Mean tone

Hey Jeff,

> Hey neat! Li'l Miss' Scale Oven now has a meantone editor too
> - just added it recently and have been putting on the final
> touches.

Great. Something of a coincidence. Do I guess right that
you've added an option for stretched octave meantone scales?

I've tried out your stretched mean tone scale in FTS just
to hear what it sounds like and look forward to doing
some improvising in it.

Robert

Hey Jeff,

> You know me well enough to know that that's true!
>
> Fight the octave!

You'll prob. be pleased to hear that I've followed your
example and added an option to make non octave mean-tones
to FTS.

Also another idea I just had today - instead of just having a fraction
of a syntonic comma as the only option, I've put a
drop list of commas there, so you can have half a septimal
comma, or a third of a undecimal diesis or whatever -
also you can enter your own comma to work with.

The nice thing about that is that it doesn't clutter up
the dialog much - just changed a text field into a
drop list - then I added the amount of the comma at the end
like this:

char *szMeanToneCommas[]=
{
"Syntonic comma 81/80"
,"Septimal comma 64/63"
,"Undecimal diesis 33/32"
,"Pythagorean comma 3^12/2^7"
,"Kleisema 5^6/3^5"
,"Skhisma 5*2^13/3^8"
}

so then when one of these is selected you can just read the comma
from the end of the string (going back to just after the first
non numerical & !='/' character), and then user can also edit
the text to make their own commas, and if they follow the
same format with the comma at the end, can include
its name there as well.

Those who are making mean tone scales with the standard syntonic comma
needn't even look at the drop list, and will just see

Syntonic comma 81/80
which reminds one of the amount of the comma .

Robert

--- In MakeMicroMusic@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hey Jeff,
>
> > You know me well enough to know that that's true!
> >
> > Fight the octave!
>
> You'll prob. be pleased to hear that I've followed your
> example and added an option to make non octave mean-tones
> to FTS.
>
> Also another idea I just had today - instead of just having a
fraction
> of a syntonic comma as the only option, I've put a
> drop list of commas there, so you can have half a septimal
> comma, or a third of a undecimal diesis or whatever -
> also you can enter your own comma to work with.

those would make even more sense for constructing scales that are
actually *in* the linear temperament you're applying the measurement
to.

for example, fractions of a syntonic comma are the quantity of
interest from a meantone tuning because the syntonic comma is
precisely what is tempered out in the tuning itself. you have to
distribute exactly one syntonic comma out of the intervals in the
tuning, so how do you do it? however you do it, you have to divide
that "one" into "fractions". hence the conventional
nomenclatures/constructions.

fractions of a schisma are what are relevant for schismic tunings and
scales (usually 12, 17, 29, 41, or 53 notes).

fractions of a kleisma are what are relevant for kleismic tunings and
scales (see, for example, my post on hanson today on the tuning list,
and http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm)

and so on. the same goes for all 5-limit linear temperaments, or
those based on any two prime numbers (besides the interval of
equivalence) -- this could for example be {3,7} instead of
{2,7} . . . i believe manuel has implemented a large number of these
linear temperaments into scala, right manuel?

-paul, registered user of fts :]

Hi Paul,

The way I was thinking about it is just that you make the
mean tone scale by contracting fifths by a quarter of a comma
in order to get the 5/4 pure. So if one contracted by
fractions of other commas, maybe you'd get the 7/4 pure
or the 11/9 etc.

So e.g. if you want 7/4 pure you want to contract the 16/9
by a septimal comma, so you want to contract 4/3 by half
a septimal comma, so that means you want to expand 3/2
by half a septimal comma, giving:

209.1 8/7 253.2 64/49 484.4 512/343 715.6 4096/2401 946.8 7/4 1178.0 2/1

Similarly if you want the 11/9 pure, you want to expand the 32/27
by a undecimal comma, so you want to contract the 3/2s
by a third of a undecimal comma giving

1189.0 168.4 11/9 336.8 515.8 162/121 684.2 673.6 18/11 1032.0 1021.0 2/1

I think those make some sense do they not?

However, I imagine the Kleisma and Skhisma comma probably don't make
so much sense here as 3^8 and 3^5 are quite remote in the
circle of fifths and not particularly linked to the circle of
12 that generates the pythagorean twelve tone scale...

Particularly, they wouldn't lead to meantone scales with interesting
low number ratios in them which I suppose is the point of interest here.

The Pythagorean comma makes sense because you get equal temperament
by tempering by a twelth of a pythagorean comma, so I'll leave that in.

I've made the octave stretch so that it is by a multiple of the comma,
so that means one can do that 1/7th comma octave stretch, and
1/7th comma fifth contraction you described.

Thanks,

Robert

--- In MakeMicroMusic@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> The way I was thinking about it is just that you make the
> mean tone scale by contracting fifths by a quarter of a comma
> in order to get the 5/4 pure. So if one contracted by
> fractions of other commas, maybe you'd get the 7/4 pure
> or the 11/9 etc.
>
> So e.g. if you want 7/4 pure you want to contract the 16/9
> by a septimal comma, so you want to contract 4/3 by half
> a septimal comma,

yup, this is what happens in 'paultone' tuning, margo and i have
posted a lot about this.

> so that means you want to expand 3/2
> by half a septimal comma, giving:
>
> 209.1 8/7 253.2 64/49 484.4 512/343 715.6 4096/2401 946.8 7/4
1178.0 2/1

right, but this is not normally considered meantone, as the fifths
are *widened* rather than *narrowed*.

> Similarly if you want the 11/9 pure, you want to expand the 32/27
> by a undecimal comma, so you want to contract the 3/2s
> by a third of a undecimal comma giving
>
> 1189.0 168.4 11/9 336.8 515.8 162/121 684.2 673.6 18/11
>1032.0 1021.0 2/1

here, since the fifth is less than 4/7 octave, the 12-tone scale
comes up out of order. pretty neat, though hardly anyone would call
it meantone . . . this temperament is in fact defined by the comma
obtained by dividing 11/9 by 32/27, namely 33:32.

> I think those make some sense do they not?

yes they make perfect sense, and are examples of linear temperaments
(not meantone in principle, though), in particular they are examples
of the subset of linear temperaments where the ~3:2 ("perfect fifth")
is the generator.

here's a catalogue of linear temperaments:
http://x31eq.com/catalog.htm

> However, I imagine the Kleisma and Skhisma comma probably don't make
> so much sense here as 3^8 and 3^5 are quite remote in the
> circle of fifths and not particularly linked to the circle of
> 12 that generates the pythagorean twelve tone scale...

the schisma makes perfect sense here (that is, tempering a chain of
fifths) -- 1/8-schisma and 1/9-schisma temperaments have been
proposed time and again and are particularly suited to music that
exploited the schisma, mainly written between 1420 and 1480 in
europe, and perhaps earlier in the arabic world.

but the kleisma doesn't make sense here -- the generator of kleimsic
temperament is the minor third, not the perfect fifth. so this falls
into the larger category of linear temperaments in general, though
not ones generated by the fifth. typically the scale would have 11
notes, instead of 12, per octave:

http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm

gene has an algorithm for determining the generator of any linear
temperament, you might want to get into it with us on the tuning-math
list.

> Particularly, they wouldn't lead to meantone scales with
interesting
> low number ratios in them which I suppose is the point of interest
here.

certainly not meantone, but they do indeed lead to scales with
interesting low number ratios. but you're right in that the more
complex the comma, the more notes you'll need in the scale to exploit
it.

if you're using more than two primes in the ratios you wish to
approximate (besides the equivalence-prime, usually 2), then you need
more than one comma to define your linear temperament. for example,
miracle temperament, seen as approximating the full 7-limit, can be
defined by the set of commas 225:224 and 2401:2400 . . . these are
topics for tuning-math . . .