WHY

Mike Battaglia, Brad Lehman, Tom Dent, Andreas Sparschuh.......,

Many times I read in the tuning group that tones of a good scale has to be distinguished from their corresponding equal tempered tones by a sufficient number of cents like 4. If the deviations are only about 2 cents, this figure would not be welcomed seeing that the proposed and equal tempered tones would sound about the same, that is, indistinguishable. I guess that this recommended tendency is not applied to the twelve tones of the octave. Probably I didn´t get the idea because the equal tempered scale appears to be an untouchable set, at least part of it.
Should the scale proponent employs three tones which are distanced by 5, 4 and 2 cents from the corresponding eqt tones, I don´t know which of these three tones would be considered applicable.
I ignore why the equal tempered scale is not set aside and let the proposals to be evaluated free of parameters that make restrictions; its tone frequencies could be taken as references and no more than that. Centuries have passed since the imperfection of this arithmetical scale was acknowledged, then it should not be also applied as the needed reference.

MARIO PIZARRO

Lima, July 07, 2008

On Mon, Jul 7, 2008 at 1:10 AM, Mario Pizarro <piagui@...> wrote:
>
> Mike Battaglia, Brad Lehman, Tom Dent, Andreas Sparschuh.......,
>
> Many times I read in the tuning group that tones of a good scale has to be distinguished from their corresponding equal tempered tones by a sufficient number of cents like 4. If the deviations are only about 2 cents, this figure would not be welcomed seeing that the proposed and equal tempered tones would sound about the same, that is, indistinguishable. I guess that this recommended tendency is not applied to the twelve tones of the octave. Probably I didn´t get the idea because the equal tempered scale appears to be an untouchable set, at least part of it.
> Should the scale proponent employs three tones which are distanced by 5, 4 and 2 cents from the corresponding eqt tones, I don´t know which of these three tones would be considered applicable.
> I ignore why the equal tempered scale is not set aside and let the proposals to be evaluated free of parameters that make restrictions; its tone frequencies could be taken as references and no more than that. Centuries have passed since the imperfection of this arithmetical scale was acknowledged, then it should not be also applied as the needed reference.
>
> MARIO PIZARRO

Mario,

From what I can gather, the only intervals in the Piagui scale that
will be distinguishable from equal temperament will be the ones
involving the 696 cent flat perfect fifths, and those will be
distinguishable in that they will be MORE out of tune than equal
temperament, and not less.

But, you are correct in stating that 12 tone equal temperament is no
end-all-be-all reference to compare the Piagui scale to. But if you
don't like it as a reference, what do you like? 5-limit just
intonation? The results are very similar.

-Mike

If we look at that 1776 "Marpurg H" layout (= "Piagui II")
melodically, we do get some distinguishable expressive features among
the diatonic scales.

There are three differently-patterned major scales in it, and
similarly three differently-patterned natural minor scales. By cents:

Major: Do-re-mi-fa-sol-la-ti-do

1. 0, 198, 402, 498, 702, 900, 1098, 1200.

2. 0, 198, 396, 498, 696, 900, 1098, 1200.

3. 0, 204, 402, 504, 702, 900, 1104, 1200.

1 = C major, A major, F# major, Eb major. (Large Do-mi and small
Sol-ti; small Mi-fa step and large Ti-do step)

2 = G major, E major, Db major, Bb major. (Small Do-mi and large
Sol-ti; large Mi-fa step and large Ti-do step)

3 = D major, B major, Ab major, F major. (Large Do-mi *and* large
Sol-ti; large Mi-fa step and small Ti-do step)

Natural minor: La-ti-do-re-mi-fa-sol-la

4. 0, 198, 300, 498, 702, 798, 1002, 1200.

5. 0, 198, 300, 498, 696, 798, 996, 1200.

6. 0, 204, 300, 504, 702, 804, 1002, 1200.

4 = A minor, F# minor, Eb minor, C minor. (Large Ti-do step and small
Mi-fa step)

5 = E minor, C# minor, Bb minor, G minor. (Large Ti-do step and large
Mi-fa step)

6 = B minor, G# minor, F minor, D minor. (Small Ti-do step and large
Mi-fa step)

The other "Piagui" layouts rotate this around so different scales get
these melodic characteristics. Its minor 3rds are always 300 cents,
of course.

If you're into temperaments like this that hack the Pythagorean comma
into only four equal pieces, try Bendeler's third one from c1690 (and
republished 1739). It has the tempering of 1/4 PC each at C-G-D, E-B,
and G#-D#. Since those are distributed asymmetrically, there is a
wider range of melodic differences than any of the "Piagui" variants
have.
- This Bendeler temperament ends up with two different sizes of major
3rds: four of them are 392 (Bb-D, F-A, C-E, and G-B), and the other
eight are 402.
- There are three different sizes of minor 3rds: 306, 300, and 294.
- There are three different sizes of whole steps: C-D at 192, five
others at 204, and the remaining six at 198.
- And there are three sizes of semitones: B-C at 108, five others at
96, and the remaining six at 102.

Brad Lehman