JUST INTONATION SCALE

DEAR FRIENDS:

THIS AFTERNOON I DERIVED A 12 TONE JI SCALE. SOME DATA FOLLOW:

E = 1.25 = 5/4 = 386.313 CENTS

G = 1.5 = 3/2 = 701.955 CENTS

F = 1.33333333 = 4/3 = 498.045 CENTS

2C = 2 = 1200 CENTS
------------------------------------------------

--TWELVE PURE FIFTHS ( 701.955 CENTS)

--SYMMETRICAL DISTRIBUTION OF TWO EQUAL GROUPS OF FIVE TONE INTERVAL FACTORS PLUS TWO CENTRAL ONES WHICH ALSO WORK IN THE GROUPS.

--EIGHT DIFFERENT MAJOR THIRDS IN THE OCTAVE, VARYING 0.025 CENTS.

--THE HELP GIVEN BY THE PROGRESSION OF MUSICAL CELLS TO GIVE ALL THE DATA WAS AMAZING.

TOMORROW I WILL START THE MANUFACTURE OF THE JI GUITAR, FOR THE PRESENT I ONLY HAVE ITS NAME:

"THE PIAJI GUITAR"

THANKS !!!!!!!!!!!!!!!

MARIO

LIMA, NOVEMBER 10
<<<<<<<<<<<<<<<<

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>
> DEAR FRIENDS:
>
> THIS AFTERNOON I DERIVED A 12 TONE JI SCALE. SOME DATA FOLLOW:
>
> E = 1.25 = 5/4 = 386.313 CENTS
>
> G = 1.5 = 3/2 = 701.955 CENTS
>
> F = 1.33333333 = 4/3 = 498.045 CENTS
>
> 2C = 2 = 1200 CENTS
> ------------------------------------------------
>
> --TWELVE PURE FIFTHS ( 701.955 CENTS)
>
> --SYMMETRICAL DISTRIBUTION OF TWO EQUAL GROUPS OF FIVE TONE INTERVAL FACTORS PLUS TWO CENTRAL ONES WHICH ALSO WORK IN THE GROUPS.
>
> --EIGHT DIFFERENT MAJOR THIRDS IN THE OCTAVE, VARYING 0.025 CENTS.

Could you please tell us what all 12 pitches of the scale are?

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@> wrote:
> >
> > DEAR FRIENDS:
> >
> > THIS AFTERNOON I DERIVED A 12 TONE JI SCALE. SOME DATA FOLLOW:
> >
> > E = 1.25 = 5/4 = 386.313 CENTS
> >
> > G = 1.5 = 3/2 = 701.955 CENTS
> >
> > F = 1.33333333 = 4/3 = 498.045 CENTS
> >
> > 2C = 2 = 1200 CENTS
> > ------------------------------------------------
> >
> > --TWELVE PURE FIFTHS ( 701.955 CENTS)
> >
> > --SYMMETRICAL DISTRIBUTION OF TWO EQUAL GROUPS OF FIVE TONE INTERVAL FACTORS PLUS TWO CENTRAL ONES WHICH ALSO WORK IN THE GROUPS.
> >
> > --EIGHT DIFFERENT MAJOR THIRDS IN THE OCTAVE, VARYING 0.025 CENTS.
>
> Could you please tell us what all 12 pitches of the scale are?

Well, Mr. Pizarro did send me a spreadsheet of the 12 pitches. It has some inconsistencies in it, but the specific inconsistencies that appear are interesting.

Here are the JI ratios from the spreadsheet (unedited):

1 ------------- C
*135/128 --C#
*9/8 --------- D
*32/27----- Eb
*5/4 --------- E
*4/3 -------- F
*64/45 ---- F#
*16384/10935 - G
*128/81-- Ab
*27/16---- A
*16/9 ------Bb
*15/8 ------ B
*2---------- 2C

BTW, thanks for using ratios, Mario. It really makes everything easier when talking about JI scales.

There's another column of the spreadsheet which lists the steps as JI intervals (again unedited):

135/128-- C#/C
16/15---D/C#
256/243-- Eb/D
135/128 - E/Eb
16/15 ---- F/E
16/15 ---- F#/F
256/243 --G/F#
135/128 - Ab/G
16/15 --A/Ab
135/128 -- Bb/A
135/128 -- B/Bb
16/15 -- 2C/B

Notice that the steps agree *almost* perfectly with the pitches, but there is a discrepancy - A would be 2048/1215 rather than 27/16 according to the steps. This difference is exactly a schisma (32805/32768)! Also note that the pitch given for G is a schisma flat of 3/2.

Since these schisma differences are popping up here and there, it makes sense to temper this scale to schismatic (helmholtz) temperament. Here's what we get:

! piaji-helmholtz.scl
!
"PiaJI" scale tempered to TOP helmholtz
12
!
91.26203
203.50821
294.77024
386.03227
498.27845
610.52464
701.78667
793.04869
905.29488
996.55691
1087.81893
1200.06512

Note that this has a simple description in terms of helmholtz temperament - helmholtz[12] 3|8 #6, a particularly simple helmholtz[12] MODMOS. (The notes of this scale in Pythagorean notation are C Db D Eb Fb F F# G Ab A Bb Cb C.)

Okay, now that we know what the scale is (either of the two slightly different JI versions, or perhaps both of them if schisma discrepancies are to be ignored in implied helmholtz temperament), let's see how the claims made about it stand up:

> > --TWELVE PURE FIFTHS ( 701.955 CENTS)

Patently false. Since this is not an equal temperament, the fifths are not all equal, so some of them are impure. In this scale there are 3 serious wolf fifths - two a comma flat of 3/2 (~681 cents) and one a comma sharp (~723 cents; all of this is ignoring schisma discrepancies, which only decrease the number of perfectly pure fifths).

> > --SYMMETRICAL DISTRIBUTION OF TWO EQUAL GROUPS OF FIVE TONE INTERVAL FACTORS PLUS TWO CENTRAL ONES WHICH ALSO WORK IN THE GROUPS.

It's unclear what this means, exactly, but it seems unlikely to be true. The particular helmholtz[12] MODMOS this tempers down to is not symmetrical.

> > --EIGHT DIFFERENT MAJOR THIRDS IN THE OCTAVE, VARYING 0.025 CENTS.

Also false. Of the 12 different 4-step intervals, eight are either 81/64 or a schisma off from that, and four are either 5/4 or a schisma off from that. So there are indeed eight intervals within a schisma of a Pythagorean "major third" 81/64, but a schisma is 1.95 cents, not 0.025 cents. I have no idea where the 0.025 cents figure comes from.

In conclusion, the scale in question is a detempering of a helmholtz[12] MODMOS, and it certainly does not have all the amazing properties claimed of it, such as 12 pure fifths.

Keenan