From: tuning@onelist.com
To: Joe Monzo
Date: Tue, 1 Dec 1998 11:06:19 -0500 (EST)
Subject: TUNING digest 1598
TUNING Digest 1598
Topics covered in this issue include:
1) All, then best, 7-limit scales with 31 consonances
2) Re: All, then best, 7-limit scales with 31 consonances
3) Re: Corrections on my "What is a Wolf?"
4) Reply to Reinhard on Subharmonics
5) Symmetry
6) Re: Orpheus and the Undertones, Subharmonics, Utonalities etc.
7) Lumma's challenge
8) Re: Lumma's challenge
9) Re: Lumma's challenge
10) Finale xen-cidentals
11) Re: Lumma's challenge
----------------------------------------------------------------------
Topic No. 1
Date: Mon, 30 Nov 1998 10:28:37 -0600 (CST)
On Sat, 28 Nov 1998, Paul H. Erlich wrote [1595.10]:
There are twelve possible orientations for this shape in the 7-limit
3-d space, only three of which are Euler genuses (geni?). That does
not, however, exhaust the possible scales which have 31 7-limit
consonances. Those can be completely enumerated by butting up two
hexanies against each other (six ways to do that) and then adding two
"stellating" points/tetrads--that is, notes which form complete tetrads
with one of the 12 exposed triangles (faces of the octahedra which are
the hexanies).
This gives 6 * 12 * 11 / 2 or 396 different scales.
Unfortunately most of them are very uneven. There are only three whose
smallest scale degree is not smaller than 25:24; they approximate 12TET
reasonably well. This is the (IMHO) best one:
You can get the other two (a) by substituting 25:16 for 8:5, or (b) by
inverting this one (which, transposed, can also be thought of as
substituting 25:16 for 8:5 and 105:64 for 42:25). (a) is more
symmetrical (same shape within the lattice as the original
Euler genus)
but has a diameter of 4, whereas this one and (b) have diameters of 3.
Furthermore, the most complex (quaternary) interval in (a) is actually a
scale step: 42:25 / 25:16 = 672:625. Yech!
------------------------------
Topic No. 2
Date: Mon, 30 Nov 1998 10:47:09 -0600 (CST)
On Mon, 30 Nov 1998, Paul Hahn wrote:
OTOH, one way in which (a) has the advantage over this and (b) is that
(a) is completely "covered" by its four tetrads, whereas the other two
are not.
------------------------------
Topic No. 3
Date: Mon, 30 Nov 1998 11:36:24 -0800 (PST)
Hello, there.
Having read over my "What is a Wolf?" article -- after posting, as
Murphy's Law would have it -- I'd like to offer two important
corrections, not to exclude more from other participants here.
First, I wonder if the following passage might have been a kind of
intonational "Freudian slip" in one direction or another:
Of course, a "pure octave"
is usually regarded as being 2:1, not 3:2
(a pure fifth). Could I have been still focusing on Pythagorean tuning
with its pure fifths (the topic of some previous paragraphs?), or
possibly the just or tempered
Bohlen-Pierce scale with its 3:1
"tritave" in place of the usual 2:1 octave?
My other error concerned an "augmented fifth" sonority found in
16th-17th century music, where I mistakenly gave the size of a minor
sixth in 1/4-comma meantone
tuning as ~890 cents (actually the size of
a major sixth in this temperament) rather than the correct ~814
cents (a just 8:5, the octave
complement
of the just 5:4 on which this
tuning is based).
Here is a corrected diagram showing the sizes in cents of intervals in the
combination in question d-bb-f#', plus a diagram of the alternative
d-bb-gb' showing the diminished fourth (or eleventh) between the outer
voices. Note that I've inexactly rounded this diminished eleventh to 1628
cents in order to make the arithmetic look consistent, although the minor
sixths d-bb and bb-gb' are actually about 813.69 cents each, and the
diminished eleventh d-gb' thus roughly 1627.38 cents, closer to 1627 than
to 1628:
Unfortunately, not only did I err here on the math, but spuriously
suggested that adding a Gb key to a meantone keyboard would permit one
to "avoid" the Wolf augmented fifth bb-f#' by replacing it with
bb-gb'. While this "solution" would succeed with bb-gb' as a simple
interval, changing d-bb-f#' to d-bb-gb' would replace the evocative
bb-f#' between the upper voices of the first sonority with a
diminished eleventh d-gb' between the outer voices, likely much less
pleasing in this stylistic context.
Having fairly muddled this meantone example, maybe a few examples of
how adding notes to the keyboard can solve certain Wolf problems
might not be out of place. First, let's consider the case where a
single extra note could resolve a Wolf diminished fourth:
Here adding a D# key permits replacing the diminished fourth and
augmented second of the first example with a regular major third and
minor third in the second. As usual, I've taken liberties with the
rounded value in cents for a 1/4-comma
meantone fifth -- 697 cents in
the first example, a less accurate 696 cents in the second (actually
~696.58 cents) -- to make the arithmetic look consistent in each
example.
In another kind of situation, an extra Ab key could correct both a
diminshed fourth (Wolf M3) and a diminished sixth (Wolf fifth):
In yet another case, two extra keys, Ab and Db, would permit a
convenient Wolf correction, as would a single more remote E# key:
While a tuning system such as Nicola Vicentino's close approximation
of 31-tone equal temperament
in 1555 (virtually the same as 1/4-comma
meantone) can open these and many other choices, my original example
set about an impossible task.
No matter how many keys one has available in an octave, it seems
mathematically impossible to construct a sonority with a just major
third and minor sixth above the lowest voice where the two upper
voices also form some regular meantone interval.
Most respectfully,
Margo Schulter
------------------------------
Topic No. 4
Date: Mon, 30 Nov 1998 15:02:06 -0500
Absolutely not! The subharmonic series consists of frequencies in the
proportions 1, 1/2, 1/3, 1/4, 1/5, 1/6 . . . . Acoustically, the most
relevant features are
The difference tones of the first six members of the subharmonic series
are
1/2, 2/3, 3/4, 4/5, 5/6, 1/6, 1/4, 3/10, 1/3, 1/12, 2/15, 1/6, 1/20,
1/12, 1/30,
many of which are outside the subharmonic series. Meanwhile, the
difference tones of the first six members of the harmonic series (1, 2,
3, 4, 5, 6) are
1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5
which are all inside the harmonic series. Therefore difference tones are
a distinctly otonal phenomenon. Summation tones are too.
You are confusing the appelations "major" and "minor", which mean one
thing when applied to thirds, with their synonyms "otonal
" and "utonal"
when applied to triads. A 6/5 is no more otonal than utonal, and the
same is true of 5/4. Any single interval appears just as prominently in
the harmonic series as it does in the subharmonic series. It is only
chords of 3 or more notes that can be classified as "otonal" or
"utonal".
------------------------------
Topic No. 5
Date: Mon, 30 Nov 1998 15:05:33 -0800
Bohlen's tuning didn't start out as an equal-temperament,
and even if it
did, Schoenberg "derived" 12-tone equal temperament
from harmonic series
and not subharmonic series. Bohlen is a man, not a tuning. If you're not
familiar with his views, read his web site and papers.
He certainly does not view it as a threat, and does not even mention his
own tuning in this article. He does, however, focus on difference tones,
and thus rates a major triad as much more simple than a minor triad.
------------------------------
Topic No. 7
Date: Mon, 30 Nov 1998 17:45:53 -0500
If you allow a mistuning of 6 cents, the usual 12-out-of-31-tET scale
has 39 consonant 7-limit intervals, all of which are unmistakeable
representations of JI intervals. It is also equivalent to the standard
keyboard tuning from 1500 to 1700 (and most organs through 1850).
If you allow a mistuning of 17 cents, a maximally-even 12-out-of-22-tET
has 50 consonant 7-limit intervals. There is one ambiguity relative to
JI: 600 cents can be either 7/5 or 10/7. That's no good for two-part
harmony, but all the 600-cents intervals in the scale occur in several
7-limit tetrads and triads, so fuller harmony will eliminate the
ambiguity.
If you allow a mistuning of 27 cents, a non-maximally-even
12-out-of-15tET has 60 consonant 7-limit intervals. There are several
ambiguities relative to JI and I'm not sure that all of them can be
resolved by fuller harmony (I doubt it).
Back to JI. This is the JI scale which was our first example of one with
31 exact 7-limit consonances.
I wrote,
Carl Lumma wrote,
I mean exactly the opposite. I want to end up with a different set of
pitches, not the same set of pitches. Since the figure has no axes or
planes of symmetry, all congruent orientations of the lattice lead to
different scales. The figure is symmetrical about the central point
Not true. Otherwise it wouldn't be a lattice.
Paul Hahn wrote,
What does that mean? Being 7-limit implies no such restrictions.
There are certainly more than three orientations. Paul H., let's stick
to fixed directions for the axes (more work, I know, but this stuff is
confusing enough for most people).
The original scale:
rotated 120 degrees in the 3-5 plane (let's arbitrarily use 1:1 as the
pivot):
rotated -120 degrees in the 3-5 plane:
rotated 120 degrees in the 3-7 plane:
and many more . . .
------------------------------
Topic No. 8
Date: Mon, 30 Nov 1998 17:07:57 -0600 (CST)
On Mon, 30 Nov 1998, Paul H. Erlich wrote:
That wasn't me, nor was any of the other stuff you quoted in that
message.
------------------------------
Topic No. 9
Date: Mon, 30 Nov 1998 17:14:04 -0600 (CST)
On Mon, 30 Nov 1998, Paul H. Erlich wrote:
As I indicated in my message from this morning, there are twelve
possible orientations.
------------------------------
Topic No. 10
Date: Tue, 01 Dec 1998 01:19:40 +0000
For all you Finale-ists on this list:
In case you didn't already know, the Tamburo font has a few useful
"xen-cidentals" which I've successfully used (all except the backwards
flat, which doesn't suit me).
These are the Macintosh keystrokes; hopefully someone can translate
these into Windows ones:
i = 1/2 flat (backwards and filled-in-black flat symbol)
------------------------------
Topic No. 11
Date: Mon, 30 Nov 1998 23:39:29 -0800 (PST)
On Mon, 30 Nov 1998, Paul H. Erlich wrote:
Actually, I wrote the following:
Umm, would you believe I was thinking in terms of a grid instead of a
triangular lattice?
I picture the tuning in question as being the intersectiors of a 2x2x3
lattice, with each direction representing a ratio.
Each of the three
ratios can be the one which is three long, and there are four different
sets of ratios which can be used:
{3,5,7} (the original given)
{1:3,5:3,7:3}
{3:5,1:5,7:5}
{3:7,5:7,1:7}
So there are a total of 12 orientations, if I understand correctly now.
9-limit tunings can't be reoriented quite so cavalierly, at least not
without potentially destroying some consonances,
since 3*3 is a 9-limit
consonance but 5*5 isn't.
-Bram
------------------------------
End of TUNING Digest 1598
I welcome feedback about this webpage: corrections, improvements, good links.
by Paul Hahn
by Paul Hahn
by "M. Schulter"
by "Paul H. Erlich"
by Carl Lumma
by "Paul H. Erlich"
by "Paul H. Erlich"
by Paul Hahn
by Paul Hahn
by Harold Fortuin
by bram
From: Paul Hahn
To: Tuning Digest
Subject: All, then best, 7-limit scales with 31 consonances
Message-ID:
35:24-------35:16------105:64
.-'/ \'-. .-'/ \'-. .-'/
5:3--/---\--5:4--/---\-15:8 /
/|\ / \ /|\ / \ /| /
/ | / \ | / \ | /
/ |/ \ / \|/ \ / \|/
/ 7:6---------7:4--------21:16
/.-' '-.\ /.-' '-.\ /.-'
4:3---------1:1---------3:2
[me:]
By my count this has 31 7-limit consonances.
Guys, wouldn't any rotation and/or reflection of this figure in the
tetrahedral/octahedral lattice work too?
42:25------21:20-------21:16
\'-. .-'/ \'-. .-'/ \'-.
\ 6:5--/---\--3:2--/---\-15:8
\ /|\ / \ /|\ / \ /|
\ | / \ | / \ |
/ \|/ \ / \|/ \ / \|
/ 7:5---------7:4--------35:32
/.-' '-.\ /.-' '-.\ /.-'
8:5---------1:1---------5:4
--pH
From: Paul Hahn
To: Tuning Digest
Subject: Re: All, then best, 7-limit scales with 31 consonances
Message-ID:
There are only three whose
smallest scale degree is not smaller than 25:24; they approximate 12TET
reasonably well. This is the (IMHO) best one:
42:25------21:20-------21:16
\'-. .-'/ \'-. .-'/ \'-.
\ 6:5--/---\--3:2--/---\-15:8
\ /|\ / \ /|\ / \ /|
\ | / \ | / \ |
/ \|/ \ / \|/ \ / \|
/ 7:5---------7:4--------35:32
/.-' '-.\ /.-' '-.\ /.-'
8:5---------1:1---------5:4
You can get the other two (a) by substituting 25:16 for 8:5, or (b) by
inverting this one (which, transposed, can also be thought of as
substituting 25:16 for 8:5 and 105:64 for 42:25).
--pH
From: "M. Schulter"
To: Tuning Digest
Subject: Re: Corrections on my "What is a Wolf?"
Message-ID:
In tunings of the later 15th-17th centuries, we run into a different
kind of "Wolf" complication: the fact that three pure major thirds
of 5:4 or ~386.31 cents each
do not add up to a pure 3:2 octave,
but fall about 41.06 cents short.
f#' gb'
(1586,772) (1628,814)
bb bb
(814) (814)
d d
f#' f#'
(697,270) (696,310)
eb' d#'
(427) (386)
b b
eb' eb'
(737,310) (696,310)
c' c'
(427) (386)
g# ab
g# ab g#
(697,270) (696,310) (696,310)
f f e#
(427) (386) (386)
c# db c#
mschulter@value.net
From: "Paul H. Erlich"
To: Tuning Digest
Subject: Reply to Reinhard on Subharmonics
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B654D@MARS>
Now about "subharmonics": are these properly the same as "difference
tones," thus produceable as the mathematical difference heard as a
result
of 2 higher pitches rubbing up against each other?
(a) the simple-integer
ratios between all pairs of tones, and
(b) all tones have a common overtone (at 1) if they are harmonic (e.g.,
voice, bowed strings, brass instruments).
Isn't the octave, what Partch once called the "aura," carved up in
actual
music to include the 6/5 from the very the same organic core as the
5/4?
Could there even be a major withour a minor?
From: Carl Lumma
To:
To: Tuning Digest
Subject: Re: Orpheus and the Undertones, Subharmonics, Utonalities etc.
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B654E@MARS>
Partch credits Riemann and many others in preceding his =
"utonality" concept. The
concept has an intersting status =
around here, with a few giving it little to no importance =
(e.g., Heinz Bohlen)
I'm not sure what to make of that suggestion. Being an
equal-temperament,
clearly Bohlen's tuning approximates the same subharmonics as it does
harmonics, and equally accurately.
I
hope that I am not to infer from this fragment of Heinz Bohlen's
article, that
he views 88CET
as a competitor, or in any other way a threat, to his
tuning.
From: "Paul H. Erlich"
To: Tuning Digest
Subject: Lumma's challenge
Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6551@MARS>
35:24-------35:16------105:64
.-'/ \'-. .-'/ \'-. .-'/
5:3--/---\--5:4--/---\-15:8 /
/|\ / \ /|\ / \ /| /
/ | / \ | / \ | /
/ |/ \ / \|/ \ / \|/
/ 7:6---------7:4--------21:16
/.-' '-.\ /.-' '-.\ /.-'
4:3---------1:1---------3:2
Guys, wouldn't any rotation and/or reflection of this figure in the
tetrahedral/octahedral lattice work too?
Should. (I presume you mean only those about which it is symmetrical)
Just moving a structure in the
lattice (called translation?)
can change the number of intervals
the restriction of it being 7-limit makes
non-orthogonal rotations fail.
The other two orientations are -
35:24-------35:16------105:64
.-'/ \'-. .-'/ \'-. .-'/
5:3--/---\--5:4--/---\-15:8 /
/|\ / \ /|\ / \ /| /
/ | / \ | / \ | /
/ |/ \ / \|/ \ / \|/
/ 7:6---------7:4--------21:16
/.-' '-.\ /.-' '-.\ /.-'
4:3---------1:1---------3:2
35:27-------35:18
\'-. .-'/ \
\ 10:9--/---\--5:3
\ |\ / \ /|\
\ | / \ | \
\|/ \ / \| \
14:9---------7:6 \
\'-.\ /.-'/ \'-.\
\ 4:3--/---\--1:1
\ |\ / \ /|\
\ | / \ | \
\|/ \ / \| \
28:15-\---/--7:5 \
'-.\ /.-' '-.\
8:5---------6:5
5:4
/|\
/ | \
/ | \
/ 7:4 \
/.-'/ \'-.\
1:1--/---\--3:2
/|\ / \ /|
/ | / \ |
/ |/ \ / \|
/ 7:5--------21:20
/.-'/ \'-.\ /.-'/
8:5--/---\--6:5 /
|\ / \ /| /
| / \ | /
|/ \ / \|/
28:25-------42:25
\'-.\ /.-'/
\ 48:25 /
\ | /
\ | /
\|/
168:125
35:27-------35:18
\'-. .-' / \'-.
\ 10:9--/---\--5:3
\ |\'/. .\'/|\'-.
\ | / 40:21-\-|-\-10:7
\|/ \ |\ / \| \ /|\
14:9---------7:6 \ | \
'-.\|/.\' '-.\| \
4:3---------1:1 \
'-.\ /.-' '-.\
8:7--------12:7
From: Paul Hahn
To: Tuning Digest
Subject: Re: Lumma's challenge
Message-ID:
Paul Hahn wrote,
the restriction of it being 7-limit makes
non-orthogonal rotations fail.
What does that mean? Being 7-limit implies no such restrictions.
--pH
From: Paul Hahn
To: Tuning Digest
Subject: Re: Lumma's challenge
Message-ID:
There are certainly more than three orientations.
[bigsnip]
and many more . . .
--pH
From: Harold Fortuin
To: Tuning Digest
Subject: Finale xen-cidentals
Message-ID: <3663442C.52E1@wavefront.com>
shift-i = 1.5 flat (regular flat fused to the above)
option-m = 1/2 sharp (1 stem, instead of 2)
shift-option-n = 1.5 sharp (3 stems, instead of 2)
shift-b = backward flat (backwards and not filled in)
From: bram
To: Tuning Digest
Subject: Re: Lumma's challenge
Message-ID:
Paul Hahn wrote,
the restriction of it being 7-limit makes
non-orthogonal rotations fail.
What does that mean? Being 7-limit implies no such restrictions.
*************************
or try some definitions.
Let me know if you don't understand something.
