RE: Tuning and Temperament and Pianovelle 2500

>}Here's the set up for the Meantone:
>
>}Pitch Alteration from EQ [1/64 of a semitone]
>}C 0
>}C# -15
>}D -4
>}D# +7
>}E -9
>}F +2
>}F# -13
>}G -2
>}G# -17
>}A -7
>}A# 4
>}B -11
>
>}Any comments?
>
>This looks like 1/4-comma meantone from Eb to G#. That's probably a good
>default for what "Meantone" means without any further information.

>My ignorant question: what does "1/4 comma" mean? What's a comma?

Here's the history: The earlier "pythagorean" major scale was defined
by stacking exact 3:2-frequency-ratio perfect fifths: one downward from
the root to define the subdominant (e.g., F in the key of C), and then five
upward to define, successively, the dominant (G), supertonic (D),
submediant (A), mediant (E), and leading tone (B).

If you do that, you end up with an E defined as an 81:64 ratio (3:2 to
the fourth power, divided by 4 to drop it down into the octave where we
started) relative to the root. 81:64 however, beats against the more
intuitive 5:4 (i.e., 80:64) major third. The pitch difference between the
two is the syntonic comma, a ratio of 81:80.

The premises behind quarter-comma meantone are that:
1. We want the mediant (the major third above the tonic) to be 5:4.
2. We want all (true) perfect fifths to be the same size.
3. We are willing to slightly compromise the ideality of the perfect fifth
to make 1 and 2 possible.
So the quarter-comma meantone solution is to build a scale in a circle of
identical fifths that are not the ideal 3:2 ratio, but flat of that ideal
by 1/4 of that 81:80 comma. That way, when you stack up each of four of
those quarter-comma flat fifths, you drop the pitch by 1/4 comma for the G
(in the key of C), 2/4 comma flat for the D, 3/4 flat for the A, and then
when you get to the E, you are 4/4 (i.e., 1) of a comma flat of that 81:64
major third, which is (by definition) the more ideal 5:4 major third.

All meantone tunings are based upon a circle of consistent-sized fifths
(or more often a "broken" circle of fifths). The difference between the
various meantone tunings is the size (i.e., tempering) of that fifth. That
circle frequently starts on Eb, going on to Bb, F, C, G, etc., on to G#.
The circle is frequently broken at the diminished sixth between G# and Eb.
If that diminished sixth were treated as if it were a perfect fifth (e.g.,
as if it were G# to D#, or Ab to Eb, rather than G# to Eb), then it would
be perceived as the "wolf" fifth, because it's WAAAAYY out of tune from the
ideal 3:2 perfect fifth.

The other solution is to continue the circle of fifths, allowing D# to
be a different pitch from Eb, or, going the other way, Ab to be a different
pitch from G#. It turns out that, purely by mathematical luck, if you
continue that to a total of 31 fifths (knocking the pitches down by an
octave as needed to keep them within the original octave's span) you end up
very close to where you started - only about 6 cents off. (A cent is a
unit of measurement for very small differences in pitch, one cent being
100th of the usual 12TET half-step.) That's another way of saying the
31TET is extremely close to the completed quarter-comma meantone system.

The coincidence between third-comma meantone and 19TET is even closer
still. Third-comma meantone uses a fifth tempered down by 1/3 of a comma
so that the submediant (e.g. A in the key of C) has a 5:3 ratio relative to
the tonic rather than 27:16 as you would get if you were to use exact 3:2
perfect fifths. It turns out that if you stack up 19 third-comma-flat
fifths (dropping pitches down by octaves as needed to keep the pitches in
same the octave as where you started), you end up less than 1 cent from
where you started!

Bob Lee wrote:

> Second question: I notice that the tuning above has the "in tune" fourths
> tuned to 502 cents. I'm tuning a pedal steel with the fourths at 503 cents.
> I really like it. What should I call this temperament?

For a start, you're mistaking 1/64ths of a semitone for cents. 1/4
comma meantone has a fourth of about 503.4 cents. Your temperament,
then, isn't far off. The fourths and fifths will be better, but the
minor thirds will be worse. I've got it tuned up on my keyboard,
and it does sound good. Given the accuracy of most tuning meters,
you can probably call it 1/4 comma meantone for the difference it
makes. More accurately, it's 0.230416 comma meantone.


Now, on the meantone thread, I do now own a guitar and I'm thinking
about how to refret it. I've decided it has to be a meantone
temperament, with 19 frets per octave and a bonus fret at the top.
At the moment, I'm planning on a tone of 160.8 moct, which makes a
fifth of about 696.5 cents. Comments would be welcome, but I'm
fairly independently minded, so I'll probably go ahead with this
anyway.