Nanotemperament chart revisions

-------------------------------------
Corrections on nanotemperaments
(with thanks to Paul Erlich)
-------------------------------------

May I express great appreciation to Paul Erlich, whose article on
22-tet in the new Xenharmonikon 17 has many insights on tuning and
harmonic organization, for calling to my attention some omissions in a
recent posting on synthesizer nanotemperaments for meantone tunings.

Specifically, my meantone chart ranged from around 1/3-comma meantone
on the narrow or "negative" side to 17-tet on the wide or "positive"
side. However, extending the positive portion of the table a bit
further not only remedies the omission of 22-tet, but also gives a
pleasing symmetry to the whole. Incidentally, it may be worth noting
that the amount of temperament applied to the fifth in 22-tet (about
7.14 cents) is actually _less_ than in 1/3-comma meantone (7.17 cents)
or 19-tet (7.22 cents) -- only the _direction_ is different.

Also, Paul Erlich helpfully pointed out that I had included
information for 17-tet, but not for 19-tet or 31-tet, very close to
1/3-comma and 1/4-comma respectively. This provides yet another
motivation for a revision.

Additionally, implementing these suggestions gave me a chance to find
a slight error in the stated size of fifth for 1/3-comma meantone, and
also to note the omission of a 512-tet nanotemperament in the region
between 17-tet and 22-tet:

603(2)-604(2) 707.23 +5.27
*301(1)-302(3) " "

To explain briefly: the first line, for 1024-tet, defines a tuning in
which a major third is built from two fifths of 603 steps and two
fifths of 604 steps: this produces an average fifth size of 707.23
cents, or 5.27 cents _wider_ (positive sign) than just. The next line
shows an equivalent 512-tet tuning with one fifth of 301 steps and
three fifths of 302 steps, producing the same average fifth size. Such
"equivalent" tunings should produce major thirds of the same size,
although other intervals may vary.

As in original post, the left columns are for 768-tet synthesizers,
and the right for 512-tet or 1024-tet. Since the chart is rather long,
maybe the best approach for now is just to post the revised regions of
the chart -- not that there might not be more errata.


----------------------------------------------------------------------
Meantone nanotemperaments for synthesizers
----------------------------------------------------------------------
768-tet 512-tet/1024-tet
(* = 512-tet steps)
Nanopattern cents temp+/- Nanopattern cents temp+/-
444 693.75 -8.21 ------- 592 693.75 -8.21
*296 " "
592(3)-593(1) 694.04 -7.91
444(3)-445(1) 694.14 -7.81
592(2)-593(2) 694.34 -7.62
*296(3)-297(1) " "
444(2)-445(2) 694.53 -7.42
592(1)-593(3) 694.63 -7.33
----------------------------------------------------------------------
19-tet = 694.74 cents = -7.22 cents
Salinas 1/3-comma meantone = 694.79 cents = -7.17 cents
----------------------------------------------------------------------
444(1)-445(3) 694.92 -7.03 ------- 593 694.92 -7.03
*296(2)-297(2) " "
593(3)-594(1) 695.74 -6.74
445 695.31 -6.64

.............................


445(2)-446(2) 696.09 -5.86 -------- 594 696.09 -5.86
*297 " "
594(3)-595(1) 696.39 -5.57
445(1)-446(3) 696.48 -5.47
----------------------------------------------------------------------
1/4-comma = 696.58 cents = -5.38 cents
----------------------------------------------------------------------
594(2)-595(2) 696.68 -5.28
*297(3)-298(1) " "
(Erlich's 1/4-comma for 512-tet)
----------------------------------------------------------------------
31-tet = 696.77 cents = -5.18 cents
----------------------------------------------------------------------
446 696.88 -5.08
594(1)-595(3) 696.97 -4.98
446(3)-447(1) 697.27 -4.69 ------- 595 697.27 -4.69
*297(2)-298(2) " "

.............................


451(2)-452(2) 705.47 +3.51 -------- 602 705.47 +3.51
*301 " "
602(3)-603(1) 705.76 +3.81
451(1)-452(3) 705.86 +3.90
----------------------------------------------------------------------
17-tet = 705.88 cents = +3.93 cents
----------------------------------------------------------------------
602(2)-603(2) 706.05 +4.10
*301(3)-302(1) " "
452 706.25 +4.29
602(1)-603(3) 706.35 +4.39
452(3)-453(1) 706.64 +4.69 -------- 603 706.64 +4.69
*301(2)-302(2) " "
603(3)-604(1) 706.93 +4.98
452(2)-453(2) 707.03 +5.08
603(2)-604(2) 707.23 +5.27
*301(1)-302(3) " "
452(1)-453(3) 707.42 +5.47
603(1)-604(3) 707.52 +5.56
453 707.81 +5.86 -------- 604 707.81 +5.86
*302 " "
604(3)-605(1) 708.11 +6.15
453(3)-454(1) 708.20 +6.25
604(2)-605(2) 708.40 +6.44
*302(3)-303(1) " "
453(2)-454(2) 708.59 +6.64
604(1)-605(3) 708.69 +6.74
453(1)-454(3) 708.98 +7.03 -------- 605 708.98 +7.03
*302(2)-303(2) " "
----------------------------------------------------------------------
Erlich 22-tet = 709.09 cents = +7.14 cents
----------------------------------------------------------------------
605(3)-606(1) 709.28 +7.32
454 709.38 +7.42
605(2)-606(2) 709.57 +7.62
*302(1)-303(3) " "
454(3)-455(1) 709.77 +7.81
605(1)-606(3) 709.86 +7.91
454(2)-455(2) 710.16 +8.20 -------- 606 710.16 +8.20
*303 " "
______________________________________________________________________
----------------------------------------------------------------------

Most appreciatively,

Margo Schulter
mschulter@value.net
7 August 1998