Mathieu on minor thirds

When I first found my own voice inside a just minor triad, I couldn't
believe it was so--well, so (arggh! I can scarcely say the dreaded
word, but here it goes)--so...*happy*. There. We are told from the
beginning that minor is sad, the designated mode for angst and
funerals. Well, to be honest, the equal-tempered version of the minor
third is rather sad. The equal-tempered major third down from G to Eb
is too *wide*, making the minor third between C and Eb too *narrow*,
or flat. So piano minor is flat and sounds dull--the fire is out of
it. But minor thirds in just intonation, and the minor keys they
support, are swift and burning. They have the gypsy left in them, and
do some leaping kind of dance.

--W. A. Mathieu, Harmonic Experience

If he is talking about the 6/5, then i agree fully.

Gene Ward Smith wrote:

> When I first found my own voice inside a just minor triad, I couldn't
> believe it was so--well, so (arggh! I can scarcely say the dreaded
> word, but here it goes)--so...*happy*. There. We are told from the
> beginning that minor is sad, the designated mode for angst and
> funerals. Well, to be honest, the equal-tempered version of the minor
> third is rather sad. The equal-tempered major third down from G to Eb
> is too *wide*, making the minor third between C and Eb too *narrow*,
> or flat. So piano minor is flat and sounds dull--the fire is out of
> it. But minor thirds in just intonation, and the minor keys they
> support, are swift and burning. They have the gypsy left in them, and
> do some leaping kind of dance.
>
> --W. A. Mathieu, Harmonic Experience
>
>
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-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
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--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> If he is talking about the 6/5, then i agree fully.

He is; that's why it's sharp in 12-equal.

It occurs to me that maybe I'm not doomed just because Mathieu thinks
twelve notes of 5-limit 12-et define a lattice, because he also thinks
1 is a prime number. If I can correctly assume most musicians are not
going to follow this usage simply because Mathieu says so, maybe there
is hope for "lattice" also. I'd be interested to learn what people
here think a "lattice" ought to refer to, and whether twelve notes of
5-limit 12-et can be one.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> > If he is talking about the 6/5, then i agree fully.
>
> He is; that's why it's sharp in 12-equal.

Meaning 12-et is flat.

yes and 6/5 do sound happy. If one was going to use 5 limit intervals i
would still pick the 3 limit 32/27 for the minor third ( i think off the
cuff)

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> >
> > > If he is talking about the 6/5, then i agree fully.
> >
> > He is; that's why it's sharp in 12-equal.
>
> Meaning 12-et is flat.
>
>
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>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

Gene Ward Smith wrote:

> It occurs to me that maybe I'm not doomed just because Mathieu thinks
> twelve notes of 5-limit 12-et define a lattice, because he also thinks
> 1 is a prime number. If I can correctly assume most musicians are not
> going to follow this usage simply because Mathieu says so, maybe there
> is hope for "lattice" also. I'd be interested to learn what people
> here think a "lattice" ought to refer to, and whether twelve notes of
> 5-limit 12-et can be one.

I can't find where Mathieu says either of these things. Do you have references? The only problem I can see with his use of "lattice" is that he confuses it with "structure". My reference being:

"We should here differentiate between the terms `structure' and `lattice' which are often used synonymously. By `structure' we mean the actual pattern of the arrangement of atoms in space. The term 'lattice', although loosely used to denote the atoms in a crystal, should be reserved to describe the network of points which show the simple translation vectors on which a structure is based."
H.M.Rosenberg, "The Solid State",
Oxford University Press, 3rd Edition 1988, p.2

Translate that into musical terms, and cut him some slack for using "lattice" as a synonym for "structure", and I can't find any problems with Mathieu's usage. I can't find where he calls ET a lattice, but I wouldn't disagree with that.

Page 27 of Harmonic Experience clearly implies that 1 is *not* a prime number. Page 164 explicitly states that a lattice should strictly be infinite.

Graham

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:

> I can't find where Mathieu says either of these things. Do you have
> references?

On pp 64-65 we find "A *prime number* is a positive whole number that
has factors only of itself and one. There is an infinite array of
prime numbers, but only the first few (1, 2, 3, 5, 7, 11) have obvious
musical significance." This definition is clear, explicit, and wrong.

Later on he discusses music in the prime limits, which includes the
1-limit, where you bang away at a unison.

> Page 27 of Harmonic Experience clearly implies that 1 is *not* a prime
> number.

Page 27 says five is the next prime after three, and two the prime
before it. It does not say one is not a prime.

Page 164 explicitly states that a lattice should strictly be
> infinite.

Chapter 10 is called "The Five-Limit Lattice of Twelve Notes"; this is
also the chapter that misdefines prime numbers. On page 65 he says
"Later we extend the lattice outward in both dimensions, but so far we
have confined our field to a specific field of twelve notes. Hence a
'five-limit lattice of notes'". People who think vague, murky
defintions are good and precise ones bad might want to ponder the
implications. What is a "field" of notes? Is the specific "field" in
question something like a Fokker block (the Ellis duodene seems to fit
the discussion pretty well, and Ellis is someone he is quite familiar
with) or are we assuming temperament, so the lattice is rolled up into
a torus (donut?) We can't really know unless he tells us.

Gene Ward Smith wrote:

> On pp 64-65 we find "A *prime number* is a positive whole number that
> has factors only of itself and one. There is an infinite array of
> prime numbers, but only the first few (1, 2, 3, 5, 7, 11) have obvious
> musical significance." This definition is clear, explicit, and wrong.

Oh yes, you've got him there. That's what you get for taking math definitions from music theory texts.

> Page 27 says five is the next prime after three, and two the prime
> before it. It does not say one is not a prime.

It says "...the lower primes, three and two."

> Chapter 10 is called "The Five-Limit Lattice of Twelve Notes"; this is
> also the chapter that misdefines prime numbers. On page 65 he says
> "Later we extend the lattice outward in both dimensions, but so far we
> have confined our field to a specific field of twelve notes. Hence a
> 'five-limit lattice of notes'". People who think vague, murky
> defintions are good and precise ones bad might want to ponder the
> implications. What is a "field" of notes? Is the specific "field" in
> question something like a Fokker block (the Ellis duodene seems to fit
> the discussion pretty well, and Ellis is someone he is quite familiar
> with) or are we assuming temperament, so the lattice is rolled up into
> a torus (donut?) We can't really know unless he tells us.

Yes, this is confusing "lattice" with "structure". That's a common usage, though strictly incorrect (as he says).

I think "field" is like a measure word. It doesn't mean anything, but it has to be there for the sentence to be grammatical. Or perhaps it refers to the "rich field of possibilities" from p.61 where he shows the "lattice of 12 notes".

He means just intonation. He says at the top of that paragraph "Spines of fifths and spines of thirds form a /five-limit lattice/ in just intonation." This part of the book is all about just intonation. He doesn't get round to temperament until chapter 20.

Graham