Meantone nanotemperament chart

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Meantone nanotemperament chart:
Some options for synthesizers
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Meantone, strictly speaking, denotes tunings where all fifths (except
possibly one) are tempered by an identical amount. In practice, the
term refers to tunings where these fifths have "essentially" the same
size. In the 16th or 18th century, the technical and human limitations
of organ and harpsichord tuning dictated this pragmatic qualification
of the ideal definition. Today, the limitations of digital synthe-
sizers have the same result.

Here I would like to focus on the range of meantone tunings for such
synthesizers offering 512, 768, or 1024 tuning steps per octave. Since
these synthesizers approximate the scales of 512-tone, 768-tone, and
1024-tone equal temperament (512-tet, 768-tet, 1024-tet), we may refer
to them as "512-tet," "768-tet," and "1024-tet" devices. Optimizing
the flexibility of these instruments for exploring the meantone
spectrum involves a process which I call _nanotemperament_.

Nanotemperament is the technique of mixing interval sizes that may
vary by one tuning unit in order to correct for cumulative errors or
to realize an _average_ interval size falling somewhere between the
available steps.[1]

An example may help both to explain the technique and to make the
chart appearing below more comprehensible.

Beginning with a choice example indeed, let us suppose that we wish to
tune Ervin Wilson's "metameantone" temperament, where each fifth is
equal to 695.6304372 cents -- a tempering of 6.324563665 cents. For
the following comparisons, let us use a rounded size of 695.63 cents.

Unfortunately, our best approximations of this metameantone fifth in
512-tet, 768-tet, 1024-tet are indeed quite approximate:

512-tet 768-tet 1024-tet
steps cents var+/- steps cents var+/- steps cents var+/-
296 693.75 -1.88 445 695.31 -0.32 593 694.92 -0.71
297 696.09 +0.46 446 696.87 +1.24 594 696.09 +0.46

In strict meantone tuning, each regular major third is derived from
four fifths of identical size. Thus if we must use a single size of
fifth, our major thirds will vary from their intended size of about
382.52 cents by four times the indicated variance. These thirds will
be about 1.28 cents narrow in 768-tet using our best approximation of
a 445-step fifth, and 1.84 cents wide in 512-tet or 1024-tet using a
297-step fifth or 594-step fifth respectively.

Fortunately, we can get considerably more accurate results than these
by a process of nanotempering, or mixing the two sizes of fifths most
closely approximating our desired interval size in order to balance
out some of the differences. Here is a nanotemperament table for the
the same metameantone tuning, with 512-tet and 1024-tet temperaments
listed in the same column, since the former are a subset of the
latter. A dashed line between the columns shows that a 768-tet and
1024-tet tuning share the same average size for the fifth, ditto marks
(") show that a 512-tet tuning (marked with an * symbol) is equivalent
to the 1024-tet tuning on the previous line of the chart:

768-tet 512-tet/1024-tet
(* = 512-tet steps)
steps cents var+/- steps cents var+/-
444(1)-445(3) 694.92 -0.71 -------- 593 694.92 -0.71
*296(2)-297(2) " "
593(3)-594(1) 695.21 -0.42
445 695.31 -0.32
593(2)-594(2) 695.51 -0.12
*296(1)-297(3) " "
445(3)-445(1) 695.70 +0.07
593(1)-594(3) 695.80 +0.17
445(2)-446(2) 696.09 +0.46 ------- 594 696.09 +0.46
*297 " "
594(3)-595(1) 696.39 +0.76
445(1)-446(3) 696.48 +0.85
594(2)-595(2) 696.68 +1.05
*297(3)-298(1) " "
446 696.87 +1.24

Here the notation in the 768-tet column of "445(1)-446(3)" translates:
"Tune one fifth of 445 steps and three fifths of 446 steps in each
chain of four generating a major third." As it turns out, this tuning
produces an _average_ fifth of 695.70 cents, only 0.07 cents wider
than the ideal metameantone fifth.

In 1024-tet, similarly, tuning two 593-step fifths and two 594-step
fifths yields an average fifth of 695.51 cents, only 0.12 cents
narrower than our goal. We get the same average size of fifth in
512-tet by tuning one 296-step fifth and three 297-step fifths.

What follows is a very tentative adumbration of the meantone spectrum
as realized by these devices, or actually the portion of the spectrum
beginning around 1/3 syntonic comma of temperament, moving through
characteristic Renaissance and Baroque tunings (1/3-1/6 comma) and the
"well-tempered" zone surrounding 12-tet (1/7-1/14 comma) to
Pythagorean intonation ("0-comma meantone") and beyond, taking our
survey a bit further than the region of 17-tet (roughly 5/27-comma of
tempering in the _wide_ direction).

Note that 512-tet, 768-tet, and 1024-tet share some nanotemperaments,
an expected result since these numbers of steps are all divisible by
256.[2] A dashed line in the middle of the chart shows points where
768-tet and 1024-tet options coincide. One 1024-tet nanotemperament
out of two has a 512-tet equivalent, which is listed immediately below
it with ditto (") marks to show this equivalence.[3]

At various points in the chart, historical and recent meantone
temperaments of interest are indicated as landmarks or points of
reference; my warm thanks to John Chalmers for lists including some of
these temperaments and many others of interest from the viewpoint of
just intonation systems and xenharmonics.


----------------------------------------------------------------------
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
----------------------------------------------------------------------
Salinas 1/3-comma meantone = 694.59 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
----------------------------------------------------------------------
Harrison/LucyTuning (M3 1200/pi cents) = 695.49 cents = -6.46 cents
----------------------------------------------------------------------
593(2)-594(2) 695.51 -6.45
*296(1)-297(3) " "
----------------------------------------------------------------------
Wilson metameantone = 695.63 cents = -6.32 cents
----------------------------------------------------------------------
445(3)-446(1) 695.70 -6.25
593(1)-594(3) 695.80 -6.15
----------------------------------------------------------------------
Zarlino 2/7-comma = 695.81 cents = -6.14 cents
----------------------------------------------------------------------
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)
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) " "
595(3)-596(1) 697.56 -4.40
446(2)-447(2) 697.66 -4.30
----------------------------------------------------------------------
1/5-comma = 697.65 cents = -4.30 cents
----------------------------------------------------------------------
595(2)-596(2) 697.85 -4.10
*297(1)-298(3) " "
446(1)-447(3) 698.05 -3.91
595(1)-596(3) 698.14 -3.81
----------------------------------------------------------------------
1/6-comma = 698.37 cents = -3.58 cents
----------------------------------------------------------------------
447 698.44 -3.52 -------- 596 698.44 -3.52
*298 " "
596(3)-597(1) 698.73 -3.22
447(3)-448(1) 698.83 -3.13
----------------------------------------------------------------------
1/7 comma = 698.88 cents = -3.07 cents
----------------------------------------------------------------------
596(2)-597(2) 699.02 -2.93
*298(3)-299(1) " "
----------------------------------------------------------------------
"New Temp" (listed by Chalmers) = 699.14 cents = -2.82 cents
----------------------------------------------------------------------
447(2)-448(2) 699.22 -2.74
----------------------------------------------------------------------
1/8 comma = 699.27 cents = -2.69 cents
----------------------------------------------------------------------
596(1)-597(3) 699.32 -2.64
----------------------------------------------------------------------
1/9 comma = 699.57 cents = -2.38 cents
----------------------------------------------------------------------
447(1)-448(3) 699.61 -2.35 -------- 597 699.61 -2.35
*298(2)-299(2) " "
----------------------------------------------------------------------
Twelfth 5th pure = 1/11 Pythagorean comma = 699.82 cents = -2.13 cents
1/10 comma = 699.80 cents = -2.15 cents
----------------------------------------------------------------------
597(3)-598(1) 699.90 -2.05
448 700.00 -1.96
(12-tet, ~1/11 comma)
597(2)-598(2) 700.20 -1.76
*298(1)-299(3) " "
448(3)-449(1) 700.39 -1.56
597(1)-598(3) 700.49 -1.47
448(2)-449(2) 700.78 -1.17 -------- 598 700.78 -1.17
*299 " "
598(3)-599(1) 701.07 -0.88
448(1)-449(3) 701.17 -0.78
598(2)-599(2) 701.37 -0.59
*299(3)-300(1) " "
449 701.56 -0.39
598(1)-599(3) 701.66 -0.29
----------------------------------------------------------------------
Helmholtz 1/8 schisma (pure schisma M3) = 701.71 cents = -0.24 cents
1/9 schisma (pure schisma m3) = 701.74 cents = -0.22 cents
----------------------------------------------------------------------
449(3)-450(1) 701.95 -0.0019 -------- 599 701.95 -0.0019
*299(2)-300(2) " "
----------------------------------------------------------------------
Pythagorean just intonation = 701.955 cents
----------------------------------------------------------------------
599(3)-600(1) 702.24 +0.29
----------------------------------------------------------------------
Positive Lucy (schisma M3 1200/pi) = 702.25 cents = +0.30 cents
----------------------------------------------------------------------
449(2)-450(2) 702.34 +0.39
599(2)-600(2) 702.54 +0.58
*299(1)-300(3) " "
449(1)-450(3) 702.73 +0.78
599(1)-600(3) 702.83 +0.88
450 703.13 +1.17 -------- 600 703.13 +1.17
*300 " "
600(3)-601(1) 703.42 +1.46
450(3)-451(1) 703.52 +1.56
600(2)-601(2) 703.71 +1.76
*300(3)-301(1) " "
450(2)-451(2) 703.91 +1.95
600(1)-601(3) 704.00 +2.04
450(1)-451(3) 704.30 +2.34 -------- 601 704.30 +2.34
*300(2)-301(2) " "
601(3)-602(1) 704.59 +2.63
451 704.69 +2.73
601(2)-602(2) 704.88 +2.93
*300(1)-301(3) " "
451(3)-452(1) 705.08 +3.12
601(1)-602(3) 705.18 +3.22
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
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 " "
______________________________________________________________________
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Notes
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1. While the term "nanotemperament" or "nano-temperament" _may_ be
new, the process itself and its applications to meantone tunings for
synthesizers have been documented by Paul Ehrlich for some time. I
would like warmly to thank this microtonalist and scholar for his
generous encouragement, and for his many contributions to this and
other areas of tuning theory and xenharmonics.

2. This pattern reminds me a bit of late 14th-century European
compositions combining melodic lines in duple and triple meters, with
the rhythmic units converging every so often.

3. The "equivalence" here is not exact: the _average_ size of the
fifths, and thus the size of the regular major thirds, are indeed
shared by our "equivalent" tunings, but sizes both of specific fifths
and of other intervals will vary.

Margo Schulter
mschulter@value.net
29 July 1998

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End of TUNING Digest 1489
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