"Just" tunings on digital pianos

A question. Yesterday I was checking out a Yamaha digital piano at the local
Guitar Center, which had Equal, Pythagorean, Meantone, Werckmeister and
Kirnberger temperaments. It also had "JustMajor" and "JustMinor", which like
the other tunings (except Equal), could be centered in any of the 12 notes
in the octave.

I assume this is all 5-limit, and I did notice some very narrow chromatic
steps, like E-flat to E in the key of C which should be about 71 cents.
Also, why is there both a major and minor just tuning? And is a Roland or
Korg or Kurzweil any different?

(This info might've been in the user's manual, which I didn't have handy.)

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

/tuning/topicId_46925.html#46925

> A question. Yesterday I was checking out a Yamaha digital piano at
the local
> Guitar Center, which had Equal, Pythagorean, Meantone, Werckmeister
and
> Kirnberger temperaments. It also had "JustMajor" and "JustMinor",
which like
> the other tunings (except Equal), could be centered in any of the
12 notes
> in the octave.
>
> I assume this is all 5-limit, and I did notice some very narrow
chromatic
> steps, like E-flat to E in the key of C which should be about 71
cents.
> Also, why is there both a major and minor just tuning? And is a
Roland or
> Korg or Kurzweil any different?
>
> (This info might've been in the user's manual, which I didn't have
handy.)

***Well, the basis for these has to be the "regular" Ptolemy Diatonic:

0: 1/1 0.000 unison, perfect prime
1: 9/8 203.910 major whole tone
2: 5/4 386.314 major third
3: 4/3 498.045 perfect fourth
4: 3/2 701.955 perfect fifth
5: 5/3 884.359 major sixth, BP sixth
6: 15/8 1088.269 classic major seventh
7: 2/1 1200.000 octave

I think the *minor* one is built starting on the "relative
minor" "A". The assignment of semitones is a little arbitrary since,
one can't have *everything* just, hence meantone and such like.

I'll let some of our "theory heads" fill you in a little more here...

J. Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

> ***Well, the basis for these has to be the "regular" Ptolemy
Diatonic:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 9/8 203.910 major whole tone
> 2: 5/4 386.314 major third
> 3: 4/3 498.045 perfect fourth
> 4: 3/2 701.955 perfect fifth
> 5: 5/3 884.359 major sixth, BP sixth
> 6: 15/8 1088.269 classic major seventh
> 7: 2/1 1200.000 octave

"intense" was the name i think ptolemy used for this one, since he
actually specified a dozen or two different diatonic scales. plus,
wouldn't have have used a mode name too if he meant for the scale to
be arranged like the "modern major scale" relative to 1/1?

> I think the *minor* one is built starting on the "relative
> minor" "A". The assignment of semitones is a little arbitrary
since,
> one can't have *everything* just, hence meantone and such like.

even the diatonic above is somewhat arbitrary, since the "supertonic
triad" formed by the second, fourth, and sixth scale degrees is
not "just" in the same sense, and attempting to fix it will break
some other "justness" -- hence meantone and such like.

> I'll let some of our "theory heads" fill you in a little more
>here...

due to the arbitrariness even in a single diatonic, there's quite a
wide variety of 12-tone just scales out there, even in the 5-limit.
i'm sure you can figure out which ones your keyboard is using by
listening to all the intervals and hearing which are "just" and which
aren't.

hi paul and Joe,

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> > ***Well, the basis for these has to be the "regular"
> > Ptolemy Diatonic:
> >
> > 0: 1/1 0.000 unison, perfect prime
> > 1: 9/8 203.910 major whole tone
> > 2: 5/4 386.314 major third
> > 3: 4/3 498.045 perfect fourth
> > 4: 3/2 701.955 perfect fifth
> > 5: 5/3 884.359 major sixth, BP sixth
> > 6: 15/8 1088.269 classic major seventh
> > 7: 2/1 1200.000 octave
>
> "intense" was the name i think ptolemy used for this one,
> since he actually specified a dozen or two different diatonic
> scales. plus, wouldn't have have used a mode name too if he
> meant for the scale to be arranged like the "modern major scale"
> relative to 1/1?

Ptolemy's terminology stems from Aristoxenos, who was interested
in avoided ratios and other means of string-length measurement.
Aristoxenos instead preferred to use only the ear and the
sense of hearing as the criterion for judging tuning, and he
therefore spoke of string *tension* instead of length.

thus, the English translation of his _syntonon_ should be
"tense" rather than "intense", and that of _malakon_ should
be "relaxed" rather than "soft" ... Partch's use of the former
translations notwithstanding.

in response to the initial question in this thread: the
variablity of the "supertonic" (II) degree of the diatonic
scale, and the desire for 4:5:6 or 1/(4:5:6) harmonic triads,
are the primary reasons why the tunings are different
depending on the key.

-monz

oops ...

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> Ptolemy's terminology stems from Aristoxenos, who was interested
> in avoided ratios and other means of string-length measurement.

that should be: "... interested in *avoiding* ratios ..."

> Aristoxenos instead preferred to use only the ear and the
> sense of hearing as the criterion for judging tuning, and he
> therefore spoke of string *tension* instead of length.
>
> thus, the English translation of his _syntonon_ should be
> "tense" rather than "intense", and that of _malakon_ should
> be "relaxed" rather than "soft" ... Partch's use of the former
> translations notwithstanding.

and here i meant to say: "Partch's use of the *latter*
translations notwithstanding". i'm hoping that *other*
tuning theorists agree that the proper terms here are
"tense" and "relaxed". the others don't make as much sense.

-monz