Re: Digest Number 213

John Chalmers wrote:

>I was intrigued by the recent discussion between Dale Scott and Paul
>Erlich about circulating temperaments. James Murray Barbour himself
>suggested a Temperament by "Regularly Varied Fifths" which is similar in
>concept to Paul's save that Barbour used a linear increment and Paul a
>trigonometric formula. Barbour's tuning is given on pages 181-183 of
>"Tuning and Temperament."
>
>Barbour started with the fift D-A and added an increment to each fifth
>around the circle in both directions. The increments are 1 +2 +3 +4 +5 +
>6 + 5 +4 +3 +2 +1 = 36 parts (from 700 cents). D-A is 3 parts flatter
>than 700; B-F# and F-C will have 700 cents and Ab-Eb will be 3 parts
>sharper than equal. Barbour recommends the part be 1 cent.
>
>Barbour's tuning is tempered to a greater extent than Erlich's as [Barbour's]
>flattest fifth (D-A) has 697 cents and his sharpest, (Ab-Eb, has 703.
>The best thirds (C-E, G-B) have 392 cents and the worst ( Gb-Bb, Db-F)
>408. A part of 1.75 cents makes the best thirds just, but the worst
>about 414 cents.

For the clock-based temperaments I reported, I chose the amount of variation in
fifth size that made the largest fifths Pythagorean in order to avoid "harmonic
waste". Since Barbour didn't avoid harmonic waste, I'll go ahead propose a variant
that matches Barbour's worst and best major thirds:

FIFTHS

Barbour Erlich
f-c 700 700
c-g 699 698.76
g-d 698 697.86
d-a 697 697.52
a-e 698 697.86
e-b 699 698.76
b-f# 700 700
f#-c# 701 701.24
c#-g# 702 702.14
g#-eb 703 702.48
eb-bb 702 702.14
bb-f 701 701.24

MAJOR THIRDS

Barbour Erlich
f-a 394 394.14
c-e 392 392
g-b 392 392
d-f# 394 394.14
a-c# 398 397.86
e-g# 402 402.14
b-eb 406 405.86
f#-bb 408 408
c#-f 408 408
g#-c 406 405.86
eb-g 402 402.14
bb-f 398 397.86

I get essentially the same major thirds as Barbour but my fifths are "rounder".

>Barbour doesn't describe the fifths and thirds of his tuning in as much
>detail as Paul did (and I don't have time today to do so myself, alas),
>but the chromatic octave is 0 92 197 297 392 500 591 699 794 894 999
>1091 and 1200 cents.

Mine is 0 92 196.62 296.62 392 500 590.76 698.76 794.14 894.14 998.76 1090.76 1200.