Pythagorean 3rds

Thanks to those who posted comments about the almost perfect 3rds that arise from the spiral of 5ths; and I'm still thinking about that phenomenon. I mean, who really knows why, but it is very interesting to me how the D# at 318 cents, and the Fb at 384 seemed to be generally overlooked in European tuning systems over the years. Yes, their ratios were pretty high up there, but in purely practical terms, since they are only 2 cents off the simple 6/5 and 5/4 ratios, it seems to me that these notes would have been sought after more. Two cents, when playing a guitar, for instance, is absolutely a non existant interval...the pitch on a guitar string changes just by the amount of pressure one puts when their finger presses down, so perhaps theoretically, 2 cents is meaningful, but not when one is actually playing.
And I know European musicians used the keyboard as a standard for tuning, overall, but even there 2 cents is nothing. I've often read that there is a deviation of several cents for most any acoustic instrument, so being precisely "in tune" is most likely impossible. Room temperature alone can make any acoustic axe go in and out of tune pretty easily. So, most ANY tuning is theoretical in that sense, you just get as close as you can at the moment, and hope it stays there. When I do clasical guitar gigs, I constantly need to retune, so while I'm playing, the pitch of my notes is often not exactly at a precise eq temp. I guess this isn't exactly news that will change the world, but it points up the difference between intellectual theory, and actual playing, in a simple way.
And, it's interesting...with fixed pitch instruments, like keyboards or the guitar family, it's very difficult, if not impossible, to adjust pitch while playing, whereas ouds, sarods, and violins can be tweaked while playing, just by the feel of where one puts their finger. The longer I study tuning theory, the deeper it gets, you won't get bored in this field....best...HHH

--- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...> wrote:
> Yes, their ratios were pretty high up there,
> but in purely practical terms, since they are only 2 cents off the
simple
> 6/5 and 5/4 ratios, it seems to me that these notes would have been
sought
> after more.

How many notes to an octave do you regard as practical on a guitar?
One way of doing this system is to use 53 equal, but that's a lot of
notes.

On 1/22/06, Gene Ward Smith <gwsmith@svpal.org> wrote:
> --- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...> wrote:
> > Yes, their ratios were pretty high up there,
> > but in purely practical terms, since they are only 2 cents off the
> simple
> > 6/5 and 5/4 ratios, it seems to me that these notes would have been
> sought
> > after more.
>
> How many notes to an octave do you regard as practical on a guitar?
> One way of doing this system is to use 53 equal, but that's a lot of
> notes.

Yes, this is the main problem. With meantone temperament, only 5
successive notes are needed to get a major or minor chord. With
schismatic temperament, you need 10 notes.

Keenan

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...>
wrote:

> > Yes, their ratios were pretty high up there,
> > but in purely practical terms, since they are only 2 cents
> > off the simple 6/5 and 5/4 ratios, it seems to me that
> > these notes would have been sought after more.
>
> How many notes to an octave do you regard as practical on a
> guitar? One way of doing this system is to use 53 equal, but
> that's a lot of notes.

I got a live in-person view of Neil's guitar collection
when i visited him in Colorado 2 weeks ago, and i'm still
amazed that he can play 31-edo guitars, let alone 34-edo.
I have one of Ivor Darreg's 31-edo guitars, and i find
the frets to be so close together that it's almost useless
for me to play anything but little experiments. But then,
i'm nowhere near the guitarist that Neil is.

Eduardo Sabat-Garibaldi plays an instrument he calls
the "dinarra" (sp?), which AFAIK is basically a guitar
with 53 frets per octave. He used to be a member here
... don't know if he still is.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Neil,

--- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...> wrote:
>
> Thanks to those who posted comments about the almost
> perfect 3rds that arise from the spiral of 5ths; and I'm
> still thinking about that phenomenon.

I have a webpage about Groven's 36-tone schismic (or
schismatic ... i think the jury is still out on which
of those two terms is preferred) temperament -- and
further down the page i mention Helmholtz's 24-tone
version which is a subset of Groven's. Both of these
tunings are based on tempering out the small ~2-cent
skhisma difference you're discussing here.

http://tonalsoft.com/monzo/groven/groven.htm

> I mean, who really knows why, but it is very interesting
> to me how the D# at 318 cents, and the Fb at 384 seemed
> to be generally overlooked in European tuning systems over
> the years. Yes, their ratios were pretty high up there,
> but in purely practical terms, since they are only 2 cents
> off the simple 6/5 and 5/4 ratios, it seems to me that these
> notes would have been sought after more.

Actually, according to Mark Lindley:

Lindley, Mark and Ronald Turner-Smith. 1993.
_Mathematical Models of Musical Scales: A New Approach_.
Orpheus-Schriftenreihe zu Grundfragen der Musik vol. 66,
Verlag für systematische Musikwissenschaft,
Bonn-Bad Godesberg, 308 pages.

... during the 1400s European compsosers did recognize
these pitch relationships and did begin using pythagorean
Gb, Db, and Ab as substitutes for quasi-JI F#, C#, and G#
in D, A, and E-major chords respectively, in their
keyboard compositions.

This was at a time when pythagorean tuning was still
the only "officially sanctioned" one ... Ramos

http://tonalsoft.com/monzo/ramos/ramos.htm

in 1482 was the first theorist of "modern" times to
describe a monochord division which used 5 (or indeed,
any prime higher than 3) as a factor.

The acceptance of 5-limit ratios was most likely
conditioned at least in part by the earlier acceptance
of the pythagorean schismic equivalents described by
Lindley.

And a somewhat of a digression, note that the
acceptance of 5-limit ratios and the resulting
recognition of the problems of commatic drift and
shift, led very rapidly to the introduction of
meantone temperament, which was already described
(non-mathematically, but the description is still
mostly pretty clear) as early as 1511 by Schlick.

-monz
http://tonalsoft.com
Tonescape microtonal music software

On 1/23/06, monz <monz@tonalsoft.com> wrote:
[snip]
> I have a webpage about Groven's 36-tone schismic (or
> schismatic ... i think the jury is still out on which
> of those two terms is preferred) temperament -- and
[snip]

My vote is definitely for "schismatic". The stem of the Greek word
"schisma" is "schismat-", so the adjective form is "schismatikos", or
"schismatic" in English.

That's also why the proper plural of "schisma" is "schismata".

Keenan

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Neil Haverstick" <microstick@m...>
wrote:
> > Yes, their ratios were pretty high up there,
> > but in purely practical terms, since they are only 2 cents off the
> simple
> > 6/5 and 5/4 ratios, it seems to me that these notes would have been
> sought
> > after more.
>
> How many notes to an octave do you regard as practical on a guitar?
> One way of doing this system is to use 53 equal, but that's a lot of
> notes.

Eduardo Sabat-Garibaldi and his students use 53 notes in a chain of 1/9-
schisma-narrowed fifths. Takes quite a while to adjust to this density
of frets, but ultimately the first fourth of the neck is certainly
playable; beyond that, you'll need particularly pointy callouses or
fingernails . . .

--- In tuning@yahoogroups.com, Keenan Pepper <keenanpepper@g...> wrote:
>
> On 1/23/06, monz <monz@t...> wrote:
> [snip]
> > I have a webpage about Groven's 36-tone schismic (or
> > schismatic ... i think the jury is still out on which
> > of those two terms is preferred) temperament -- and
> [snip]
>
> My vote is definitely for "schismatic". The stem of the Greek word
> "schisma" is "schismat-", so the adjective form is "schismatikos", or
> "schismatic" in English.

Thanks for explaining this, Keenan -- I never studied Greek!
So "schismatic" it is.

> That's also why the proper plural of "schisma" is "schismata".
>
> Keenan
>