Yahoo Tuning Groups Ultimate Backup tuning Fwd: Re: tuning books

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >The Structure of Recognizable Diatonic Tunings
> >Easley Blackwood

According to Blackwood, a recognizeable diatonic tuning has the
following characteristics:

(1) The fifth is between 4/7 and 3/5 of an octave, not inclusive.

(2) It tempers out 81/80 ("Theorem 44")

Hence, a recognizable diatonic tuning is a tuning *for the meantone
temperament*, so this means something more than just a tuning with a
fifth in that range. Blackwood recognizes the septimal interval
properties of meantone, so the definition can be taken to mean a
tuning for septimal meantone.

If h5 = <5 8 12 15| and h7 = <7 11 16 19| then a recognizable diatonic
tuning, from the above, is p*(h5/5)+q*(h7/7), where p+q=1 and p,q>0.
Blackwood instead puts things in terms of his parameter R; if h2=
h7-h5 = <2 3 4 4| then a recogizable diatonic tuning is proportional
to h2 + Rh5. When R=1 this is h7, and as R tends to infinity it
approaches being proportional to h5, so in terms of R, the
recognizable range is 1 to infinity.

Blackwood introduces the notation pR + q for meantone intervals, where
if u is a 7-limit interval, then p = h5(u) and q = h2(u); this
expresses every meantone interval uniquely. Since h2(2) = 2 and
h5(2)=5, the octave is represented by 5R+2, and hence the tuning of
the interval u by (pR+q)/(5R+2) (as a fraction of an octave; so 1200
times that amount in cents.) Since h2(3/2)=1 and h5(3/2)=3, the fifth
is 3R+1 in this notation, and so R corresponds to a fifth of f =
(3R+1)/(5R+2); solving for R in terms of f gives R = (2f-1)/(3-5f).

Aside from recognizable diatonic tunings, Blackwood also defines
acceptable diatonic tunings, which have R values from (roughly, at
least) R=1.5 to R=2.2. Subsituting these in the expression for the
fifth in terms of R, this is a fifth from 11/19 to 38/65, or about 1/3
comma to 1/52 comma meantone. However, Blackwood recongizes that 65
(and 53) is really a schismatic temperament system.

Mathematically speaking, Blackwood proves some elementary number
theory results (including the so-called "fundamental theorem of
arithmetic", or prime factorization) instead of merely citing them. He
uses "log" to mean base 2^(1/1200), or in other words cents, which
makes sense, but most unfortunately uses "2 exp x" to mean 2^x and not
e^x.

I don't agree with some of Blackwood's characterizations of intervals,
and particularly object to his characterization of 7/6. First he says
it has the sound of a minor third, but with a distinct character of
its own. Then he says it produces "an ambivalent reaction in a trained
musician", and later on claims it is "quite discordent", which I think
is baloney.

Blackwood discusses JI a little, talking about the comma pumps which
are likely to arise if you try to do meantone pieces in it, and how
these become exaggerated if you use 22 or 34 to approximate meantone.
The transformation of a meantone piece to a supermajor one doesn't
seem to have occurred to him, but he's not a fan of 9/7.

🔗Carl Lumma <ekin@lumma.org>

>>>> Which are the "big name" tunings books
>>>
>>> On the Sensations of Tone
>>> Hermann von Helmholtz
>>>
>>> An Elementary Treatise on Musical Intervals and Temperament
>>> R.H.M. Bosanquet, London, 1876.
>>> ed. Rudolph Rasch, The Diapason Press, Utrecht, 1987.
>>>
>>> Genesis of a Music
>>> Harry Partch
>>>
>>> The Structure of Recognizable Diatonic Tunings
>>> Easley Blackwood
>> //
>>
>> -Carl
>
>Jorgensen outta be on that list, no?

I aimed this list at historically-significant works which attempt
to extend (or at least explain) common-practice tuning systems,
and which are readable (not just references). I haven't read
Jorgensen, but I expect it would fail at least two of these goals.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>> >The Structure of Recognizable Diatonic Tunings
>> >Easley Blackwood
>
>According to Blackwood, a recognizeable diatonic tuning has the
>following characteristics:
>
>(1) The fifth is between 4/7 and 3/5 of an octave, not inclusive.
>
>(2) It tempers out 81/80 ("Theorem 44")
>
>Hence, a recognizable diatonic tuning is a tuning *for the meantone
>temperament*, so this means something more than just a tuning with a
>fifth in that range. Blackwood recognizes the septimal interval
>properties of meantone, so the definition can be taken to mean a
>tuning for septimal meantone.
>
>If h5 = <5 8 12 15| and h7 = <7 11 16 19| then a recognizable diatonic
>tuning, from the above, is p*(h5/5)+q*(h7/7), where p+q=1 and p,q>0.
>Blackwood instead puts things in terms of his parameter R; if h2=
>h7-h5 = <2 3 4 4| then a recogizable diatonic tuning is proportional
>to h2 + Rh5. When R=1 this is h7, and as R tends to infinity it
>approaches being proportional to h5, so in terms of R, the
>recognizable range is 1 to infinity.
>
>Blackwood introduces the notation pR + q for meantone intervals, where
>if u is a 7-limit interval, then p = h5(u) and q = h2(u); this
>expresses every meantone interval uniquely. Since h2(2) = 2 and
>h5(2)=5, the octave is represented by 5R+2, and hence the tuning of
>the interval u by (pR+q)/(5R+2) (as a fraction of an octave; so 1200
>times that amount in cents.) Since h2(3/2)=1 and h5(3/2)=3, the fifth
>is 3R+1 in this notation, and so R corresponds to a fifth of f =
>(3R+1)/(5R+2); solving for R in terms of f gives R = (2f-1)/(3-5f).
>
>Aside from recognizable diatonic tunings, Blackwood also defines
>acceptable diatonic tunings, which have R values from (roughly, at
>least) R=1.5 to R=2.2. Subsituting these in the expression for the
>fifth in terms of R, this is a fifth from 11/19 to 38/65, or about 1/3
>comma to 1/52 comma meantone. However, Blackwood recongizes that 65
>(and 53) is really a schismatic temperament system.
>
>Mathematically speaking, Blackwood proves some elementary number
>theory results (including the so-called "fundamental theorem of
>arithmetic", or prime factorization) instead of merely citing them. He
>uses "log" to mean base 2^(1/1200), or in other words cents, which
>makes sense, but most unfortunately uses "2 exp x" to mean 2^x and not
>e^x.
>
>I don't agree with some of Blackwood's characterizations of intervals,
>and particularly object to his characterization of 7/6. First he says
>it has the sound of a minor third, but with a distinct character of
>its own. Then he says it produces "an ambivalent reaction in a trained
>musician", and later on claims it is "quite discordent", which I think
>is baloney.
>
>Blackwood discusses JI a little, talking about the comma pumps which
>are likely to arise if you try to do meantone pieces in it, and how
>these become exaggerated if you use 22 or 34 to approximate meantone.
>The transformation of a meantone piece to a supermajor one doesn't
>seem to have occurred to him, but he's not a fan of 9/7.

Thanks for the summary, Gene. I don't have Blackwood's book, but
I crammed all I could in the New York public library once.

-Carl

Fwd: Re: tuning books

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>