Topic: Rational 12 and 19 nearly equal

2 scales

Thread (1 messages)

From: Gene Ward Smith (2004-07-28)
Subject: Rational 12 and 19 nearly equal

Below I give a 12-note 7-limit well-temperament obtained by Hahn
reducing a chain of fifths according to 7-limit 72-et, meaning via the
commas 225/224, 1029/1024 and 4375/4374. The result has three 112/75
meantone fifths, a 125000/83349 quasi-pure fifth (it's flat by an
interval of 250047/250000, which is less than a third of a cent) and
eight pure fifths. As a well-temperament the main problem with it is
that it doesn't seem to help the thirds much; rearranging the fifths
so that the meantone fifths were in the same part of the chain would
seem to be a good plan if we wanted a well-temperament with sweeter
home keys.

I also give a 19-note 7-limit pseudo 19-equal which is the Hahn
reduction via the commas of 171-et of a chain of minor thirds. It is
interesting for being rational and having a lot of pure minor thirds.

! rat12.scl
72-et Hahn reduced 12-fairly-equal well-temperament
12
!
200/189
9/8
25/21
63/50
4/3
625/441
3/2
100/63
42/25
16/9
189/100
2

! rat19.scl
171-et Hahn reduced 7-limit 19-almost-equal
19
!
28/27
672/625
125/112
125/108
6/5
56/45
1323/1024
75/56
25/18
36/25
112/75
2048/1323
45/28
5/3
216/125
224/125
625/336
27/14
2