Yahoo Tuning Groups Ultimate Backup tuning-math names and definitions: meantone

let's start the whole project with perhaps the
most familiar family of tunings.

i already have a page about meantone at

i realize that some of the categories below require
data for specific flavors of meantone, and not the
whole family itself. but if there is a way to put
in a range of data that does cover the whole family,
it think that is good.

please keep in mind that this is an extremely rough
draft that's coming right off the top of my head.
i'm real busy with other things but while i'm in the
thick of working on the Encyclopaedia, i might as well
use the opportunity to create a whole slew of webpages
covering the names that everyone is complaining about.

so that at least then familiarity can breed contempt ...

;-)

fill in the blanks, and adjust, correct, argue etc.
as much as possible ...

family name: meantone
period: 2:1 ratio
generator:
wedge product:
wedgie:
unison-vectors:
monzos, multimonzos:
vals, multivals:
badness:
MOS:
DE:
propriety:
consistency:
characteristic interval(s):
x-chordal interval structure (tetrachord, pentachord, etc.):

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> family name: meantone
> period: 2:1 ratio
> generator: a flat fifth or sharp fourth
> wedge product: eh?

5-limit: 81/80
mapping: [<1 2 4|, <0 -1 -4|]
poptimal generator: 34/81
MOS: 5, 7, 12, 19, 31, 50

7-limit:

name: meantone, standard septimal meantone, 12&19
wedgie: <1 4 10 4 13 12|
mapping: [<1 2 4 7|, <0 -1 -4 -10|] (fourth generaor)
7&9 limit copoptimal generator: 5^(1/4) fifth (1/4 comma meantone)
7 limit poptimal: 86/205
9 limit poptimal: 47/112
TM basis: {81/80, 126/125}
MOS: 5, 7, 12, 19, 31, 50

name: dominant, dominant sevenths, 5&12
wedgie: <1 4 -2 4 -6 -16|
mapping: [<1 2 4 2|, <0 -1 -4 2|]
7&9 limit copoptimal generator: 29/70
TM basis: {36/35, 64/63}
MOS: 5, 7, 12, 17, 29

name: flattone, 19&26
wedgie: <1 4 -9 4 -17 -32|
mapping: [<1 2 4 -1|, <0 -1 -4 9|]
7 limit poptimal: 46/109
9 limit poptimal: 27/64
TM basis: {81/80, 525/512}
MOS: 5, 7, 12, 19, 26, 45

11 limit

name: meantone, meanpop, 31&19
wedgie: <1 4 10 -13 4 13 -24 12 -44 -71|
mapping: [<1 2 4 7 -2|, <0 -1 -4 -10 13|]
poptimal generator: 47/112
TM basis: {81/80, 126/125, 385/384}
MOS: 5, 7, 12, 19, 31, 50, 81

name: huygens, fokker, 31&12
wedgie: <1 4 10 18 4 13 25 12 28 16|
mapping: [<1 2 4 7 11|, <0 -1 -4 -10 -18|]
poptimal generator: 13/31
TM basis: {81/80, 126/125, 99/98}
MOS: 5, 7, 12, 19, 31, 43

name: meanertone
wedgie: <1 4 3 -1 4 2 -5 -4 -16 -13|
mapping: [<1 2 4 4 3|, <0 -1 -4 -3 1|]
poptimal generator: 13/31
TM basis: {27/25, 21/20, 33/32}
MOS: 5, 7, 12, 19, 31, 43

🔗monz <monz@attglobal.net>

thanks, Gene ... this is great!

it looks to me like this tells the whole story
... but others out there will know better than i.

can you make some like this for all of the other
named temperaments? i'm in no hurry, but folks
who are confused about the names might appreciate it.

-monz

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > family name: meantone
> > period: 2:1 ratio
> > generator: a flat fifth or sharp fourth
> > wedge product: eh?
>
> 5-limit: 81/80
> mapping: [<1 2 4|, <0 -1 -4|]
> poptimal generator: 34/81
> MOS: 5, 7, 12, 19, 31, 50
>
> 7-limit:
>
> name: meantone, standard septimal meantone, 12&19
> wedgie: <1 4 10 4 13 12|
> mapping: [<1 2 4 7|, <0 -1 -4 -10|] (fourth generaor)
> 7&9 limit copoptimal generator: 5^(1/4) fifth (1/4 comma meantone)
> 7 limit poptimal: 86/205
> 9 limit poptimal: 47/112
> TM basis: {81/80, 126/125}
> MOS: 5, 7, 12, 19, 31, 50
>
> name: dominant, dominant sevenths, 5&12
> wedgie: <1 4 -2 4 -6 -16|
> mapping: [<1 2 4 2|, <0 -1 -4 2|]
> 7&9 limit copoptimal generator: 29/70
> TM basis: {36/35, 64/63}
> MOS: 5, 7, 12, 17, 29
>
> name: flattone, 19&26
> wedgie: <1 4 -9 4 -17 -32|
> mapping: [<1 2 4 -1|, <0 -1 -4 9|]
> 7 limit poptimal: 46/109
> 9 limit poptimal: 27/64
> TM basis: {81/80, 525/512}
> MOS: 5, 7, 12, 19, 26, 45
>
> 11 limit
>
> name: meantone, meanpop, 31&19
> wedgie: <1 4 10 -13 4 13 -24 12 -44 -71|
> mapping: [<1 2 4 7 -2|, <0 -1 -4 -10 13|]
> poptimal generator: 47/112
> TM basis: {81/80, 126/125, 385/384}
> MOS: 5, 7, 12, 19, 31, 50, 81
>
> name: huygens, fokker, 31&12
> wedgie: <1 4 10 18 4 13 25 12 28 16|
> mapping: [<1 2 4 7 11|, <0 -1 -4 -10 -18|]
> poptimal generator: 13/31
> TM basis: {81/80, 126/125, 99/98}
> MOS: 5, 7, 12, 19, 31, 43
>
> name: meanertone
> wedgie: <1 4 3 -1 4 2 -5 -4 -16 -13|
> mapping: [<1 2 4 4 3|, <0 -1 -4 -3 1|]
> poptimal generator: 13/31
> TM basis: {27/25, 21/20, 33/32}
> MOS: 5, 7, 12, 19, 31, 43

🔗Gene Ward Smith <gwsmith@svpal.org>

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

This is fixed up.

family name: meantone
period: octave
generator: a flat fifth or sharp fourth

5-limit

name: meantone, 12&19
comma: 81/80
mapping: [<1 2 4|, <0 -1 -4|]
poptimal generator: 34/81
MOS: 5, 7, 12, 19, 31, 50, 81

7-limit

name: meantone, standard septimal meantone, 12&19
wedgie: <<1 4 10 4 13 12||
mapping: [<1 2 4 7|, <0 -1 -4 -10|] (fourth generaor)
7&9 limit copoptimal generator: 5^(1/4) fifth (1/4 comma meantone)
7 limit poptimal: 86/205
9 limit poptimal: 47/112
TM basis: {81/80, 126/125}
MOS: 5, 7, 12, 19, 31, 50, 81

name: dominant, dominant sevenths, 5&12
wedgie: <<1 4 -2 4 -6 -16||
mapping: [<1 2 4 2|, <0 -1 -4 2|]
7&9 limit copoptimal generator: 29/70
TM basis: {36/35, 64/63}
MOS: 5, 7, 12, 17, 29

name: flattone, 19&26
wedgie: <<1 4 -9 4 -17 -32||
mapping: [<1 2 4 -1|, <0 -1 -4 9|]
7 limit poptimal: 46/109
9 limit poptimal: 27/64
TM basis: {81/80, 525/512}
MOS: 5, 7, 12, 19, 26, 45

11 limit

name: meantone, meanpop, 19&31
wedgie: <<1 4 10 -13 4 13 -24 12 -44 -71||
mapping: [<1 2 4 7 -2|, <0 -1 -4 -10 13|]
poptimal generator: 47/112
TM basis: {81/80, 126/125, 385/384}
MOS: 5, 7, 12, 19, 31, 50, 81

name: huygens, fokker, 12&31
wedgie: <<1 4 10 18 4 13 25 12 28 16||
mapping: [<1 2 4 7 11|, <0 -1 -4 -10 -18|]
poptimal generator: 13/31
TM basis: {81/80, 126/125, 99/98}
MOS: 5, 7, 12, 19, 31, 43

name: meanertone
wedgie: <<1 4 3 -1 4 2 -5 -4 -16 -13||
mapping: [<1 2 4 4 3|, <0 -1 -4 -3 1|]
poptimal generator: 13/31
TM basis: {21/20, 28/27, 33/32}
MOS: 5, 7, 12, 19, 31, 43

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

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

re:
http://tonalsoft.com/enc
meantone

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > thanks, Gene ... this is great!
>
> But please use the revised version.

done.

i put it at the very bottom of the page.
does anyone think it should be at the top, or
closer to the top? my feeling is that i should
leave the verbal description at the top and
let the math come at the end.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:

> name: meantone, 12&19
> comma: 81/80
> mapping: [<1 2 4|, <0 -1 -4|]
> poptimal generator: 34/81
> MOS: 5, 7, 12, 19, 31, 50, 81

This MOS list is valid for quarter-comma meantone and Kornerup's golden meantone (among others), but as we've seen, various tunings of the same temperament can have different MOS structures. In particular, meantone with a 23/55 generator/period ratio (Mozart's tuning) has a 43-note MOS (12L+31s), but not one with 50 notes. 1/5-comma meantone technically has a 43-note MOS, but it's so close to 43-ET that the step sizes are roughly equal.

🔗monz <monz@attglobal.net>

on this page (which can also be reached via a link on
the "meantone" defintion)

http://tonalsoft.com/enc/meantone-error/meantone-error.htm

i found that 136edo is some sort of optimal meantone for
the 11-limit, regarding the amount of error from JI.
(at least, using the prime-mappings i used)

i found this visually using my graphic applet.
but i don't recall anything ever being said about 136edo.

comments?

-monz

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s..
.>
> > wrote:
> >
> > re:
> > http://tonalsoft.com/enc
> > meantone
>
> It doesn't seem to be up.

looks OK to me. did you hit "refresh/reload"?

Gene, when you wrote, for example,
"7 limit poptimal: 86/205", you *did* mean
"7-limit poptimal *generator*", right?

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Gene Ward Smith wrote:
>
> > name: meantone, 12&19
> > comma: 81/80
> > mapping: [<1 2 4|, <0 -1 -4|]
> > poptimal generator: 34/81
> > MOS: 5, 7, 12, 19, 31, 50, 81
>
> This MOS list is valid for quarter-comma meantone and Kornerup's golden
> meantone (among others), but as we've seen, various tunings of the same
> temperament can have different MOS structures. In particular, meantone
> with a 23/55 generator/period ratio (Mozart's tuning) has a 43-note MOS
> (12L+31s), but not one with 50 notes. 1/5-comma meantone technically
has
> a 43-note MOS, but it's so close to 43-ET that the step sizes are
> roughly equal.

True enough; I went with the proposition that MOS for the optimized
tunings were the proper ones to list. Trying to list all of them gets
a little out of hand, and could even be confusing.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> This MOS list is valid for quarter-comma meantone and Kornerup's golden
> meantone (among others), but as we've seen, various tunings of the same
> temperament can have different MOS structures.

One possible answer to this question which works for meantone, but not
anything else, is to use 1/4-comma meantone. Meantone is provided with
a standard, classical definition of its generator, so we can do this,
but it won't work for anything else. The semiconvergents of the
1/4-comma fifth give us 5, 7, 12, 19, 31, 50, 81, 112, 143, 205 ...
for MOS, and it would certainly be possible to view these as standard.
It matches the list I gave up to the point I stopped.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> on this page (which can also be reached via a link on
> the "meantone" defintion)
>
> http://tonalsoft.com/enc/meantone-error/meantone-error.htm
>
> i found that 136edo is some sort of optimal meantone for
> the 11-limit, regarding the amount of error from JI.
> (at least, using the prime-mappings i used)

It's very near the rms optimum and hence the poptimal range for
huygens. Since I'm on semiconvergents, here is what you get for rms
(exponent 2), exponent 3, and exponent 4 optimums:

rms 7, 12, 19, 31, 43, 74, 105, 136, 167, 198, 365, 532, ...

p3 7, 12, 19, 31, 43, 74, 105, 136, 167, 198, 365, 563, ...

p4 7, 12, 19, 31, 43, 74, 105, 136, 167, 198, 229, 427, ...

167 is a convergent for both rms and p3, however, which 136 is not. A
while back we had a discussion of whether 74 was not, in fact, totally
useless; here we see it showing up on the huygens lists.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> > on this page (which can also be reached via a link on
> > the "meantone" defintion)
> >
> > http://tonalsoft.com/enc/meantone-error/meantone-error.htm
> >
> > i found that 136edo is some sort of optimal meantone for
> > the 11-limit, regarding the amount of error from JI.
> > (at least, using the prime-mappings i used)
>
> It's very near the rms optimum and hence the poptimal range for
> huygens.

Joe's web page says ``It can be seen that the lowest overall error in
the 11-limit is given by meantones in the area of 3/13-comma to 136edo
(in the version of 136edo where the "5th" is mapped to 79 degrees).''
Joe, this needs to be corrected to make clear that it is assuming the
huygens mapping; another and equally valid choice is the
meantone/meanpop mapping.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Joe's web page says ``It can be seen that the lowest overall error in
> the 11-limit is given by meantones in the area of 3/13-comma to 136edo
> (in the version of 136edo where the "5th" is mapped to 79 degrees).''
> Joe, this needs to be corrected to make clear that it is assuming the
> huygens mapping; another and equally valid choice is the
> meantone/meanpop mapping.

On the meantone page, I think the error for both meanpop and huygens
should be graphed; comparing them on a graph of just the 11-limit
versions would be instructive.

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > Gene, when you wrote, for example,
> > "7 limit poptimal: 86/205", you *did* mean
> > "7-limit poptimal *generator*", right?
>
> Right; the generator is 86/205 of an octave, or 2^(86/205).

to be precise, you really mean 86/205 of the period,
which in this case happens to be an octave ... correct?

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > Right; the generator is 86/205 of an octave, or 2^(86/205).
> >
> >
> > to be precise, you really mean 86/205 of the period,
> > which in this case happens to be an octave ... correct?
>
> No, I always intend the fractions as fractions of an octave.

OK, that answers my question with respect to your
own individual usage.

but my real question was: shouldn't the reader assume
that the generator size is given as a fraction of the
equivalence-interval? so in all of your definitions,
it is an octave, but the definitions don't necessarily
have to make it an octave. am i wrong about this?

-monz

🔗monz <monz@attglobal.net>

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Joe's web page says ``It can be seen that the lowest
> > overall error in the 11-limit is given by meantones in
> > the area of 3/13-comma to 136edo (in the version of 136edo
> > where the "5th" is mapped to 79 degrees).''
> >
> > Joe, this needs to be corrected to make clear that it is
> > assuming the huygens mapping; another and equally valid
> > choice is the meantone/meanpop mapping.
>
> On the meantone page, I think the error for both meanpop
> and huygens should be graphed; comparing them on a graph
> of just the 11-limit versions would be instructive.

OK ... i'd like to make that graph, but not sure how
to do it. can you give me some pointers?

-monz

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene,
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s..
.>
> > wrote:
> >
> > > Joe's web page says ``It can be seen that the lowest
> > > overall error in the 11-limit is given by meantones in
> > > the area of 3/13-comma to 136edo (in the version of 136edo
> > > where the "5th" is mapped to 79 degrees).''
> > >
> > > Joe, this needs to be corrected to make clear that it is
> > > assuming the huygens mapping; another and equally valid
> > > choice is the meantone/meanpop mapping.
> >
> > On the meantone page, I think the error for both meanpop
> > and huygens should be graphed; comparing them on a graph
> > of just the 11-limit versions would be instructive.
>
>
>
> OK ... i'd like to make that graph, but not sure how
> to do it. can you give me some pointers?

never mind ... i've done it.

http://tonalsoft.com/enc/meantone.htm

also, scroll further down to check out the 4 different
perspectives of helical meantone lattices that i've
just put up there.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> but my real question was: shouldn't the reader assume
> that the generator size is given as a fraction of the
> equivalence-interval?

I think it makes more sense to give both generator and period in the
same units, whether those are cents or octaves.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > but my real question was: shouldn't the reader assume
> > that the generator size is given as a fraction of the
> > equivalence-interval?
>
> I think it makes more sense to give both generator and period in
the
> same units, whether those are cents or octaves.

But if we're talking about the naming system where we give a single
fraction (with GCD of numerator and denominator being the number of
periods per octave) then I believe it should be a semi-convergent of
generator/period, not generator/1200-cents * periods_per_octave.

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

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > but my real question was: shouldn't the reader assume
> > > that the generator size is given as a fraction of the
> > > equivalence-interval?
> >
> > I think it makes more sense to give both generator and period in
> the
> > same units, whether those are cents or octaves.
>
> But if we're talking about the naming system where we give a
single
> fraction (with GCD of numerator and denominator being the number
of
> periods per octave) then I believe it should be a semi-convergent
of
> generator/period, not generator/1200-cents * periods_per_octave.

Woops! I got that all frack-to-bunt. I think I meant it should be a
(semi)convergent of generator/period, with its denominator mutiplied
by periods_per_octave, rather than a (semi)convergent of
generator/1200_cents.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> But if we're talking about the naming system where we give a single
> fraction (with GCD of numerator and denominator being the number of
> periods per octave) then I believe it should be a semi-convergent of
> generator/period, not generator/1200-cents * periods_per_octave.

I presume you don't mean a semiconvergent of the generator divided by
the period?

🔗monz <monz@attglobal.net>

hi Gene (and Dave),

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

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

>
> > But if we're talking about the naming system where we
> > give a single fraction (with GCD of numerator and denominator
> > being the number of periods per octave)then I believe it
> > should be a semi-convergent of generator/period, not
> > generator/1200-cents * periods_per_octave.
>
> I presume you don't mean a semiconvergent of the generator
> divided by the period?

no Gene, he means to describe the format of the notation
as: the generator as a fraction of the period.

-monz

🔗monz <monz@attglobal.net>

hi Gene,

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

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Joe's web page says ``It can be seen that the lowest
> > overall error in the 11-limit is given by meantones
> > in the area of 3/13-comma to 136edo (in the version of
> > 136edo where the "5th" is mapped to 79 degrees).''
> >
> > Joe, this needs to be corrected to make clear that
> > it is assuming the huygens mapping; another and equally
> > valid choice is the meantone/meanpop mapping.
>
> On the meantone page, I think the error for both meanpop
> and huygens should be graphed; comparing them on a graph
> of just the 11-limit versions would be instructive.

i had put those graphs up right after you posted this;
now i've made them into a nice mouse-over applet.

http://tonalsoft.com/enc/meantone.htm#11-map

-monz