Summary of optimal meantone tunings

Here is a summary of the meantone tunings (giving the size of fifth in
cents, the fraction of the syntonic comma by which the fifth is reduced, and
the first known advocate or reference to a TD posting by me from Brett
Barbaro's e-mail) that are optimal under various error criteria for the
three "classical" consonant interval classes: the one 3-limit interval, the
p4/p5; and the two 5-limit intervals, the m3/M6 and the M3/m6.

|Max. error|Sum-squared error|Sum-absolute error|
---------+----------+-----------------+------------------+
Inverse | 697.3465 | 696.5354 | 696.5784 |
Limit |3/14-comma| 63/250-comma | 1/4-comma |
Weighted |Drobisch??|TD 162.10 5/5/99 | Aron |
---------+----------+-----------------+------------------+
Equal | 696.5784 | 696.1648 | 696.5784 |
Weighted |1/4-comma | 7/26-comma | 1/4-comma |
| Aron | Woolhouse | Aron |
---------+----------+-----------------+------------------+
Limit | 695.9810 | 696.0187 | 696.5784 |
Weighted |5/18-comma| 175/634-comma | 1/4-comma |
| ~Smith | Erlich (TTTTTT) | Aron |
---------+----------+-----------------+------------------+
m3/M6 & | 695.8103 | 695.9332 | 696.5784 |
M3/m6 |2/7-comma | 7/25-comma | 1/4-comma |
only | Zarlino | This may be new | Aron |
---------+----------+-----------------+------------------+

The Drobisch reference is I think in Barbour. I could be remembering it
wrong.

> The Drobisch reference is I think in Barbour. I could be remembering it wrong.

3/14-comma is from Giordano Riccati (1762).

-Manuel

I wrote,

>> The Drobisch reference is I think in Barbour. I could be remembering it
wrong.

Manuel wrote,

>3/14-comma is from Giordano Riccati (1762).

Do you have a fuller reference? And what, exactly, did Drobisch advocate?

Paul wrote:
>>3/14-comma is from Giordano Riccati (1762).

>Do you have a fuller reference? And what, exactly, did Drobisch advocate?

The Ricatti (1762) reference I have read in Lindley & Turner-Smith:
_Mathematical Models of Musical Scales_. There it's said in a footnote
on p. 111 that Hensling (1710) first proposed the 50 division and
Riccati (1762) the 74 division. They don't mention 3/14-comma; since
it's close enough to the 74-tET fifth, it's possible that it came out of
my mind but I'm not sure.
Giordano Riccati was son of Giacomo Riccati, from the Riccati equations.
I can't help you with Drobisch.

Manuel Op de Coul coul@ezh.nl

I wrote:
> They don't mention 3/14-comma; since it's close enough to
> the 74-tET fifth, it's possible that it came out of my mind
> but I'm not sure.

Found it! It's in Lindley's article about temperaments in the
New Grove. Page 666: "J.B. Romieu in 1758 expressed preference
for 1/6-comma meantone because he liked the relative amount
of tempering that it allots to 5ths and 3rds. The same
aesthetic criterion worked up into a mathematical formula
led giordano Riccati in 1762 to give preference to his
own 3/14-comma system (among regular temperaments)."

In the same article he writes "Hensling" but this must
be "Henfling".

Manuel Op de Coul coul@ezh.nl

Manuel, quoting New Grove:

>The same
>aesthetic criterion worked up into a mathematical formula
>led giordano Riccati in 1762 to give preference to his
>own 3/14-comma system (among regular temperaments)."

Excellent! Thanks. Now someone needs to look at Barbour for Drobisch or
whatever.

>In the same article he writes "Hensling" but this must
>be "Henfling".

Helmholtz-Ellis also mentions "Henfling" as a proponent of 50-tET.