a deficiency of 79.5 for 31 compared to 109.7 for 41. So 31 is now much better, as I suspected!

You can also see, as has been mentioned, that 72 is very good in all cases, and its 11-limit error is less than half the size of anything else. Although, because it's so big, its deficiency is still 238.1.

>I undertstand that you like to include in the metric function then >intervals like 9/8 an others. Is that right?.
> >
Yes, for the 9 and higher limits. But you can also look at the 7-limit, where 9 isn't included.

>Paul points out that the would rather include the minor thirds. I agree, I >thought about that myself. In fact, wouldn't it be making more sense to >"a priori" select those intervals for which you want a good fit because >one plans to use them the most and the create a metric, maybe through a >weighting vector or so, that prioritizes them.
> >
Indeed, there's no one temperament that works in all situations. So decide what intervals you want, and find a tuning that approximates them efficiently.

Graham

monz@attglobal.net wrote:

>i was just pointing out that whatever the metric is
>measuring, if it's meant to be used over the whole
>tuning system, it only needs those 3 intervals in the
>5-limit, or those 6 in the 7-limit, etc., to cover
>the metric for any interval in the system.
> >
You only need 2 intervals to cover the 5-limit. The two Fokker used, for example.

Graham

Paul Erlich wrote:

>actually, 53-equal suffers from an additional problem with this type >of calculation, whether fokker's original or modified as you have >above. the problem is "inconsistency"... >
Fokker only looked at prime numbers, so consistency wasn't an issue.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> monz@a... wrote:
>
> >i was just pointing out that whatever the metric is
> >measuring, if it's meant to be used over the whole
> >tuning system, it only needs those 3 intervals in the
> >5-limit, or those 6 in the 7-limit, etc., to cover
> >the metric for any interval in the system.
> >
> >
> You only need 2 intervals to cover the 5-limit. The two Fokker
used,
> for example.

Eh? You need three. 2, 3, and 5 is the obvious choice. Octave
reduction leads to two classes, of course.

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> The intervals I have considered then are
>
> Just fifth 3/2
> Just mayor third 5/4
> Just minor third 6/5
> Harmonic seventh 7/4
> Subminor fifth 7/5
> Subminor third 7/6
> Supersecond 8/7
> Major tone 9/8
> Super major third 9/7
> Acute minor seventh 9/5
> Trumpet interval 11/10
> The 11th harmonic 11/8
> Meshaqah quartertones 11/6
> Unamed_1 11/9
> Unamed_2 11/7

Having both 7/4 and 8/7 is redundent. If you eliminate that, you have
a table of representatives for the 14 11-limit consonances, in the
terminology of this group.

hi Graham (and paul and Carlos),

> From: Graham Breed [mailto:graham@microtonal.co.uk]
> Sent: Wednesday, August 13, 2003 5:07 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: Comments about Fokker's misfit metric
>
>
> monz@attglobal.net wrote:
>
> >i was just pointing out that whatever the metric is
> >measuring, if it's meant to be used over the whole
> >tuning system, it only needs those 3 intervals in the
> >5-limit, or those 6 in the 7-limit, etc., to cover
> >the metric for any interval in the system.
> >
> >
> You only need 2 intervals to cover the 5-limit.
> The two Fokker used, for example.

well, there you go ... i wasn't following the thread
and spoke up when i should have kept my mouth shut.
forget it -- i'm out of this thread until and unless
i go back and read from the beginning.

-monz

Paul and Graham,

Thanks to both for the interesting discussion, you have obviously discussed
at least part of this before in the group but I am kind of new to it. I
hope it hasn't been much of a repetition for the community here.

And also sorry, because I should have not included the 8/7 inteval.
Luckily, taking this one out does not change much the results and the
commented conclusions still holds.

Carlos

On Thursday 14 August 2003 02:50, Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> > The intervals I have considered then are
> >
> > Just fifth 3/2
> > Just mayor third 5/4
> > Just minor third 6/5
> > Harmonic seventh 7/4
> > Subminor fifth 7/5
> > Subminor third 7/6
> > Supersecond 8/7
> > Major tone 9/8
> > Super major third 9/7
> > Acute minor seventh 9/5
> > Trumpet interval 11/10
> > The 11th harmonic 11/8
> > Meshaqah quartertones 11/6
> > Unamed_1 11/9
> > Unamed_2 11/7
>
> Having both 7/4 and 8/7 is redundent. If you eliminate that, you have
> a table of representatives for the 14 11-limit consonances, in the
> terminology of this group.
>
>
>
>
>
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>
>
>
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--- In tuning-math@yahoogroups.com, <monz@a...> wrote:

> i was just pointing out that whatever the metric is
> measuring, if it's meant to be used over the whole
> tuning system, it only needs those 3 intervals in the
> 5-limit, or those 6 in the 7-limit, etc., to cover
> the metric for any interval in the system.

that still doesn't make any sense to me. reminds me of a mysterious
part of your woolhouse page that i criticized way back, though.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> >actually, 53-equal suffers from an additional problem with this
type
> >of calculation, whether fokker's original or modified as you have
> >above. the problem is "inconsistency"...
> >
> Fokker only looked at prime numbers, so consistency wasn't an issue.
>
>
> Graham

sorry, you're right -- so it's only when minor thirds and non-prime-
harmonic primary ratios are included, for example in the analyses of
wendy carlos (!) and yunik & swift, that inconsistency calls the
results into question. stoney had his own way of enforcing
consistency, which was a bit ad hoc, but generally yields better
results than just using the best approximation to each prime.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...>
wrote:
> > The intervals I have considered then are
> >
> > Just fifth 3/2
> > Just mayor third 5/4
> > Just minor third 6/5
> > Harmonic seventh 7/4
> > Subminor fifth 7/5
> > Subminor third 7/6
> > Supersecond 8/7
> > Major tone 9/8
> > Super major third 9/7
> > Acute minor seventh 9/5
> > Trumpet interval 11/10
> > The 11th harmonic 11/8
> > Meshaqah quartertones 11/6
> > Unamed_1 11/9
> > Unamed_2 11/7
>
> Having both 7/4 and 8/7 is redundent. If you eliminate that, you
have
> a table of representatives for the 14 11-limit consonances, in the
> terminology of this group.

oops! thanks for catching that, gene!

personally, i would also include "9/6", since a complete 11-limit
hexad contains both a 3:2 and a "9:6", so the perfect fifth should be
weighted twice.

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

stoney had his own way of enforcing
> consistency, which was a bit ad hoc, but generally yields better
> results than just using the best approximation to each prime.

There's always my suggestion of using the Zeta tuning.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> stoney had his own way of enforcing
> > consistency, which was a bit ad hoc, but generally yields better
> > results than just using the best approximation to each prime.
>
> There's always my suggestion of using the Zeta tuning.

ok, i just read _prime obsession_, but i think i know even less about
the zetafunction than after reading manfred schroeder's _number
theory in science and communication_ . . .

where can i read about zeta tuning again?

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

> ok, i just read _prime obsession_, but i think i know even less
about
> the zetafunction than after reading manfred schroeder's _number
> theory in science and communication_ . . .
>
> where can i read about zeta tuning again?

Somewhere lost in the archives of this list; however it's a good one
to put up a web page on xenharmony for.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > ok, i just read _prime obsession_, but i think i know even less
> about
> > the zetafunction than after reading manfred schroeder's _number
> > theory in science and communication_ . . .
> >
> > where can i read about zeta tuning again?
>
> Somewhere lost in the archives of this list;

not lost; i just read three. i'm not sure i understand how
consistency is enforced differently than just using the best
approximations to the primes, though.

> however it's a good one
> to put up a web page on xenharmony for.

how about some graphs?

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

> not lost; i just read three. i'm not sure i understand how
> consistency is enforced differently than just using the best
> approximations to the primes, though.

All it does is supply a different, and normally preferable, answer to
the question of how to define a standard val. Choosing any val
enforces consistency.

> > however it's a good one
> > to put up a web page on xenharmony for.
>
> how about some graphs?

Why not.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > not lost; i just read three. i'm not sure i understand how
> > consistency is enforced differently than just using the best
> > approximations to the primes, though.
>
> All it does is supply a different, and normally preferable, answer
to
> the question of how to define a standard val.

yes, i just wasn't seeing how it arrived at that val.

> Choosing any val
> enforces consistency.

yup!

>personally, i would also include "9/6", since a complete 11-limit
>hexad contains both a 3:2 and a "9:6", so the perfect fifth should
>be weighted twice.

Here, here!

-Carl