Re: The Golden Mediant: Complex ratios and metastable intervals

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The Golden Mediant:
Complex ratios and metastable musical intervals
---------------------------------------------------
by Margo Schulter and David Keenan
---------------------------------------------------

One feature of European music of the 13th and 14th centuries using
Pythagorean tuning with pure fifths, is a predilection for thirds and
sixths with complex ratios. This is also a feature of "neo-Gothic"
temperaments in the most characteristic range between Pythagorean and
17-tET, with fifths somewhat wider than pure.

Pythagorean major and minor thirds at the large integer ratios of 81:64
(407 cents) and 32:27 (294 cents), and also neo-Gothic thirds at or near
14:11 (417 cents) and 13:11 (289 cents), are prized for their complex and
active quality.[1] They often serve as points of directed tension or
instability, resolving to stable 3-limit sonorities with pure or near-pure
fifths and fourths.

Since for many modern readers, as opposed to medieval or neo-Gothic
theorists, an integer ratio may imply an interval simple enough to be tuned
by ear through a "locking-in" of the partials, we emphasize that no such
implications necessarily attach to the larger integer ratios mentioned in
this paper.[2]

In a dialogue on the role of these intervals in Gothic and neo-Gothic
music, we were both intrigued by the concept that they may draw their
appeal precisely from their complexity.

We conceive of major thirds in regular tunings between Pythagorean and
17-tET as located on a gently rounded plateau region between the notch-like
valleys of 5:4 and 9:7. Minor thirds are likewise situated on a plateau of
complexity between the simpler 7:6 and 6:5.

The central portions of such plateaux might be called regions of
maximal "harmonic entropy" (Paul Erlich)[3] or "sensory dissonance"
(William Sethares)[4] or "complexity" [M.S.].[5]

Even while we were engaged in an absorbing dialogue on these plateau
regions and their mathematical and musical nature, Keenan Pepper, in a
delightful synchronicity, posted an article to the Tuning List[6] on the
application of the Golden Ratio or Phi to another area of music: the
generation of scales with particular relationships between scale steps.

One of us recognized that Pepper's Phi-related function could also be
applied to the problem of finding the region of maximum complexity
between two simpler ratios, providing a shortcut to the longer process
of successive approximation by iterating mediants.

In what follows, we show how this function, here termed the "Golden
Mediant", can be used to locate regions of maximum complexity.

Section 1 presents the classic mediant function, and Section 2 the process
of finding closer and closer approximations to such regions of maximum
complexity.

Section 3 shows how the Golden Mediant can simplify this process, yielding
results in general agreement with the local maxima of Paul Erlich's
harmonic entropy or William Sethares' sensory dissonance. Both of these
must be calculated using complicated computer algorithms and do not admit
of a simple closed-form expression like the Golden Mediant.

Section 4 briefly considers the classic mediant and Golden Mediant in
relation to the tuning of certain common unstable intervals along the most
characteristic portion of the neo-Gothic spectrum from Pythagorean to
17-tone equal temperament (17-tET).

While our perspectives may somewhat differ, we strongly agree that the
Golden Mediant provides new cardinal points of orientation in exploring the
subtle shadings of neo-Gothic intervals and tunings.

At the same time, we emphasize that the problem of locating plateau regions
may be of interest for various musics, and that we welcome the application
of the concepts here described, to a variety of styles.

---------------------------------------------------
1. The classic mediant and plateaux of complexity
---------------------------------------------------

In locating a region of complexity between two simpler ratios such as 5:4
and 9:7, one useful index is the _mediant_ of the two ratios. For any two
ratios i:j and m:n, this mediant is defined as the sum of the two
numerators over the sum of the two denominators:

(i + m)
-------
(j + n)

To illustrate this mediant formula, let us apply it to 5:4 and 9:7, finding
a size of major third which may be close to the central plateau region of
equal remoteness from both valleys:

(5 + 9)
-------
(4 + 7)

This value, as it happens, is identical with the favored 14:11 ratio for a
major third in neo-Gothic theory, giving this ratio a new mathematical
significance fitting its intriguing musical qualities of instability and
complexity in Gothic or neo-Gothic styles. This mediant is very closely
approximated, for example, by 46-tET.

Finding the mediant of 6:5 and 7:6, two simple ratios or valleys for minor
thirds, gives a similar meaning to another favored neo-Gothic ratio:

(6 + 7)
-------
(5 + 6)

This mediant is identical to the neo-Gothic 13:11, closely approximated for
example by 29-tET, a leading neo-Gothic tuning.

If we wish to make these mediant relationships of complexity explicit, or
avoid the assumption that these are to be considered as relative
_consonances_ of say an 11-limit or 13-limit just intonation, we can leave
14:11 written as (5+9):(4+7), and likewise 13:11 as (6+7):(5+6).

Here we shall refer to this mediant of (i+m):(j+n) as the "classic mediant"
to distinguish it from the Phi-based "Golden Mediant" we shall describe
below.

-----------------------------------------------
2. Refining our estimates: the Fibonacci series
-----------------------------------------------

>From one viewpoint, the point of maximum complexity between two simple
interval ratios is like the highest point of a gently rounded plateau
between valleys, a familiar metaphor in relation to Paul Erlich's and
William Sethares' studies and charts.

From another viewpoint, it might be compared to the point of equal
gravitational attraction between two planets or planetlike bodies such as
the Earth and Moon.

If an object were placed at such a point, a physicist would say it was
in a _metastable_ state. This is understood to be a special kind of
_un_stable state, one which may persist for a very long time, but not
forever, since the slightest perturbation of the object will see it
eventually tumble all the way to one side or the other. We think the
term metastable may also be descriptive of the quality of the
corresponding musical intervals.

The planetary metaphor suggests a refinement in our process of
approximating the point of metastability. Since the Earth is larger than
the Moon, and exerts a greater gravitational attraction, we find that the
point of equal attraction is actually located somewhat closer to the Moon
than to the Earth, roughly at about 3/4 of the way from the Earth to the
Moon.

Similarly, while both 4:3 and 5:4 are simple or "planetlike" ratios, the
4:3 has a greater degree of simplicity or attraction, so that we might
expect the point of maximum complexity or ambiguity to be somewhat closer
to 5:4. The classic mediant already gives us this result to some degree.
(4+5):(3+4) (or 9:7) is indeed closer to 5:4 than to 4:3.

However, in this case we find that 9:7 is itself simple enough to be a weak
attractor and greater complexity can be obtained by taking the mediant of
9:7 with the less simple of its predecessors, giving (5+9):(4+7).

This is the complex major third known in neo-Gothic theory as the
14:11, at around 417.5 cents. It is instructive to note that this
interval is about 31 cents wider than 5:4 (386.3 cents), and about 18
cents narrower than 9:7 (435.1 cents). This position of the classic
mediant somewhat closer to the less simple 9:7 fits our intuitive
expectation that the region of rough gravitational equality should be
closer to the less powerful attractor or "planet."

One might feel justified in stopping when the resulting ratio is too
complex to be considered an attractor, but if we want the _most_ complex
ratio we think the process should be continued.

The mediant of 9:7 and 14:11, is known in neo-Gothic theory as 23:18, and
is located around 424.4 cents, or about 38 cents from 5:4 and 11 cents from
9:7. The major third of 17-tET, at 423.5 cents, is quite close to this
intermediate ratio.

As we progress through the successive mediants, our approximations
gradually converge toward a limit about which they oscillate more and more
closely. At this stage we will drop the 4:3 and consider the series to have
begun with the last two attractors to appear, 5:4 and 9:7, and we look at
the pattern of successive mediants.

----------------------------------------------------------------------
Mediant Ratio Cents Dist from: 5:4 9:7
----------------------------------------------------------------------
(5+9):(4+7) 14:11 417.5 +31.2 -17.6
(9+14):(7+11) 23:18 424.4 +38.1 -10.7
(14+23):(11+18) 37:29 421.8 +35.5 -13.3
(23+37):(18+29) 60:47 422.8 +36.4 -12.3
(37+60):(29+47) 97:76 422.4 +36.1 -12.7
(60+97):(47+76) 157:123 422.53 +36.21 -12.55
(97+157):(76+123) 254:199 422.47 +36.15 -12.61
. . . . .
----------------------------------------------------------------------

Curiously, as it happens, the region between 5:4 and 9:7 seems to resemble
the Earth-Moon system in that the point of gravitational parity appears to
be situated about 3/4 of the way (in a logarithmic sense) from the more
powerful to the less powerful attractor.

As we progress through this series of approximations, our values for this
central plateau region of maximal complexity approach convergence at around
422.5 cents, or about 1 cent narrower than the 17-tET major third.

If the sides of the ratios are considered separately, each may be seen to
be a series of integers where every number after the second is the sum of
the preceding two numbers. We say they are Fibonacci-like.

Originally designed as a model for the reproduction of rabbits, the
Fibonacci series begins with the first two numbers 1, 1 -- each new member
of the series then being equal to the sum of the previous two members:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 ...

This famous number series was interestingly described by Leonard Fibonacci
of Pisa around 1200, the same era in which the composer Perotin and his
colleagues made composition for three and four voices a regular practice.
In his _Liber Abaci_, Fibonacci also introduced the decimal system of
Arabic numerals to Gothic Europe.

To apply this famous series to our problem of finding the region of maximum
complexity or gravitational balance between two simpler intervals, we begin
with a "weighted" version of the formula for the classic mediant (Section
1.1), where i:j is the simpler ratio and m:n the less simple ratio or less
strong "attractor":

(xi + ym)
---------
(xj + yn)

Here x and y are weights for the two ratios, with the second or "y" term
being applied to the _less simple_ ratio.

We show what happens when the weights are chosen to be successive members
of the Fibonacci series. This will be easier to understand if we apply it
to our problem of the zone of gravitational parity between 5:4 and 9:7,
stepping through the first few approximations near the beginning of the
Fibonacci series.

Our first two Fibonacci numbers are 1, 1, so that x=1 and y=1. This gives a
result identical to the classic mediant:

(1*5 + 1*9) (5 + 9)
----------- = ------- = 14:11
(1*4 + 1*7) (4 + 7)

Our next pair of Fibonacci numbers is 1, 2, giving us x=1, y=2. Note that
the larger value for y gives the less simple ratio more "weight," bringing
our estimate of the point of maximum complexity somewhat closer to 9:7.

(1*5 + 2*9) (5 + 18)
----------- = -------- = 23:18
(1*4 + 2*7) (4 + 14)

Our next Fibonacci pair of 2, 3 gives us x=2, y=3:

(2*5 + 3*9) (10 + 27)
----------- = --------- = 37:29
(2*4 + 3*7) (8 + 21)

As we progress through the successive Fibonacci pairs for our weights
x and y, we reproduce exactly the same successive approximations as
obtained above by iterating the mediant.

If we apply the same series of Fibonacci weights to the 6:5 (315.6 cents)
and 7:6 (266.9 cents), the 6:5 having the more powerful gravitational
attraction (or being situated in a deeper "valley"), we get these results,
starting with the classic mediant (6+7:5+6) where x=1, y=1:

----------------------------------------------------------------------
Fibonacci weights Wtd mediant Cents Dist from: 6:5 7:6
----------------------------------------------------------------------
x=1, y=1 (6+7):(5+6) 289.2 -26.4 +22.3
x=1, y=2 (6+14):(5+12) 281.4 -34.3 +14.5
x=2, y=3 (12+21):(10+18) 284.4 -31.2 +17.6
x=3, y=5 (18+35):(15+30) 283.3 -32.4 +16.4
x=5, y=8 (30+56):(25+48) 283.7 -31.9 +16.9
x=8, y=13 (48+91):(40+78) 283.56 -32.08 +16.69
x=13, y=21 (78+147):(65+126) 283.62 -32.02 +16.75
. . . . .
----------------------------------------------------------------------

Here the terms appear to be converging on a region around 283.6 cents, or
roughly 2/3 of the way from the wider 6:5 to the narrower 7:6. The central
plateau of maximum complexity or gravitational parity is again closer to
the shallower valley, or the planet with the less powerful attraction. On
the spectrum of regular neo-Gothic tunings, this region of maximal
complexity for the minor third is located between 46-tET (287.0 cents) and
17-tET (282.4 cents).

------------------------------------------------
3. A new index of complexity: the Golden Mediant
------------------------------------------------

As we have seen, the Fibonacci series of values for x and y beginning with
the classic mediant (x=1, y=1) offers us closer and closer approximations
converging on a limit which may indicate the region of maximal complexity
between two simpler intervals.

We can simplify the process by directly finding this limit itself, here
termed the "Golden Mediant," using a function like that applied to the
different area of scale generation by Keenan Pepper (see note 4).

As terms of the Fibonacci series grow larger and larger, the ratio between
any two successive terms converges on a value known as Phi, or the Golden
Ratio. Phi has the property:

1
----- = Phi - 1 or Phi^2 - Phi = 1
Phi

As a solution of the above quadratic one finds that Phi is (sqrt(5)+1)/2 or
approximately 1.61803398874989484820459.

Thus we can find the Golden Mediant for the region of maximum complexity
between two simpler intervals by setting x=1, y=Phi. For two such interval
ratios i:j and m:n where i:j is the simpler ratio:

(i + Phi * m)
GoldenMediant(i/j, m/n) = -------------
(j + Phi * n)

Interestingly, Phi itself occurs as the ratio of greatest complexity
between the major and minor sixths 5:3 and 8:5 (or between 8:5 and 13:8).

For a maximally complex major third between 5:4 and 9:7, or a maximally
complex minor third between 6:5 and 7:6, our new Phi-based function yields
these results:

----------------------------------------------------------------------
Intervals Golden Mediant Cents Dist from: i:j m:n
i:j m:n (i + m Phi):(j + n Phi)
----------------------------------------------------------------------
5:4 9:7 (5 + 9 Phi):(4 + 7 Phi) 422.5 +36.2 -12.6
----------------------------------------------------------------------
6:5 7:6 (6 + 7 Phi):(5 + 6 Phi) 283.6 -32.0 +16.7
----------------------------------------------------------------------

It is interesting to compare these results, and some others of relevance to
neo-Gothic music, with Paul Erlich's values to the nearest cent for the
regions of "maximum entropy" or complexity between these interval pairs:

----------------------------------------------------------------------
Intervals Measure Cents Dist from: i:j m:n
i:j m:n
----------------------------------------------------------------------
5:4 9:7 classic mediant 417.5 +31.2 -17.6
Golden Mediant 422.5 +36.2 -12.6
Erlich 423 +37 -12
----------------------------------------------------------------------
6:5 7:6 classic mediant 289.2 -26.4 +22.3
Golden Mediant 283.6 -32.0 +16.7
Erlich 285 -31 +18
----------------------------------------------------------------------
5:3 12:7 classic mediant 918.6 +34.3 -14.5
Golden Mediant 923.0 +38.7 -10.1
Erlich 924 +40 -9
----------------------------------------------------------------------
7:4 9:5 classic mediant 996.1 +27.3 -21.5
Golden Mediant 1001.6 +32.8 -16.0
Erlich 999 +30 -19
----------------------------------------------------------------------
5:4 6:5 classic mediant 347.5 -38.9 +31.8
Golden Mediant 339.3 -47.0 +23.7
Erlich 348 -38 +32
----------------------------------------------------------------------
9:7 4:3 classic mediant 454.2 -43.8 +19.1
Golden mediant 448.5 -49.6 +13.4
Erlich 457 -41 +22
----------------------------------------------------------------------

For the first three pairs of "valley" or "planet" intervals, separated by
the ratio of 36:35 (5:4-9:7; 6:5-7:6; 5:3-12:7), the Golden Mediant and the
maxima of Erlich's harmonic entropy coincide within 1-2 cents.

For the similar pair 7:4-9:5, they differ by about 3 cents, with Erlich's
point of maximum entropy about midway between the classic mediant (the
Pythagorean minor seventh at 16:9) and the Golden Mediant.

For the pair 5:4-6:5, where a zone of maximum complexity or "ambiguity"
might be expected to fall around the 11:9 "neutral third" (the classic
mediant), Erlich's 348 cents virtually coincides with this mediant, while
the Golden Mediant at 339.3 cents is decidedly closer to the less simple
6:5 ratio.

For the pair 9:7-4:3, where 13:10 is the classic mediant, in an area of
complexity or ambiguity where large major thirds begin to shade toward
narrow fourths, the Golden Mediant leans more toward the less simple 9:7,
while Erlich's 457 cents leans more toward the fourth.

Intonationally complex intervals of all these varieties may occur in
neo-Gothic styles. The regular thirds and sixths, as in historical Gothic
music, play vital cadential and coloristic roles in various unstable
sonorities. Minor sevenths at or near 16:9, conceived of not as especially
"complex" intervals but rather as comparatively simple ones derived from
two pure or near-pure 4:3 fourths, may play similar roles.

The two remaining interval types are less conventional "special
effects" categories. Regular neo-Gothic tunings in the range from
around 29-tET to 17-tET feature diminished fourths or alternative
major thirds (372.4-352.9 cents) and augmented seconds or alternative
minor thirds (331.0-352.9 cents) offering various intermediate
shadings converging on the "neutral third" of 17-tET.

In 29-tET, the interval of the "wide major third" at 11/29 octave,
455.2 cents, is also a "special effects" interval in the zone of
ambiguity where such thirds approach the region of narrow fourths;
this interval is close to the classic mediant or Erlich's "entropy"
maximum.

In addition to providing these comparisons and inviting readers to perform
their own listening tests, we suggest that Erlich's algorithm might
usefully be modified to use the Golden Mediant where it currently uses the
classic mediant or his more recent "limit-weighted midpoint". Of course,
having put Golden Mediants into the algorithm, we should not then be
surprised if we obtain Golden Mediants out of it.

In the case of Sethares' algorithm we find that the position of the
local maxima are too dependent on the parameters of the model to
permit any detailed comparison. In particular, one can (and must)
specify the timbre being used. Our use of the Golden Mediant (which is
a rule-of-thumb rather than a model) is intended to apply only to
"typical" harmonic timbres. In such cases we expect we would be in
general agreement with Sethares.

-----------------------------------------------------
4. Shadings of complexity and the neo-Gothic spectrum
-----------------------------------------------------

Both the historical Gothic music of Europe based on a pure Pythagorean
tuning, and also neo-Gothic temperaments in the most characteristic
range of Pythagorean to 17-tET, feature fifths at or reasonably close
to pure, and complex thirds and sixths. As the historian Carl Dahlhaus
has written, such complexity in a Gothic setting fits the role of
thirds and sixths with their "factor of instability."[7]

Using the classic mediant and Golden Mediant together with Erlich's values
for regions of maximum entropy, we can briefly survey the subtle shadings
of complexity along the spectrum from Pythagorean to 17-tET. Here we sample
three categories of unstable intervals: major and minor thirds, and major
sixths.

---------------------------------------------------------------------
Interval Classic Mediant Golden Mediant Erlich
---------------------------------------------------------------------
M3 (5:4-9:7) 417.5 422.5 423
.....................................................................
Pythagorean 407.8 - 9.7 -14.7 -15
29-tET 413.8 - 3.7 - 8.7 - 9
46-tET 417.4 - 0.1 - 5.1 - 6
17-tET 423.5 + 6.0 + 1.0 0.5
---------------------------------------------------------------------
m3 (6:5-7:6) 289.2 283.6 285
.....................................................................
Pythagorean 294.1 + 4.9 +10.9 + 9
29-tET 289.7 + 0.4 + 6.4 + 5
46-tET 287.0 - 2.3 + 3.7 + 2
17-tET 282.4 - 6.9 - 0.9 - 3
---------------------------------------------------------------------
M6 (5:3-12:7) 918.6 923.0 924
.....................................................................
Pythagorean 905.9 -12.8 -17.2 -18
29-tET 910.3 - 8.3 -12.7 -14
46-tET 913.0 - 5.6 -10.0 -9
17-tET 917.6 - 1.0 - 5.4 -6
---------------------------------------------------------------------

For the major and minor thirds, Pythagorean intervals are located on the
portion of a plateau with a shading somewhat closer to that of the simpler
or more strongly attracting "valley" or "planet": the 5:4 or 6:5 rather
than the 9:7 or 7:6.

In the especially characteristic portion of the neo-Gothic range from
around 29-tET to slightly beyond 46-tET, these intervals are at or near
their classic mediant values, (9+5):(7+4) and (6+7):(5+6), the celebrated
14:11 and 13:11 of neo-Gothic theory.

Around 17-tET, these intervals approach the point of maximum complexity as
defined either by the Golden Mediant or by Erlich's statistical model.

For major sixths, we remain on the portion of the plateau somewhat closer
to 5:3 than the classic mediant until around 17-tET, and to reach the
Golden Mediant or Erlich's region of maximal entropy, we would need to
temper the fifth by 5.7 cents. This is almost exactly the fifth of 39-tET
(707.7 cents), a tuning in what is termed the "far neo-Gothic" zone beyond
the characteristic range of Pythagorean to 17-tET.

Fine distinctions of shading within a plateau region may be reflected in
descriptions of 29-tET as "gentle," and tunings around 46-tET also as
"mild" in comparison to the "stronger" or more "avant-garde" 17-tET.[8]

Since major and minor thirds in 29-tET or 46-tET are close to the classic
mediants, while 17-tET thirds closely approximate the Golden Mediants or
Erlich's regions of maximal entropy, it would appear that the choice
between shades of complexity is a matter of musical discretion and taste.

-----
Notes
-----

1. In the interests of familiarity, D.K. has agreed to use, in this paper,
the convention of placing the larger number first in ratios for musical
intervals, despite his objections as outlined in
http://dkeenan.com/Music/ANoteOnNotation.htm.

2. On the basis of experiment, D.K. asserts that, under ordinary
conditions, 14:11 and 13:11, and likewise the Pythagorean 81:64 and 32:27,
are not directly recognizable or tuneable by ear, any more than are
tempered intervals at nearby locations on the continuum. This is in
contrast to simpler ratios such as 5:4, 6:5, 9:7 and 7:6, and possibly such
as 11:7 and 11:8.

3. See, for example, http://www.ixpres.com/interval/td/entropy.htm,
including a table of "entropy maxima" quoted in this article. Erlich's
model proposes that there is a kind of probability curve that a
listener's auditory system will perceive any given interval as fitting
one of a set of more or less simple integer ratios. Thus in the
historical meantone range near 4:5, a major third is very likely to be
perceived as this ratio; around 7:9, as in some of Erlich's music
based on 22-tET, recognition is also likely, although this is a
shallower "valley," or less strongly attractive "planet." On the
neo-Gothic plateau between these valleys or planets, such
identifications would seem very problematic, giving major thirds a
complex and intriguing quality.

4. William Sethares, _Relating Timbre and Tuning_,
http://eceserv0.ece.wisc.edu/~sethares/consemi.html

5. The term complexity is used in this paper to mean both (a) the
complexity of the ratio as given (e.g.) by the product of its two
sides when in lowest terms, and (b) the way an interval sounds to
us. We must point out that these do not always correspond, as Paul
Erlich's example of 3001:2001 makes clear.

6. Keenan Pepper, "The Other Noble Fifth," Tuning Digest [TD] 794:8,
10 September 2000.

7. Carl Dahlhaus, _Studies on the Origin of Harmonic Tonality_ (trans.
Robert O. Gjerdingen), Princeton: Princeton University Press, 1990, p. 187.
Dahlhaus specifically mentions "the complicated Pythagorean proportions
64:81 and 27:32" for major and minor thirds, emphasizing that this tuning
"should ... be understood as a musical phenomenon rather than as a
mathematically motivated acoustical defect," ibid. p. 188.

8. This subjective contrast may reflect not only the tuning of regular
intervals, but the differing qualities of diminished fourths and augmented
seconds. In the more "avant-garde" 17-tET, these are identical neutral
thirds, with cadential resolutions differing radically from a usual Gothic
flavor; the spectrum of intermediate thirds from around 29-tET to 46-tET
may involve a less dramatic contrast with other elements of intonation and
style.