beating meantones

hi,

something for the mathheads: the "gleichschwebend" thread (plus the fact that it was a good excuse to learn how to use a spreadsheet) made me have a closer look at the meantones. i understand that e.g. in two part passages it is not too nice to have one pure third/sixth and everything else beating, so you might want to find an intermediate step between third and fourth comma meantones. my math skills made me calculate the means from the fractions, (1/3+1/4)/2 being 7/24. this turned out to be the geometric division, smack in the middle of the logaritmical values, and don't know if this meantone flavor has ever been used. on the other hand, 2/7 meantone did exist. this is calculated from the integers, (3+4)/2, which have to be inverted again. this looks like two extra, not necessarily intuitive, steps to me. is that so, and why not?

2/7 is a harmonic mean of the two meantones, so there must be a "utonal" division also. how do i calculate this?

thanks,
klaus

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:

i understand that e.g. in two
> part passages it is not too nice to have one pure third/sixth and
> everything else beating, so you might want to find an intermediate
> step between third and fourth comma meantones.

One nice version of this is the Wilson meantone. If you pick a fifth
of size f>0 such that f^4 = 2f + 2, then 1-3/2, 1-5/4, 5/4-3/2 all
beat at the same rate.

my math skills made me
> calculate the means from the fractions, (1/3+1/4)/2 being 7/24.
this
> turned out to be the geometric division, smack in the middle of the
> logaritmical values, and don't know if this meantone flavor has
ever
> been used. on the other hand, 2/7 meantone did exist. this is
> calculated from the integers, (3+4)/2, which have to be inverted
> again. this looks like two extra, not necessarily intuitive, steps
to
> me. is that so, and why not?

2/7 is the mediant of 1/3 and 1/4; (1+1)/(3+4) = 2/7. This is, of
course, awfully easy to calculate!

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
>>... find an intermediate >>step between third and fourth comma meantones.
> > > One nice version of this is the Wilson meantone. If you pick a fifth > of size f>0 such that f^4 = 2f + 2, then 1-3/2, 1-5/4, 5/4-3/2 all > beat at the same rate.

(the unit being ratios, i.e. f~1.5 -- this took me some time to figure out.) thanks. my homework for tonight!

>>(1/3+1/4)/2 being 7/24. > > this > >>turned out to be the geometric division, smack in the middle of the >>logaritmical values, and don't know if this meantone flavor has >> ever been used. i goofed! it's close to the middle for the resulting fifth in cents, but it's the harmonic mean. the geometric mean is the root of the product.

on the other hand, 2/7 meantone did exist. this is
>>calculated from the integers, (3+4)/2, which have to be inverted >>again. this looks like two extra, not necessarily intuitive, steps >> to me. is that so, and why not?
> > > 2/7 is the mediant of 1/3 and 1/4; (1+1)/(3+4) = 2/7. This is, of > course, awfully easy to calculate!

but not at all what i was taught to do with fractions when i was at school. don't you know you NEVER add the denominators, and the enumerators only after normalizing everything to a common denominator? (well, i know it works and how it works. it's just contrary to my indoctrination.)

i thought i had also asked about the "utonal" version of the harmonic means. after a night's sleep, i think it's simply 1 minus the harmonic mean, in this case 17/24. the resulting fifth almost gives you 7-equal (major and minor thirds are 7 cents apart). next puzzle: 7, 17 and twice twelve -- is this a coincidence? (don't worry, i'll meditate on that myself.)

thanks,
klaus

on 7/28/03 12:49 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@z...> wrote:
>
> i understand that e.g. in two
>> part passages it is not too nice to have one pure third/sixth and
>> everything else beating, so you might want to find an intermediate
>> step between third and fourth comma meantones.
>
> One nice version of this is the Wilson meantone. If you pick a fifth
> of size f>0 such that f^4 = 2f + 2, then 1-3/2, 1-5/4, 5/4-3/2 all
> beat at the same rate.
>
> my math skills made me
>> calculate the means from the fractions, (1/3+1/4)/2 being 7/24.
> this
>> turned out to be the geometric division, smack in the middle of the
>> logaritmical values, and don't know if this meantone flavor has
> ever
>> been used. on the other hand, 2/7 meantone did exist. this is
>> calculated from the integers, (3+4)/2, which have to be inverted
>> again. this looks like two extra, not necessarily intuitive, steps
> to
>> me. is that so, and why not?
>
> 2/7 is the mediant of 1/3 and 1/4; (1+1)/(3+4) = 2/7. This is, of
> course, awfully easy to calculate!

Can this really be right? This would mean the mediant of 2/6 and 1/4 would
yield a different result (2+1)/(6+4) = 3/10. Can this "mediant" really
depend on the fraction being in lowest terms? Somehow this doesn't make
much intuitive sense.

I looked up mediant on monz's dictionary and it doesn't have such a formula.

Can someone clarify?

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Can this really be right? This would mean the mediant of 2/6 and
1/4 would
> yield a different result (2+1)/(6+4) = 3/10. Can this "mediant"
really
> depend on the fraction being in lowest terms?

It can and does. What's the problem?

> I looked up mediant on monz's dictionary and it doesn't have such a
formula.

http://mathworld.wolfram.com/Mediant.html