
The Noble Mediant:
Complex ratios and metastable musical intervals

by Margo Schulter and David Keenan

Posted to the tuning list 17Sep2000
Last updated 12Oct2007
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 "neoGothic"
temperaments in the most characteristic range between Pythagorean and
17tET, 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 neoGothic 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 3limit sonorities with
pure or nearpure fifths and fourths.
Since for many modern readers, as opposed to medieval or neoGothic
theorists, an integer ratio may imply an interval simple enough to be
tuned by ear through a "lockingin" 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 neoGothic
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
17tET as located on a gently rounded plateau region between the
notchlike 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) or "sensory dissonance"
(William Sethares) or "complexity" [M.S.].[3]
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[4] 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 Phirelated 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 "Noble
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 Noble Mediant can simplify this process,
yielding results in general agreement with the local maxima of Paul
Erlich's "harmonic entropy"[5] or William Sethares' "dissonance"[6].
Both of these must be calculated using complicated computer algorithms
and do not admit of a simple closedform expression like the Noble
Mediant.
Section 4 briefly considers the classic mediant and Noble Mediant in
relation to the tuning of certain common unstable intervals along the
most characteristic portion of the neoGothic spectrum from
Pythagorean to 17tone equal temperament (17tET).
While our perspectives may somewhat differ, we strongly agree that the
Noble Mediant provides new cardinal points of orientation in exploring
the subtle shadings of neoGothic 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 neoGothic theory, giving this ratio a new
mathematical significance fitting its intriguing musical qualities of
instability and complexity in Gothic or neoGothic styles. This
mediant is very closely approximated, for example, by 46tET.
Finding the mediant of 6:5 and 7:6, two simple ratios or valleys for
minor thirds, gives a similar meaning to another favored neoGothic
ratio:
(6 + 7)

(5 + 6)
This mediant is identical to the neoGothic 13:11, closely
approximated for example by 29tET, a leading neoGothic 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 11limit or 13limit 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 Phibased "Noble 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.
In physics, if an object were placed at such a point it would be said
to be 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 neoGothic theory as the
14:11, at around 417.508 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 neoGothic 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 17tET, 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 EarthMoon 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 17tET 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 Fibonaccilike.
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) 289.210 26.432 +22.339
x=1, y=2 (6+14):(5+12) 281.358 34.283 +14.487
x=2, y=3 (12+21):(10+18) 284.447 31.194 +17.576
x=3, y=5 (18+35):(15+30) 283.281 32.360 +16.410
x=5, y=8 (30+56):(25+48) 283.728 31.913 +16.857
x=8, y=13 (48+91):(40+78) 283.557 32.084 +16.687
x=13, y=21 (78+147):(65+126) 283.623 32.018 +16.752
. . . . .

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 neoGothic tunings, this region
of maximal complexity for the minor third is located between 46tET
(287.0 cents) and 17tET (282.4 cents).

3. A new index of complexity: the Noble 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 "Noble 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 Noble 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)
NobleMediant(i/j, m/n) = 
(j + Phi * n)
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 Phibased
function yields these results:

Intervals Noble 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 neoGothic 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
Noble Mediant 422.5 +36.2 12.6
Erlich 423 +37 12

6:5 7:6 classic mediant 289.2 26.4 +22.3
Noble Mediant 283.6 32.0 +16.7
Erlich 285 31 +18

5:3 12:7 classic mediant 918.6 +34.3 14.5
Noble Mediant 923.0 +38.7 10.1
Erlich 924 +40 9

7:4 9:5 classic mediant 996.1 +27.3 21.5
Noble Mediant 1001.6 +32.8 16.0
Erlich 999 +30 19

5:4 6:5 classic mediant 347.5 38.9 +31.8
Noble Mediant 339.3 47.0 +23.7
Erlich 348 38 +32

9:7 4:3 classic mediant 454.2 43.8 +19.1
Noble 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 (5:49:7; 6:57:6; 5:312:7), the Noble Mediant
and the maxima of Erlich's harmonic entropy coincide within 12 cents.
For the similar pair 7:49:5, 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 Noble Mediant.
For the pair 5:46:5, 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 Noble Mediant at 339.344 cents is decidedly
closer to the less simple 6:5 ratio.
For the pair 9:74:3, 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 Noble 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
neoGothic 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 nearpure 4:3 fourths, may play similar
roles.
The two remaining interval types are less conventional "special
effects" categories. Regular neoGothic tunings in the range from
around 29tET to 17tET feature diminished fourths or alternative
major thirds (372.4352.9 cents) and augmented seconds or alternative
minor thirds (331.0352.9 cents) offering various intermediate
shadings converging on the "neutral third" of 17tET.
In 29tET, 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 Noble Mediant where it currently
uses the classic mediant or his more recent "limitweighted midpoint".
Of course, having put Noble Mediants into the algorithm, we should not
then be surprised if we obtain Noble 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. Like Erlich's model, ours (which is a
ruleofthumb 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 neoGothic spectrum

Both the historical Gothic music of Europe based on a pure Pythagorean
tuning, and also neoGothic temperaments in the most characteristic
range of Pythagorean to 17tET, 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 Noble 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
17tET. Here we sample three categories of unstable intervals: major
and minor thirds, and major sixths.

Interval Classic Mediant Noble Mediant Erlich

M3 (5:49:7) 417.5 422.5 423
.....................................................................
Pythagorean 407.8  9.7 14.7 15
29tET 413.8  3.7  8.7  9
46tET 417.4  0.1  5.1  6
17tET 423.5 + 6.0 + 1.0 0.5

m3 (6:57:6) 289.2 283.6 285
.....................................................................
Pythagorean 294.1 + 4.9 +10.9 + 9
29tET 289.7 + 0.4 + 6.4 + 5
46tET 287.0  2.3 + 3.7 + 2
17tET 282.4  6.9  0.9  3

M6 (5:312:7) 918.6 923.0 924
.....................................................................
Pythagorean 905.9 12.8 17.2 18
29tET 910.3  8.3 12.7 14
46tET 913.0  5.6 10.0 9
17tET 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 4:5 or
5:6 rather than the 7:9 or 6:7.
In the especially characteristic portion of the neoGothic range from
around 29tET to slightly beyond 46tET, 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 neoGothic theory.
Around 17tET, these intervals approach the point of maximum
complexity as defined either by the Noble Mediant or by Erlich's
statistical model.
For major sixths, we remain on the portion of the plateau somewhat
closer to 3:5 than the classic mediant until around 17tET, and to
reach the Noble 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 39tET (707.7 cents), a tuning in what is termed the "far
neoGothic" zone beyond the characteristic range of Pythagorean to 17
tET.
Fine distinctions of shading within a plateau region may be reflected
in descriptions of 29tET as "gentle," and tunings around 46tET also
as "mild" in comparison to the "stronger" or more "avantgarde" 17
tET.[8]
Since major and minor thirds in 29tET or 46tET are close to the
classic mediants, while 17tET thirds closely approximate the Noble
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.

Update (October 2007)

After the initial publication of this article, Kraig Grady kindly
referred us to some earlier work by Lorne Temes [9], and the diagram
by Erv Wilson showing the noble numbers in relation to the Stern
Brocot tree [10], and more recently, some comments by Erv Wilson on
Lorne Temes' Phi neutral sixth (833.1 cents).
"... it is the worstest of the worst  and yet somehow with divinity
imbued, Lord have mercy!" [11]
One of the authors [M.S.] finds the Phisixth, a kind of "superminor
sixth" in a neoGothic setting, to be not so much "dissonant" as
often charmingly "vague" or "diffuse," evoking a certain association
with Debussy (e.g. _Nuages_) and Impressionism as they might be
realized in some parallel universe of intonation.

Update (September 2022)

It is important to note that the noble mediant is not simply a phi
weighted mediant. There are two important differences:
1. The order of the arguments is irrelevant in the case of the noble
mediant. The phi weight always goes to the more complex ratio.
2. The noble mediant is only defined for pairs of ratios where, if you
crossmultiply and subtract, you obtain +1. The result of a phi
weighted mediant is not a noble number unless this condition is
satisfied. This has been called the "unimodular" requirement, as the
calculation is like the determinant of a 2x2 matrix, and a matrix is
called unimodular if its determinant is +1.
To summarise:
When i/j is the simpler ratio,
NobleMediant(m/n, i/j) =
/ (i + Phi * m)
  if in  jm = 1;
NobleMediant(i/j, m/n) = < (j + Phi * n)

\ undefined otherwise.

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
https://dkeenan.com/Music/ANoteOnNotation.htm.
2. On the basis of experiment, D.K. asserts that, under ordinary
conditions, the bare dyads 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. 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.
4. Keenan Pepper, "The Other Noble Fifth," Tuning Digest [TD] 794:8,
10 September 2000.
5. See, for example,
http://www.tonalsoft.com/sonicarts/td/erlich/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 5:4, a major third is very likely to be
perceived as this ratio; around 9:7, as in some of Erlich's music
based on 22tET, 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.
6. William Sethares, _Relating Timbre and Tuning_,
https://eceserv0.ece.wisc.edu/~sethares/consemi.html
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.
178.
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 "avantgarde" 17tET, these are
identical neutral thirds, with cadential resolutions differing
radically from a usual Gothic flavor; the spectrum of intermediate
thirds from around 29tET to 46tET may involve a less dramatic
contrast with other elements of intonation and style.
9. Lorne Temes, "Golden Tones?" in a letter to Erv Wilson, 4 January
1970.
https://anaphoria.com/temes.PDF
10. Erv Wilson, "Scale Tree", 1994.
https://anaphoria.com/sctree.PDF
11. Erv Wilson, "Radical JustUnjust property of 72ET", 28 June 2002.
https://anaphoria.com/tres.PDF (second last page)