From: tuning@onelist.com
To: Joe Monzo
Date: Thu, 3 Dec 1998 11:10:57 -0500 (EST)
Subject: TUNING digest 1600
TUNING Digest 1600
Topics covered in this issue include:
1) Re: triangular vs. rectangular lattices
by monz@juno.com
2) Meet George...
3) Jetsons chord
4) Howlin Wolf
5) Lumma's 7-limit Delight
6) His boy Elroy . . .
7) Re: Lumma's 7-limit Delight
8) Re: Max-interval 7-limit JI scales, yet again
9) Re: Max-interval 7-limit JI scales, yet again
10) Re: Otonality/utonality: trines and triads
11) Re: m6 as 128:81 or 8:5 -- response to Gary Morrison
12) my lattices in ASCII
13) TUNING digest 1599
14) Re: m6 as 128:81 or 8:5 -- response to Gary Morrison
15) 7-limit diamond as a scale
----------------------------------------------------------------------
Topic No. 1
Date: Wed, 2 Dec 1998 13:20:42 -0800
Although I'm quite impressed by the artistry of
the ASCII lattices I've seen here lately, the fact
still remains that ASCII is woefully inadequate to
portray the kinds of complex visual mapping we're
talking about here. (I plan on eventually giving my
version of a lattice diagram for each of those presented
here in the Tuning Digest Archives on the Sonic Arts site;
only had time to do a few of them so far.)
Probably due to my stance as regards the prime-vs-odd
dichotomy (in favor of primes), my lattices are sort
of rectangular, and are definitely not triangular.
However, the complex multi-dimensional reduction
Erlich's referring to here certainly comes into play
in my diagrams and my theories, with different angles
and lengths for different prime vectors.
The main difference between Erlich's and my positions
seems to be that he places more emphasis on intervals
per se than I do. Or, put another way, my theory can
be considered to be more reductionist than his.
I'm not familiar with Tenney's work yet, and was not
aware that anyone else had ever designed a visual
representation of tonal structures where "each prime
interval is given a length proportional to the log of
the corresponding prime number". This sounds quite
similar to my lattices.
I'm very interested, in light of the ideas of "finity"
and "bridging" which I've been discussing, in further
work on these visual representations, particularly
the idea of reducing an infinite JI conception to
a finite one, and the associated cognitive processes.
In a private message, Dave Hill said:
This leads me to believe that the right brain would be
experiencing the harmonic aspect of music with an
accuracy beyond anything that can be described with
numbers (unless it's just-intonation, and really tuned
precisely to those numbers, which hardly ever happens
in real performance, with the exception of electronics),
while the left brain experiences the finity and bridging
due to its prediliction for analysis, categorization,
and pattern-recognition. Interesting area for research.
- Joe Monzo
------------------------------
Topic No. 2
Date: Wed, 02 Dec 1998 16:36:25 -0800
Paul Hahn wrote...
The rhombic dodecahedron and the cuboctahedron are duals, so they should
share the same symmetries. But I'm not sure where you're getting 48...
Paul Erlich made a very interesting post...
You say you've done this for various scales? Do you have any pics?
Carl
------------------------------
Topic No. 3
Date: Wed, 2 Dec 1998 13:51:04 -0800
[Hahn, TD#1599.4:]
Wouldn't it have been nice if my youthful TV-watching
days really were filled with microtonal chords like
that!
- Joe Monzo
------------------------------
Topic No. 4
Date: Wed, 2 Dec 1998 13:40:04 -0800
[Haverstick, TD#1599.1:]
Boy, I sure do wish I (or someone else!) could
do a thorough microtonal analysis of Howlin Wolf.
He was really one of the greats...and some of
the intervals *he* sang were far more complicated
than anything in meantone!
- Joe Monzo
------------------------------
Topic No. 5
Date: Wed, 2 Dec 1998 13:48:14 -0800
I've tried counting the consonances on these
diagrams, and I see either less or *way* more
than 30, 31, 32...I'm not sure how you're
defining "consonance".
Could you guys (Lumma, Hahn, Erlich, Bram) please
outline for us clod-heads exactly what you're
debating here?
[Hahn, TD#1599.4:]
Please, give me some web links that I can put into
the TD Archive version of this, so those of us
who aren't familiar with this material can learn
about it.
[Hahn:]
By all means, if you have the time to do it,
draw the lattices. In my opinion, it's a great
aid to understanding the theoretical concepts.
- Joe Monzo
------------------------------
Topic No. 6
Date: Wed, 2 Dec 1998 16:11:58 -0600 (CST)
On Wed, 2 Dec 1998, Carl Lumma wrote:
Yuppers.
Imagine the faces of the rhombic dodec divided into four little right
triangles. 12 faces, each divided in four, gives 48 pieces, right? And
each of those little 48 triangles can be brought to coincide with any of
the others through an appropriate rotation/reflection of the dodec
(after which the dodec will still coincide with itself as a whole, as
well). That's what "how many members of the symmetry group" means, and
that's what Paul E. was asking.
If you prefer thinking of the cubocta instead of the rhombic dodec, the
little triangles I was talking about correspond to half an edge of the
cubocta. 2 halves per edge times 24 edges still gives you 48.
------------------------------
Topic No. 7
Date: Wed, 2 Dec 1998 16:41:50 -0600 (CST)
On Wed, 2 Dec 1998 monz@juno.com wrote:
Could you guys (Lumma, Hahn, Erlich, Bram) please
outline for us clod-heads exactly what you're
debating here?
Okay. Remember the ASCII lattice I drew that you liked so much?
A 7-limit consonance is any of the drawn lines in this diagram which
directly connects one ratio to another without anything in between. In
this diagram there are eight 3:2s, six 5:4s, six 7:4s, four 6:5s, four
7:6s, and three 7:5s. 8+6+6+4+4+3 makes 31 7-limit consonances in this
scale.
Okay. When I have time.
------------------------------
Topic No. 8
Date: Wed, 2 Dec 1998 15:55:46 -0800 (PST)
I have an idea.
Consider the following 13-tone tuning, which I have split into two
diagrams, each of which is missing some notes:
This tuning has a number of interesting properties. First of all, it makes
a really interesting shape geometrically in 3-space (I forget the name of
the thing.) Second, it doesn't change when subjected to any sort of
limit-preserving rotation. Third, it's extremely consonant. Fourth, none
of the ratios has a numerator or denominator greater than ten when
multiplied by the correct power of 2 to get it within the octave. (None of
these are coincidental, by the way.)
One could make a nice 12-tone tuning by removing one of the outside
ratios. However, I think it makes a *lot* more sense musically to instead
remove unity, and play a chord consisting of 2,1, and 1/2 as a drone
(maybe throw 4 and 1/4 in there for good measure.)
Any musical piece played in this way would have the very interesting
property that it could by simple transformation be converted into 23 other
musical pieces using the same tuning, each equally consonant but with a
different flavor.
I think it's also reasonably evenly spaced.
-Bram
------------------------------
Topic No. 9
Date: Wed, 2 Dec 1998 16:40:34 -0800 (PST)
I messed up my numbers a bit [previous post].
The corrected diagrams are as follows:
Ok ... I think I got 'em right that time.
-Bram
------------------------------
Topic No. 10
Date: Wed, 2 Dec 1998 17:46:14 -0800 (PST)
Hello, there.
In a recent article (Tuning Digest 1599), Joe Monzo discusses the
relationship between the concepts of triads, modality vs. the
major/minor system, and what has been called otonality/utonality.
Maybe as one way of marking the 60th anniversary of Joseph Yasser's
famous series on "Medieval Quartal Harmony," I'd like to note that the
otonal/utonal concept might be applied to Gothic as well as later
harmonic styles based on "complete" or "perfect" sonorities involving
three (or more) voices and intervals.
For readers unfamiliar with the term trine, I should explain that
it's an English term I've derived from the trina harmoniae perfectio
of Johannes de Grocheio (c. 1300), who notes that the polyphony of his
day is based on a "perfect" sonority of three voices and intervals:
outer octave, lower fifth, and upper fourth. This sonority is
analogous to the later 5-limit triad, and the still later tetrads and
more complex constructions of 7-limit and higher systems.
Interestingly, one could argue that an "otonality/utonality" contrast
might apply not only to Renaissance-Romantic music based on the triad
(with an ideal 5-limit tuning of 4:5:6), but also to Gothic music
based on the trine (with an ideal 3-limit tuning of 2:3:4).
In both systems, we can take the same three intervals which define a
"complete" sonority, and arrange them in two different ways:
Here I use a short form of notation outer|lower-upper. Thus 8|5-4
might be read, "the sonority with an outer octave `split' by a third
voice into a lower fifth and upper fourth."
In fact, composers of the 13th century use both forms of the trine,
but with the 8|5-4 form (fifth below fourth) clearly preferred for
conclusions, and 8|4-5 felt as less smooth and conclusive. Around
1300, Coussemaker's Anonymous I notes this distinction in three-voice
sonorities, and Jacobus of Liege (possibly the same writer at a more
advanced age, c. 1325) discusses it at length.
Similarly, by the mid-16th century epoch of Vicentino and Zarlino,
composers and theorists alike find in a Renaissance setting that
5|M3-m3 seems more conclusive than 5|m3-M3.
While medieval and Renaissance theory focuses on string-ratios rather
than the later concept of frequency-ratios, one might from a later
viewpoint analyse the "otonal" 8|5-4 and 5|M3-m3 as 2:3:4 and 4:5:6,
and the "utonal" 8|4-5 and 5|m3-M3 as 1/2:1/3:1/4 and 1/4:1/5:1/6.
Yasser plays around with some of these parallels, and although my
analysis of Gothic polyphony is rather different in detail, I am much
indebted to his basic insight that medieval harmony should indeed be
approached as an independent system worthy of appreciation in its own
terms.
Most respectfully,
Margo Schulter
------------------------------
Topic No. 11
Date: Wed, 2 Dec 1998 18:25:19 -0800 (PST)
Recently Gary Morrison asked an interesting philosophical question: in
a medieval setting, would I consider a Pythagorean m6 (128:81, ~792.18
cents) as a variant of 8:5 (~814.69 cents)?
Here there could really be at least three points of view.
From a stylistic point of view, I would say that a Pythagorean 128:81
is the expected size for a 3-limit minor sixth, just as 8:5 is in
Renaissance 1/4-comma meantone, or 800 cents in
12-tet keyboard music.
If we consider the problem from the viewpoint of "native language,"
then I guess that I've "grown up" mainly on 12-tet, despite my more
recent fascinations with Pythagorean and meantone as well as n-tet's.
Finally, I've seen the argument here that there may be a basic
tendency of listeners across cultural traditions to hear small integer
ratios as "basic," and from this view one might argue that even in a
setting where 128:81 or 800/1200 octave is the norm for a minor sixth,
these intervals are in some sense heard as "quasi-8:5 ratios."
One complication with the minor sixth is that even in the "pure" 8:5
form, there is an acoustical tension between the third partial of the
fundamental and the second partial of the upper note, these partials
forming a 16:15 semitone:
One might argue that this semitonal friction could explain, in a
3-limit setting, why M3 (81:64) is regarded as relatively concordant
but m6 (128:81) as a strong discord in much 13th-century theory,
although both ratios look comparably complex.
Most respectfully,
Margo Schulter
------------------------------
Topic No. 12
Date: Wed, 2 Dec 1998 21:08:03 -0800
Since everyone's drawing ASCII lattices
these days, just thought I'd toss this out
to the list.
Here's an 11-note scale (most of the 7-limit
tonality diamond) in the triangular style used
by Paul Hahn, Carl Lumma, and Paul Erlich:
Simplifying this structure into a rectangular
form keeps the same notes, but eliminates the
illustration of some of the relationships:
After reading Erlich's description of Tenney's
lattices with proportional vector-lengths, I
made an attempt at drawing my own lattice design
in ASCII. In my design, which is similar to the
rectangular form above, lengths are proportional
to the prime's place in the prime series, and
angles are representative of circular "octave"
pitch deployment for each interval. (Prime-factor
notation and Semitone values are also given with
the ratios.)
I'd be interested to see an analysis of this
(along the lines of recent posts) and also
what variations you guys come up with using
this format.
- Joe Monzo
------------------------------
Topic No. 13
Date: Thu, 3 Dec 1998 06:57:36 -0500
Dave Hill wrote [1599.8:]
Okay, but the other things were far from equal. On early keyboards, the
guage used for the wire was narrower, so they would have behaved more lik=
e
ordinary strings than does modern piano wire. Further, I recall seeing
sonograms from both IRCAM and Robert Cogan showing that the amplitude of
the partials above the fundamental in harpsichords, clavichords and
fortepianos was, overall, higher than in modern pianos, so I would suspec=
t
that sensitivity to tuning vis-a-vis the overtone stucture would have
actually been higher. Even adding a caveat for the complication that
historical production techniques probably led to a more uneven distributi=
on
of mass along the wire, I am generally convinced that sensitivity to
intonation was greater then.
I may also add, as a bit of music-cultural speculation, that the creation=
of the the tuning profession probably had a net effect of densensitizing
players to the quality of keyboard intonation. Whereas in earlier
generations the keyboard player (organists largely excepted) would have
been responsible for his or her own tuning, the institution of the
professional tuner removed this element of performance practice from the
players' responsibility and tuning, as far as the player was concerned, w=
as
"found", not chosen.
This situation has changed somewhat today. In addition to players of earl=
y
instruments (who are almost universally able to tune themselves), there a=
re
a few piano players out there (Uchido, Jarrett, Nurit Tilles, Cecil Lytle=
,
Michael Harrison) who have learned to use tuning hammers. Unfortunately,
there are still a lot of recording studios (e.g. European radio stations)=
with firm contracts with their piano technicians which forbid recordings
from being made on any of these instruments unless tuned by the house
technician. =
------------------------------
Topic No. 14
Date: Thu, 03 Dec 1998 07:45:37 -0500
Actually, although the context was Medieval, I intended the question in
a general sense. And I intended it more as an auditory-intuitiveness
question more than a philosophical or theoretical one.
Using a related 3-limit interval as another example, I personally have
never managed to attribute any intuitive meaning to 81:64. To me it sounds
like an off-5:4 much more than anything meaningful in itself. It's just
too complicated a pitch relationship very close to a vastly more obvious
one.
There is another possible explanation though: Exactly opposite of
Margo, I have almost no time whatsoever with 81:64, or 3-limit tunings in
general. Historically, I've been much more interested in the other end of
the spectrum: 7s, 11s, 13s, and such. In fact, I may even be able to
count the total number of times I've intentionally confronted myself with
81:64 on my fingers and toes. If I were to listen to it more an intuitive
meaning for it might become apparent.
------------------------------
Topic No. 15
Date: Thu, 3 Dec 1998 08:55:25 -0600 (CST)
On Wed, 2 Dec 1998, bram wrote:
This tuning has a number of interesting properties. First of all, it makes
a really interesting shape geometrically in 3-space (I forget the name of
the thing.)
Cuboctahedron.
One could make a nice 12-tone tuning by removing one of the outside
ratios.
These are the group of 31-consonance scales I mentioned earlier.
I think it's also reasonably evenly spaced.
To use a Clintonism, it depends on what your idea of "reasonably even"
is. 8-)> The largest single-step interval in your scale is 64:49; the
smallest is 50:49--that's a pretty big difference. The largest two-step
interval is 4:3; the smallest is 21:20. And so on. I wouldn't call
that "reasonably even".
------------------------------
End of TUNING Digest 1600
I welcome feedback about this webpage: corrections, improvements, good links.
by Carl Lumma
by monz@juno.com
by monz@juno.com
by monz@juno.com
by Paul Hahn
by Paul Hahn
by bram
by bram
by "M. Schulter"
by "M. Schulter"
by monz@juno.com
by Daniel Wolf
by Gary Morrison
by Paul Hahn
From: monz@juno.com
To: Tuning Digest
Subject: Re: triangular vs. rectangular lattices
Message-ID: <19981202.132047.-147713.6.monz@juno.com>
Can we all agree that the triangular lattice is
more useful? There is nothing to prevent us form
defining a city-block dissonance metric like the
Tenney Harmonic Distance function on the triangular
lattice, where 6:5 would be represented by a single
segment and thus have a smaller Harmonic Distance
than 15:8, which is represented by two segments
(Tenney's HD is defined on a rectangular matrix
and thus gives these two intervals equal dissonance).
I've suggested giving each interval a length proportional
to the log of its odd limit. I came up with this before
I heard of Tenney's idea, where each prime interval
is given a length proportional to the log of the
corresponding prime number, and before I saw any
3-d triangular lattices in music theory.
There was an article in Scientific American on
follow up research to split–brain research performed in
the 1960s. It seems the two brain halves generally
perceive the same things very differently, with the
right brain responding more to the raw stimuli and the
left brain processing the inputextensively and
interpreting it according to learned structures,
classifications etc. In individuals who have had
their brains severed into two halves (treatment for
intractable epilepsy), I believe the responses of the
two halves of the brain are sometimes very different,
with one half responding positively and the other
responding negatively to the same inputs (presented in
different halves of the visual field - I don't know
about auditory work done here).
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
From: Carl Lumma
To: Tuning Digest
Subject: Meet George...
Message-ID: <19981202213538906.AAA209@nietzsche>
Forty-eight. (Simple way to figure it: the Voronoi cell is a rhombic
dodecahedron, which has 12 faces each of which is fourfold symmetric.)
The original genus could only have twelve distinct orientations because
it itself was fourfold symmetrical. However, if you use an assymmetrical
shape, like the "Jetsons" chord 1:1 5:4 7:5 3:2, it has 48 distinct
rotations and reflections.
For example, a diatonic or extended
meantone scale, when scaled to three
dimensions, comes out as a helix, with the chain of fifths winding around
the helix so that one full turn corresponds to 3-4 fifths, putting all the
notes of each consonant triad near to one another. The result is the
simplest and most informative diagram of the diatonic scale or extended
meantone tuning that I can imagine.
From: monz@juno.com
To: Tuning Digest
Subject: Jetsons chord
Message-ID: <19981202.135122.-147713.9.monz@juno.com>
if you use an assymmetrical shape, like the
"Jetsons" chord 1:1 5:4 7:5 3:2,...]
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
From: monz@juno.com
To: Tuning Digest
Subject: Howlin Wolf
Message-ID: <19981202.135122.-147713.7.monz@juno.com>
I had a dream last night that Howlin Wolf
walked in ...
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
From: monz@juno.com
To: Tuning Digest
Subject: Lumma's 7-limit Delight
Message-ID: <19981202.135122.-147713.8.monz@juno.com>
...the Voronoi cell is a rhombic dodecahedron,
which has 12 faces each of which is fourfold
symmetric.)
I was going to ASCII-draw them all, but then
I thought better of it. 8-)>
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
From: Paul Hahn
To: Tuning Digest
Subject: His boy Elroy . . .
Message-ID:
Paul Hahn wrote...
Forty-eight. (Simple way to figure it: the Voronoi cell is a rhombic
dodecahedron, which has 12 faces each of which is fourfold symmetric.)
The original genus could only have twelve distinct orientations because
it itself was fourfold symmetrical. However, if you use an assymmetrical
shape, like the "Jetsons" chord 1:1 5:4 7:5 3:2, it has 48 distinct
rotations and reflections.
The rhombic dodecahedron and the cuboctahedron are duals, so they should
share the same symmetries.
But I'm not sure where you're getting 48...
--pH
From: Paul Hahn
To: Tuning Digest
Subject: Re: Lumma's 7-limit Delight
Message-ID:
I've tried counting the consonances on these
diagrams, and I see either less or *way* more
than 30, 31, 32...I'm not sure how you're
defining "consonance".
35:24-------35:16------105:64
.-'/ \'-. .-'/ \'-. .-'/
5:3--/---\--5:4--/---\-15:8 /
/|\ / \ /|\ / \ /| /
/ | / \ | / \ | /
/ |/ \ / \|/ \ / \|/
/ 7:6---------7:4--------21:16
/.-' '-.\ /.-' '-.\ /.-'
4:3---------1:1---------3:2
[Hahn:]
I was going to ASCII-draw [all 48 orientations of the Jetsons chord],
but then I thought better of it. 8-)>
By all means, if you have the time to do it,
draw the lattices.
--pH
From: bram
To: Tuning Digest
Subject: Re: Max-interval 7-limit JI scales, yet again
Message-ID:
5:4---------5:3
/ \'-. .-'/ \
/ \ 7:4 / \
/ \ /|\ / \
/ X | X \
/ / \|/ \ \
4:3------/--1:1--\------3:2
\'-. /.-'/ \'-.\ .-'/
\ 7:5---------7:6 /
\ | / \ | /
\ | / \ | /
\|/ \|/
6:5---------8:5
5:4---------5:3
/|\ /|\
/ | \ / | \
/ | \ / | \
/ 10:7--\---/--7:6 \
/.-' \'-.\ /.-'/ '-.\
4:3---------1:1---------3:2
\ \ /|\ / /
\ X | X /
\ / \|/ \ /
\ / 8:7 \ /
\ /.-' '-.\ /
6:5---------8:5
From: bram
To: Tuning Digest
Subject: Re: Max-interval 7-limit JI scales, yet again
Message-ID:
5:4---------3:2
/ \'-. .-'/ \
/ \ 7:4 / \
/ \ /|\ / \
/ X | X \
/ / \|/ \ \
5:3------/--1:1--\------6:5
\'-. /.-'/ \'-.\ .-'/
\ 7:6---------7:5 /
\ | / \ | /
\ | / \ | /
\|/ \|/
4:3---------8:5
5:4---------3:2
/|\ /|\
/ | \ / | \
/ | \ / | \
/ 10:7--\---/-12:7 \
/.-' \'-.\ /.-'/ '-.\
5:3---------1:1---------6:5
\ \ /|\ / /
\ X | X /
\ / \|/ \ /
\ / 8:7 \ /
\ /.-' '-.\ /
4:3---------8:5
From: "M. Schulter"
To: Tuning Digest
Subject: Re: Otonality/utonality: trines and triads
Message-ID:
I've pointed out here recently that the dualistic
major/minor system could only happen, as Partch
observed in Genesis, [p. 88-90] because of the
inherent dual relationship of a two-member ratio,
and [p 109-118] the use of at least three tones
to delimit a tonality.
3-limit trine (8 + 5 + 4) 5-limit triad (5 + M3 + m3)
|d' |d' |g |g
| 4 | 5 | m3 | M3
8 |a 8 |g 5 |e 5 |eb
| 5 | 4 | M3 | m3
|d |d |c |c
8|5-4 8|4-5 5|M3-m3 5|m3-M3
mschulter@value.net
From: "M. Schulter"
To: Tuning Digest
Subject: Re: m6 as 128:81 or 8:5 -- response to Gary Morrison
Message-ID:
c' 8 c'' 16
e 5 b' 15
mschulter@value.net
From: monz@juno.com
To: Tuning Digest
Subject: my lattices in ASCII
Message-ID: <19981202.210807.-147713.11.monz@juno.com>
5:3---------5:4
/|\ /|\
/ | \ / | \
/ | \ / | \
/ 7:6---------7:4 \
/.-' '-.\ /.-' '-.\
4:3---------1:1---------3:2
\'-. .-'/ \'-. .-'/
\ 8:7--/---\--12:7 /
\ | / \ | /
\ | / \ | /
\|/ \|/
8:5---------6:5
5:3---------5:4
/ /
/ /
/ /
/ 7:6---------7:4
/.-' /.-'
4:3---------1:1---------3:2
.-'/ .-'/
8:7--/------12:7 /
/ /
/ /
/ /
8:5---------6:5
12:7
3^1*7^-1____ 3:2
9.33 `---._____ 3^1_
/ 7.02`-._
/ 5:4 / `-._ 6:5
/ 5^1_ / `3^1*5^-1
8:7 3.86 `-._ / 3.16
7^-1_____/ `-._ 1:1 /
2.31 /`-----._____`n^0_____ / 7:4
/ 0.00 `-._`-----._/______7^1
5:3 / `-._ 8:5 9.69
3^-1*5^1 / `5^-1 /
8.84 `-_ / 8:14 /
`-._ 4:3 /
`3^-1_____ 7/6
4.98 `----._____ 3^-1*7^1
2.67
monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
From: Daniel Wolf
To: Tuning Digest
Subject: TUNING digest 1599
Message-ID: <199812030657_MC2-624B-F786@compuserve.com>
Note 2: - an observation: although on a modern piano
the differences in character between just or mean
tone music and equal tempered music are striking to
many people, these differences are likely to have
been less marked on early pianos, particularly those
in existence before about 1800, which were not
strong enough to hold a bank of strings under such
high tension as strings on a modern piano are kept
at. At lower tension, other things being equal, the
inharmonicity of the partials is greater and as a
result, the stretching of the partials and
consequent "tuning ambiguity" is greater for
such pianos having less mechanical strength and
whose strings were kept at lower tension.
From: Gary Morrison
To: Tuning Digest
Subject: Re: m6 as 128:81 or 8:5 -- response to Gary Morrison
Message-ID: <366687E9.B8D5F1FB@texas.net>
Recently Gary Morrison asked an interesting philosophical question: in
a medieval setting, would I consider a Pythagorean m6 (128:81, ~792.18
cents) as a variant of 8:5 (~814.69 cents)?
From: Paul Hahn
To: Tuning Digest
Subject: 7-limit diamond as a scale
Message-ID:
> 5:4---------3:2
> / \'-. .-'/ \
> / \ 7:4 / \
> / \ /|\ / \
> / X | X \
> / / \|/ \ \
> 5:3------/--1:1--\------6:5
> \'-. /.-'/ \'-.\ .-'/
> \ 7:6---------7:5 /
> \ | / \ | /
> \ | / \ | /
> \|/ \|/
> 4:3---------8:5
>
>
> 5:4---------3:2
> /|\ /|\
> / | \ / | \
> / | \ / | \
> / 10:7--\---/-12:7 \
> /.-' \'-.\ /.-'/ '-.\
> 5:3---------1:1---------6:5
> \ \ /|\ / /
> \ X | X /
> \ / \|/ \ /
> \ / 8:7 \ /
> \ /.-' '-.\ /
> 4:3---------8:5
[snip]
[snip]
--pH
*************************
or try some definitions.
Let me know if you don't understand something.
