Mailing list post
From: Gene Ward Smith (2004-02-16)
Subject: Re: Number 64 as a 19-note well temperament
--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> From the big list of 114 7-limit temperaments:
>
> Number 64
>
> [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
> TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
> TOP generators [1202.900537, 570.4479508]
> bad: 69.388565 comp: 4.891080 err: 2.900537
>
> This scale is very close to 19-ET with a generator of about 9/19 (an
> augmented fourth). It has two sizes of recognizable fifths (694.457
> cents and 706.863 cents). Because of the slightly sharpened octaves, the
> "wolf" fifths are actually slightly better than the normal fifths.
You do have some rather flat major thirds along with really good ones,
but it seems promising. Since the generator is a 7/5 not a 3/2, when
used as a meantone it's a little odd. The TM basis for the
temperament, in case we are interested, is {225/224, 1029/1000}.
! miller19.scl
Herman Miller circulating based on {225/224, 1029/1000}
19
!
62.004635
124.009271
186.013906
248.018542
322.429409
384.434045
446.438680
508.443315
570.447951
632.452586
694.457222
756.461857
818.466492
880.471128
954.881995
1016.886631
1078.891266
1140.895902
1202.900537
Full thread (3 messages)
From: Herman Miller (2004-02-15)
Subject: Number 64 as a 19-note well temperament
From the big list of 114 7-limit temperaments:
Number 64
[3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
TOP generators [1202.900537, 570.4479508]
bad: 69.388565 comp: 4.891080 err: 2.900537
This scale is very close to 19-ET with a generator of about 9/19 (an
augmented fourth). It has two sizes of recognizable fifths (694.457
cents and 706.863 cents). Because of the slightly sharpened octaves, the
"wolf" fifths are actually slightly better than the normal fifths. The
"wolf" major thirds are still in the reasonable range, at 372.028 cents,
and the regular major thirds are within a couple cents of just at
384.434 cents. The "wolf" 7th harmonic at 942.476 cents is worse than
the 19-ET equivalent, but the regular 7:4 approximation is reasonable at
954.882 cents. So this looks like a good candidate for a 19-note
well-tempered scale.
From: Gene Ward Smith (2004-02-16)
Subject: Re: Number 64 as a 19-note well temperament
--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
Herman, this is great. I've tried to cook up 19-tone well-
temperaments, and never got one I thought was worth posting.
From: Gene Ward Smith (2004-02-16)
Subject: Re: Number 64 as a 19-note well temperament
--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> From the big list of 114 7-limit temperaments:
>
> Number 64
>
> [3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
> TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
> TOP generators [1202.900537, 570.4479508]
> bad: 69.388565 comp: 4.891080 err: 2.900537
>
> This scale is very close to 19-ET with a generator of about 9/19 (an
> augmented fourth). It has two sizes of recognizable fifths (694.457
> cents and 706.863 cents). Because of the slightly sharpened octaves, the
> "wolf" fifths are actually slightly better than the normal fifths.
You do have some rather flat major thirds along with really good ones,
but it seems promising. Since the generator is a 7/5 not a 3/2, when
used as a meantone it's a little odd. The TM basis for the
temperament, in case we are interested, is {225/224, 1029/1000}.
! miller19.scl
Herman Miller circulating based on {225/224, 1029/1000}
19
!
62.004635
124.009271
186.013906
248.018542
322.429409
384.434045
446.438680
508.443315
570.447951
632.452586
694.457222
756.461857
818.466492
880.471128
954.881995
1016.886631
1078.891266
1140.895902
1202.900537