Tuning knowledge again

Hey Tom Dent...easy there, feller...I think you may have overreacted a bit to my post about the lack of knowledge I see in musicians relating to tuning, and why there are 12 tones in an octave. When I said that nobody in my audiences knew a thing about tuning, I was trying to make a point...I wasn't expecting a scientific dissertation on tunings when I asked why there are 12 eq tones in an octave; I was sort of fishing, to see just who knows what about anything as far as tuning goes, and if anyone had come up with some sort of answer that was remotely in the ballpark, that would have been super cool. But, as I had, sadly, expected, nobody had a clue...and if they had, and wanted to chat about it, that would have been fine with me, as I like interplay when I do a class. Actually, the answer I was sort of hoping to hear was: there's 12 eq notes because Europeans were looking for a way to find a tuning system that would apply to music that modulated through a lot of chord changes, without sounding TOO much out of tune...and that's why 12 eq came into being, from what I've learned over the years.
And yes, there are certainly other 12 tone systems, meantone and well being foremost...I know that, but since the infamous 12eq is the dominant system these days, it is a good place to start when talking about temperaments. I'm pretty informal, and tend to come from a more "street" take on art, if you will...that's the way I am, and it works pretty well for me in most situations, especially when talking to people who know very little about music theory, or tunings.
Anyway, your comments are appreciated, it's good to get different viewpoints, it keeps one focused on why they believe the way they do...best...Hstick
www.microstick.net

--- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...>
wrote:

> <snip>
>
> Actually, the answer I was sort of hoping to hear was:
> there's 12 eq notes because Europeans were looking for a
> way to find a tuning system that would apply to music that
> modulated through a lot of chord changes, without sounding
> TOO much out of tune...

yes, this is also an important reason for the acceptance
of 12-edo. more could be said on this point ...
inspired by this discussion, i've been adding some
historical info to the new-and-improved Encyclopedia
12-edo webpage (it will be online in a couple of weeks).

In connection with Neil's comment, it is worth pointing out
that beginning with Mozart, and continuing more strongly with
Beethoven and Schubert, 12-edo's membership in the augmented
temperament family began to be exploited more and more;
the signpost of this technique is the use of pairs of
enharmonically equivalent notes (i.e., notes which are
"spelled" differently but have the same pitch in 12-edo)
as common-tones. Then during later eras, Scriabin,
Stravinsky, and other composers began exploiting 12-edo's
membership in the diminished family with the use of the
octatonic diminished scale.

As composers sought new means of organizing musical pitches,
the desire grew to explore these "new" temperament families
and to modulate from one to another, a procedure which is
facilitated by 12-edo's membership in so many different
families.

(the smaller the cardinality of an EDO, the more unison-vectors
it will temper out, and thus the more families it will
belong to. but as already noted, 12 is the smallest cardinality
which provides recognizable approximations to the diatonic scale
and major/minor/modal system.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> In connection with Neil's comment, it is worth pointing out
> that beginning with Mozart, and continuing more strongly with
> Beethoven and Schubert, 12-edo's membership in the augmented
> temperament family began to be exploited more and more;
> the signpost of this technique is the use of pairs of
> enharmonically equivalent notes (i.e., notes which are
> "spelled" differently but have the same pitch in 12-edo)
> as common-tones.

i should have specified that the enharmonically equivalent
pairs of notes are a diesis apart, ratio 128/125 =
2,3,5-monzo [7 0, -3> = ~41 cents, in JI; thus, in any
tuning of the augmented family (among the others are
9, 15, 18, and 21-edo) the augmented-7th becomes the same
pitch as the octave.

> Then during later eras, Scriabin,
> Stravinsky, and other composers began exploiting 12-edo's
> membership in the diminished family with the use of the
> octatonic diminished scale.

and the unison-vector tempered out here is the "major diesis"
of ratio 648/625 = 2,3,5-monzo [3, 4, -4 > = ~62.5 cents in JI;
thus, in any tuning of the diminished family (among the
others are 8, 16, 24, and 28-edo) the diminished-9th is
the same pitch as the octave.

-monz

Neil wrote, in reply to Tom:
________________________________________________________________________
Date: Thu, 21 Apr 2005 12:07:57 -0600
From: "Neil Haverstick" <microstick@...>
Subject: Tuning knowledge again

.... Actually, the answer I was sort of hoping
to hear was: there's 12 eq notes because Europeans were looking for a way to
find a tuning system that would apply to music that modulated through a lot
of chord changes, without sounding TOO much out of tune...and that's why 12
eq came into being, from what I've learned over the years.
________________________________________________________________________
[YA] And that's exactly what they taught me in school -
in the eighth year, to be exact - that although pure ratios
or a system with "average" thirds (meantone) worked very
well while modulation was limited, the increasing tendencies
of late Classical composers, eg Beethoven -
a) to modulate to ever more remote keys, and
b) to use increasing chromaticism,
both pushed towards using equal semitones.

That's if I remember correctly (and I have an excellent
memory for ideas, if not for names & dates ...)

Regards,
Yahya

--
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Monz,

You wrote much concerning 12-EDO's membership in so many
tuning families, including the augmented and diminished, and
concluded with:

"... 12 is the smallest cardinality which provides recognizable
approximations to the diatonic scale and major/minor/modal
system."

Perhaps 12-EDO truly is a "miracle" temperament ... :-)

After 12, what are the next-larger cardinalities n that give
us in n-EDO all these benefits of 12-EDO:

1. recognizable approximations to ecclesiastical modes
2. recognizable approximations to major diatonic scale
3. recognizable approximations to minor diatonic scale
4. a useful augmented scale
5. a useful diminished scale
?

Regards,
Yahya

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hi Yayha,

--- In tuning@yahoogroups.com,
"Yahya Abdal-Aziz" <yahya@m...>
wrote:
>
> Monz,
>
> You wrote much concerning
> 12-EDO's membership in so
> many tuning families,
> including the augmented and
> diminished, and concluded
> with:
>
> "... 12 is the smallest
> cardinality which provides
> recognizable approximations
> to the diatonic scale and
> major/minor/modal system."
>
> Perhaps 12-EDO truly is a "miracle" temperament ... :-)
>
> After 12, what are the next-larger cardinalities n that give
> us in n-EDO all these benefits of 12-EDO:
>
> 1. recognizable approximations to ecclesiastical modes
> 2. recognizable approximations to major diatonic scale
> 3. recognizable approximations to minor diatonic scale
> 4. a useful augmented scale
> 5. a useful diminished scale
> ?

well, yes ... and there's even more!

the families we've discussed here lately to which
12-edo belongs are:

. meantone
. augmented
. diminished

but 12-edo also belongs to the following temperament families:

. schismic
. aristoxenean
. diaschismic

and there are others!

and i know that you said "miracle" in jest ... but just so
that newbie readers are not led astray: no, 12-edo is *not*
a member of the "miracle" temperament family. the name
"miracle" is specifically meant to indicate that tunings of
that family give good approximations to 11-limit JI, and
while 12 is great for 3-limit and decent for 5, it's not so
good for 7 and terrible for 11.

the new version of the Tonalsoft Encyclopedia entry
for 12-edo includes a 5-limit bingo-card lattice which
shows, outlined in green, various different periodicity-blocks
which may be constructed in 12-edo. i've put it here as well:

/tuning/files/monz/12edo_3-
5-space_bingo_pbs.gif

delete the line-break, OR

http://tinyurl.com/7w9sk

as for which other temperaments can meet your criteria ...
i leave that to others to discover ... maybe Gene knows.

-monz