JI and Hindemith

Hi all, I'm currently reading Hindemith's first "Craft" book, where he's talking about his theories of harmonic construction and classification. I was rather shocked to find that he basically derives the 12-tET scale from JI principles -- in fact, that the 5-limit just chromatic scale that he derives is identical to the one that Partch derives in Genesis (ignoring the F-sharp/G-flat conundrum, which Partch resolves arbitrarily and which Hindemith politely ignores). It makes me wonder, ultimately, if Hindemith's "Craft" couldn't be extrapolated to a higher prime limit.
Of course, life gets difficult right away, since any JI system is going to provide a much higher number of unique intervals than 12-tET. Hindemith classifies the 12-tET intervals in terms of their relative dissonance -- and since he's only got 11 intervals to classify (not counting octave and unison), it's a short list. Then, he classifies all harmonies according to their interval content. I can't even imagine what an interval list for Partch's Monophonic Fabric would look like... and then you have to take into account false consonances, etc. Then, the chord classification scheme that follows would be monumental... and probably not worth the time...
It seems like some sort of Grand Unifying Theory is possible, but it would take some serious man-hours to find... Eric

hi Eric,

--- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
wrote:

> Hi all,
>
> I'm currently reading Hindemith's first "Craft" book,
> where he's talking about his theories of harmonic construction
> and classification. I was rather shocked to find that he
> basically derives the 12-tET scale from JI principles --
> in fact, that the 5-limit just chromatic scale that he
> derives is identical to the one that Partch derives in
> Genesis (ignoring the F-sharp/G-flat conundrum, which
> Partch resolves arbitrarily and which Hindemith politely
> ignores).

i'm surprised that you're familiar with Partch's _Genesis_
and didn't already know about Hindemath's derivation of
12-et -- Partch describes it on p. 420-22 and even gives
a diagram!

(unless Partch only put it in the 2nd edition and you
read the 1st ... which i'd guess is unlikely.)

> It makes me wonder, ultimately, if Hindemith's "Craft"
> couldn't be extrapolated to a higher prime limit.
>
> Of course, life gets difficult right away, since
> any JI system is going to provide a much higher number
> of unique intervals than 12-tET. Hindemith classifies
> the 12-tET intervals in terms of their relative dissonance
> -- and since he's only got 11 intervals to classify
> (not counting octave and unison), it's a short list.
> Then, he classifies all harmonies according to their
> interval content. I can't even imagine what an interval
> list for Partch's Monophonic Fabric would look like...

again, you don't have to imagine it ... it's in _Genesis_!!

Appendix 1, which covers p 461-63 of the 2nd edition,
is a list in ascending order of "Cents Values of Intervals",
subtitled:

"Including the 340 Intervals Narrower than a 2/1
Found in the Monophonic Fabric"

> and then you have to take into account false consonances,
> etc. Then, the chord classification scheme that follows
> would be monumental... and probably not worth the time...
> It seems like some sort of Grand Unifying Theory is
> possible, but it would take some serious man-hours to find...
>
> Eric

not quite sure what you mean about a "Grand Unifying Theory",
but i suspect that it's very similar to what i was working
on 20 years ago. at that time, i constructed 29
Tonality Diamonds centered on each of the notes in Partch's
11-limit Expanded Tonality Diamond. i was using them to
experiment with pieces in strict JI which could modulate.

playing with those 29 Tonality Diamonds led me not too
long after into constructing the earliest prototypes of
my lattice-diagrams, which i called "matrices" (until i
learned of the work of others -- Johnston, Fokker, Wilson
-- who all called them "lattices").

then, still not knowing about Wilson's work, i spent the
next several years wracking my brain over how to portray
structures of >3 dimensions in a diagram, finally succeeding
in 1998.

there's been a lot of work posted here and on tuning-math
over the last few years, which concerns the math and
visualization of multidimensional structures representing
tuning systems. and very recently there's been a lot of
progress on learning how JI and temperaments intersect.

maybe if i get a chance i'll make a lattice of Hindemith's
5-limit JI explanation of 12-et ... and then some
higher-limit reinterpretations of it.

-monz

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

> hi Eric,
>
>
> --- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
> wrote:
>
> > Hi all,
> >
> > I'm currently reading Hindemith's first "Craft" book,
> > where he's talking about his theories of harmonic construction
> > and classification. I was rather shocked to find that he
> > basically derives the 12-tET scale from JI principles --
> > in fact, that the 5-limit just chromatic scale that he
> > derives is identical to the one that Partch derives in
> > Genesis (ignoring the F-sharp/G-flat conundrum, which
> > Partch resolves arbitrarily and which Hindemith politely
> > ignores).
>
>
>
> i'm surprised that you're familiar with Partch's _Genesis_
> and didn't already know about Hindemath's derivation of
> 12-et -- Partch describes it on p. 420-22 and even gives
> a diagram!
>
> (unless Partch only put it in the 2nd edition and you
> read the 1st ... which i'd guess is unlikely.)
>
>
>
>
>
>
>
>
> > It makes me wonder, ultimately, if Hindemith's "Craft"
> > couldn't be extrapolated to a higher prime limit.

in fact, Partch says that at another point Hindemith implies
these between-degree ratios:

C-Db-D-Eb-E
16/15 17/16 18/17 19/18

which is basically the reverse of Ganassi's tuning:

http://tonalsoft.com/enc/index2.htm?../monzo/ganassi/ganassi.htm

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> hi Eric,
>
>
> --- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
> wrote:
>
> > Hi all,
> >
> > I'm currently reading Hindemith's first "Craft" book,
> > where he's talking about his theories of harmonic construction
> > and classification. I was rather shocked to find that he
> > basically derives the 12-tET scale from JI principles --
> > in fact, that the 5-limit just chromatic scale that he
> > derives is identical to the one that Partch derives in
> > Genesis (ignoring the F-sharp/G-flat conundrum, which
> > Partch resolves arbitrarily and which Hindemith politely
> > ignores).
>
>
>
> i'm surprised that you're familiar with Partch's _Genesis_
> and didn't already know about Hindemath's derivation of
> 12-et -- Partch describes it on p. 420-22 and even gives
> a diagram!
>
> <snip>
>
> maybe if i get a chance i'll make a lattice of Hindemith's
> 5-limit JI explanation of 12-et

done.

http://tonalsoft.com/enc/index2.htm?hindemith-12et-5limit-ji.htm

my scanner isn't working ... would someone be kind enough
to scan Partch's diagram and email it to me?

-monz

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

> http://tonalsoft.com/enc/index2.htm?hindemith-12et-5limit-ji.htm

It seems to me all that is being said here is that

(1) 12-et can be described as a tempering of The Malcolm Monochord or
its inverse, New Albion.

(2) The fundamental theorem of arithmetic--that any positive rational
number can be uniquely factored into primes--is really true.

There's also a suggestion that Hindemith is willing, Peter Sault
style, to switch between Malcolm and New Albion. Point number one is
of course true for any epimorphic 12-scale, and point number two
removes any necessity of showing that a rational number is the product
of primes in any particular case, since this is always true.

--- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
wrote:
> Hi all,
>
> I'm currently reading Hindemith's first "Craft" book, where
he's talking
> about his theories of harmonic construction and classification. I
was
> rather shocked to find that he basically derives the 12-tET scale
from JI
> principles -- in fact, that the 5-limit just chromatic scale that
he derives
> is identical to the one that Partch derives in Genesis (ignoring
the
> F-sharp/G-flat conundrum, which Partch resolves arbitrarily and
which
> Hindemith politely ignores).

[snip]

I don't have either of these books available at the moment...
could you please explain what this conundrum is?

thanks,
Jeff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:

> --- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
> wrote:
>
> > Hi all,
> >
> > I'm currently reading Hindemith's first "Craft"
> > book, where he's talking about his theories of harmonic
> > construction and classification. I was rather shocked
> > to find that he basically derives the 12-tET scale
> > from JI principles -- in fact, that the 5-limit
> > just chromatic scale that he derives is identical
> > to the one that Partch derives in Genesis (ignoring
> > the F-sharp/G-flat conundrum, which Partch resolves
> > arbitrarily and which Hindemith politely ignores).
>
> [snip]
>
> I don't have either of these books available at the moment...
> could you please explain what this conundrum is?
>
> thanks,
> Jeff

i've never read Hindemith's _Craft of Musical Composition_
(a lack that i really ought to remedy), so i can only
go by Partch's diagram.

according to Partch, assuming "C" as 1/1, Hindemith invokes
both 45/32 (F#) and 64/45 (Gb) for the 6th degree of the
12-tone scale.

if you look at the lattice-diagram on my webpage, you'll
see 13 orange cubes on the lattice and not 12.

http://tonalsoft.com/enc/index2.htm?hindemith-12et-5limit-ji.htm

on my the bingo-card page, i made a lattices for several ETs
which show a tiling where all the 5-limit representations
are as close as possible (in "taxicab metric", i.e., steps
along the 3- and 5-axes) to 1/1.

http://tonalsoft.com/enc/index2.htm?bingo.htm&12et-hindemith

on the 12-et tiling, all degrees of 12edo except for the 6th
are either 1 or 2 taxicab steps away from 1/1.

45/32 and 64/45, which are both represented by 6 degrees
of 12edo, are also both 3 steps away.

so using the taxicab criteria, one must make an arbitrary
decision which of the two falls within the 12-et
periodicity-block. choosing 45/32 gives the Malcolm monochord,
and choosing 64/45 gives New Albion.

-monz

This is just my opinion, but I think Hindemith's chords sound awful,
compared to, for instance, Stravinky's sense of vertical sonority.
(You can hear it even in his (ie Igor's) serial music.) How does
Hindemith's theory of dissonance rate? Kelly

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@yahoogroups.com, "Eric T Knechtges" <knechtge@m...>
> wrote:
> > Hi all,
> >
> > I'm currently reading Hindemith's first "Craft" book, where
> he's talking
> > about his theories of harmonic construction and classification.
I
> was
> > rather shocked to find that he basically derives the 12-tET scale
> from JI
> > principles -- in fact, that the 5-limit just chromatic scale that
> he derives
> > is identical to the one that Partch derives in Genesis (ignoring
> the
> > F-sharp/G-flat conundrum, which Partch resolves arbitrarily and
> which
> > Hindemith politely ignores).
>
> [snip]
>
> I don't have either of these books available at the moment...
> could you please explain what this conundrum is?
>
> thanks,
> Jeff

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

[snip]
> i've never read Hindemith's _Craft of Musical Composition_
> (a lack that i really ought to remedy), so i can only
> go by Partch's diagram.
>
> according to Partch, assuming "C" as 1/1, Hindemith invokes
> both 45/32 (F#) and 64/45 (Gb) for the 6th degree of the
> 12-tone scale.
>
> if you look at the lattice-diagram on my webpage, you'll
> see 13 orange cubes on the lattice and not 12.
>
> http://tonalsoft.com/enc/index2.htm?hindemith-12et-5limit-ji.htm
>
>
> on my the bingo-card page, i made a lattices for several ETs
> which show a tiling where all the 5-limit representations
> are as close as possible (in "taxicab metric", i.e., steps
> along the 3- and 5-axes) to 1/1.
>
> http://tonalsoft.com/enc/index2.htm?bingo.htm&12et-hindemith
>
> on the 12-et tiling, all degrees of 12edo except for the 6th
> are either 1 or 2 taxicab steps away from 1/1.
>
> 45/32 and 64/45, which are both represented by 6 degrees
> of 12edo, are also both 3 steps away.
>
> so using the taxicab criteria, one must make an arbitrary
> decision which of the two falls within the 12-et
> periodicity-block. choosing 45/32 gives the Malcolm monochord,
> and choosing 64/45 gives New Albion.
>
>
>
> -monz

Thanks for the information, Monz. I looked at the diagrams briefly
yesterday, but since I am away from home for the next few days, I
will study them in more detail when I get back. (I'm in the San Diego
area, and as I guess you know, the weather is too nice to stay
inside working at the computer...:) )

Of course, one question that springs to mind is why use the
taxicab criteria then?

--Jeff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:

> Thanks for the information, Monz. I looked at the diagrams
> briefly yesterday, but since I am away from home for the next
> few days, I will study them in more detail when I get back.
> (I'm in the San Diego area,

really?! then we should hang out some time!

> and as I guess you know, the weather is too nice to stay
> inside working at the computer...:) )

yeah ... unfortunately i miss a lot of that beautiful
weather because i *am* inside at the computer all day
and night! :(

> Of course, one question that springs to mind is why use the
> taxicab criteria then?

the important thing to remember is that when you outline
a periodicity-block with unison-vectors as its boundaries,
those boundaries can be moved around on the lattice.

at certain places, the boundaries will lie exactly on
a pair of points. i don't know how to validly say this
mathematically, since a point has no dimensions, but
it's something like imagining that half of each of those
points lies within the periodicity-block, and the other
half lies outside it.

so all you have to do is move the bounding unsion-vectors
slightly, and you get one of those points completely
within the periodicity-block and the other one completely
outside it.

i realize i didn't really answer your question "why
use the taxicab metric in the first place" ... i'll
leave that to someone else who can give a good answer.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
>
> > Thanks for the information, Monz. I looked at the diagrams
> > briefly yesterday, but since I am away from home for the next
> > few days, I will study them in more detail when I get back.
> > (I'm in the San Diego area,
>
>
>
> really?! then we should hang out some time!
>

That might be fun...then I'd actually meet in person someone
on this list! But I'm afraid that will have to wait until the
next time I come out, since I'm only here a couple more days and
don't have much free time. :(

[snip]

> the important thing to remember is that when you outline
> a periodicity-block with unison-vectors as its boundaries,
> those boundaries can be moved around on the lattice.
>
> at certain places, the boundaries will lie exactly on
> a pair of points. i don't know how to validly say this
> mathematically, since a point has no dimensions, but
> it's something like imagining that half of each of those
> points lies within the periodicity-block, and the other
> half lies outside it.
>
> so all you have to do is move the bounding unsion-vectors
> slightly, and you get one of those points completely
> within the periodicity-block and the other one completely
> outside it.

Yeah, its things like that that make me a little hesitant to
really embrace periodicity blocks and lattices...

>
> i realize i didn't really answer your question "why
> use the taxicab metric in the first place" ... i'll
> leave that to someone else who can give a good answer.
>

This is another thing that makes me hesitant... no one seems
to ever give a good answer. I guess I'll just have to keep
fooling around with music theory and maybe someday it will all
make sense...

--Jeff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:

> > i realize i didn't really answer your question "why
> > use the taxicab metric in the first place" ... i'll
> > leave that to someone else who can give a good answer.
> >
>
> This is another thing that makes me hesitant... no one seems
> to ever give a good answer. I guess I'll just have to keep
> fooling around with music theory and maybe someday it will all
> make sense...

Paul Erlich should really be the one to answer this, but he is in
seclusion. The point about using the weighted taxicab distance of the
Tenney metric is that it sorts things according to increasing
complexity in a way which makes sense. It is equivalent to using the
product of the numerator and denominator to determine how complex a
ratio is, and that has an obvious attraction as a system.

In the 5-limit, within an octave, for example, Tenney height (the
product of the numerator and denominator) sorts things as 1, 2, 3/2,
4/3, 5/3, 5/4, 6/5, 8/5, 9/5, 9/8, 10/9, 15/8, 16/9, 16/15 ... The
ordering seems quite sensible.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Paul Erlich should really be the one to answer this, but he is in
> seclusion. The point about using the weighted taxicab distance of
the
> Tenney metric is that it sorts things according to increasing
> complexity in a way which makes sense. It is equivalent to using
the
> product of the numerator and denominator to determine how complex a
> ratio is, and that has an obvious attraction as a system.
>
> In the 5-limit, within an octave, for example, Tenney height (the
> product of the numerator and denominator) sorts things as 1, 2,
3/2,
> 4/3, 5/3, 5/4, 6/5, 8/5, 9/5, 9/8, 10/9, 15/8, 16/9, 16/15 ... The
> ordering seems quite sensible.

Thanks, Gene.
__jeff