andrew s. allen - distance in microtonal ratio-based scales

7/23/2019 Andrew S. Allen - Distance in Microtonal Ratio-based Scales

http://slidepdf.com/reader/full/andrew-s-allen-distance-in-microtonal-ratio-based-scales 1/16

Andrew S. Allen

20th Century Modalities

J. Martins

Distance in Microtonal Ratio-based Scales

Example 1. Transcription of Shri Camel, I.

After a single listening of Terry Riley's ingenious Shri Camel, for improvised de-tuned organ

with a built-in tape delay system, one could approach a huge array of possible inquiries. One could

mention the stylistic traits, reminiscent of the large structures of Indian classical music and progressive

rock, full of delicate, florid ornamentations alongside rock'n'roll grooves. Another could posit this piece



as being on the edge of both “Western” and “Non-Western,” being self-exotic and exotic

simultaneously. But for most listeners used to listening to organ music of the Bach variety, the most

immediate fascination might be that of the peculiar tuning Riley employs. Right from the beginning,

Riley begins the first movement with what some might call a “wrong-note” dissonance in the opening

melody. To people with the expectation of hearing “evenness” and “unity” in the sound of a keyboard

instrument, Riley's performance leaves a strange taste. For many music theorists interested in what

properties exist in a given work, there has been reluctance at best to try to examine music that does not

fit into the accepted, canonical tuning system of equal temperament, especially works by contemporary

composers, where even less scholarly work has been endeavored. This introductory paper proposes a

methodology to examine how distance is treated in the works of contemporary composers in ratio-

based tuning systems by comparing the complexity of the ratios in a given system and determining how

tones may play a subconscious functional role, using analysis examples of Terry Riley, Harry Partch

and the author.

Basic Acoustics Primer

What I would like to do before continuing any further in any acoustical theory discussion is to

define the terms I will be using in this article. If the reader is already well-acquainted with the science

of acoustics then he or she may skip to “Distance and Integral Sums.”

At the core level of sound, there is only pressure in time. When changes in the pressure of air

(either contracting or expanding) move across a space, this is called a soundwave. When this

soundwave has random pressures, it is called noise. When a soundwave moves across a space and the

contracting and expanding of air pressure form a cycle that repeats, that which is repeats is called a

period and the act of repeating this period is called a cycle. When a cycle occurs x times per second,

this number is called the frequency. If a frequency is between roughly 20 and 20000 times per second

(hertz), we say it has an audible pitch.



Distance and Integral Sums

Distance is here defined as a relative measurement between two points in one-dimensional

space. It uses multiplicative rather than additive quantification. With regards to pitch, this has to do

with a comparison, or ratio between two tones. With this kind of ratio-based measurement, one must

access the complexity/simplicity of the ratio's integers (the integral sum) to determine the amount of

distance that separates these two tones. This integral sum can be placed on a linear scale, along with the

integer sums of various other ratios, where the smallest sums are the most closely related in terms of

frequency content and the relationship between the harmonic (sometimes called overtone) series of the

given two tones. For example, the simplest ratio (and closest distance) would be that of two identical

tones; 1:1; which produces integral sum 2. The next simplest ratio would that of two tones spanning a

distance of an octave apart; 2:1. The next would be two tones a distance of two octaves apart and so

forth and only after two more octave spans would we even begin to see intervals other than the octave

come into play. However, when dealing with “scaler” content that is usually limited to the span of a

single octave, the theorem needs to express the distance in terms that describe these relations using

octave equivalency. To express this idea in mathematical terms:

where x and y are a comparative ratio (r ) between two tones, and i is the integral sum:

1!r !2, x

y=r , x" y=i

Expressing the distances of a 5-limit just 12-tone scale in this way yields the following results

seen in Ex.2. This simple theorem states that complex ratios create complex intervals and that this

proof can be used to structure the order of relational distance in a given collection of pitches. This

proof is necessary to state in order to illustrate the need to describe relative and not absolute distance.

Therefore, in this methodology, the half-step between C and Db, though it is the smallest absolute

frequency distance creates the second-most furthest relative distance when in this just 12-tone system.



Example 2. Distance in Justly-tuned 12-tone Scale I

scaleorder

1

1(C)

16

15(Db)

9

8(D)

6

5(Eb)

5

4(E)

4

3(F)

45

32(F#)

3

2(G)

8

5(Ab)

5

3(A)

16

9(Bb)

15

8(B)

2

1(C)

integralsum

2 31 17 11 9 7 77 5 13 8 25 23 3

distanceorder

1

1(C)

2

1(C)

3

2(G)

4

3(F)

5

3(A)

5

4(E)

6

5(Eb)

8

5(Ab)

9

8(D)

15

8(B)

16

9(Bb)

16

15(Db)

45

32(F#)

integralsum

2 3 5 7 8 9 11 13 17 23 25 31 77

Before anyone feels an inclination to discredit my sly attempt at ratio-ing tonal function, Ex. 3

shows how these relations work when using a just system that employs only the simplest ratios to

create a similar 12-tone space that employs simple ratios that more closely relate to the fundamental

pitch at the cost of a loss of much of the 12-tone equal temperament accuracy. It should be clear from

this table that pitch-labeling in any just-system are more based on proximity rather than function.

Example 3. Distance in Justly-tuned 12-tone Scale II

scaleorder 1

1(C)

87

(Db)

76

(D)

65

(Eb)

54

(E)

43

(F)

75

(F#)

32

(G)

53

(Ab)

74

(A)

95

(Bb)

116

(B)

21

(C)

integralsum

2 15 13 11 9 7 12 5 8 11 14 17 3

distanceorder

1

1

(C)

2

1

(C)

3

2(G)

4

3(F)

5

3(Ab)

5

4(E)

6

5 ,

7

4(Eb) & (A)

7

5(F#)

7

6(D)

9

5(Bb)

8

7(Db)

11

6(B)

integral sum 2 3 5 7 8 9 11 12 13 14 15 17

As stated before, the “just 12-tone scale” is an approximation of the equal-tempered system by

employing just ratios and therefore, this kind of analysis can not readily be used to describe the 12th

root musics with much insight, mostly because the ratios formed in 12-tone equal temperament are all

very complex and because of that, integral sum analysis will not yield the types of implications



inherent in that system. However, I do propose that this idea of relative distance in ratio-based systems

can provide insight on interesting relationships in a way that may provide a starting point for further

analytical work on musics outside of the equal-tempered canon, mainly because of how these

relationships of simple to complex ratios form connections in sound through the harmonic series.

Acoustic Proofs of Integral Sum-derived Distance

Before we return to the Riley, let's discuss another property of the integral sums. This theory is

very readily pronounced in the harmonic series (overtone series), which charts all the harmonically-

related frequencies (partials) to a given fundamental. Note Ex.4 which shows the harmonic series as

founding on C2.

Example 4. The harmonic series, starting on C2.

Each numbered partial in this series is a multiple of the fundamental frequency. The diamond noteheads

represent pitches that are rather far away in pitch from their respective notations on the grand staff, but

for sake of conformity to a simple notation system, are placed within a 12-tone context.

For ease of demonstration, let us assume we are feeling rather French and have decided on a flat

tuning, so that partial one (C2) equals a frequency of exactly 100 Hertz (Hz). To generate the next

partials (C3,G3,C4,E4...), one simply needs to multiply this fundamental frequency by 2, 3, 4, 5 and so

forth, which produce content of 200Hz, 300Hz, 400Hz, and 500Hz frequencies, respectively. These

harmonically-related frequencies have wavelength periods that are divisions of the fundamental.

(Partial 1 divides the fundamental into one period, partial 2 divides the fundamental into two periods,

and so forth). Acoustically, this creates frequencies that repeat their periods at exactly the same time



each time, creating harmonic unity in the series. Very often, two different fundamentals can share a few

similar frequencies, which implies unity between the two tones. Below, in example 5, is a chart of the

first twelve partials of the harmonic series of each of the pitches in Riley's Shri Camel, I. Anthem of the

Trinity.

This graph lists the scale collection in order of distance. The superscripts next to the Hertz

indicates which tones share that given frequency in their harmonic series. As the complexity of the

ratio increases, the occurrence of commonalities between harmonic frequencies becomes more and

more sporadic. “G” shares a harmonic frequency with “C” every two partials. “F” shares a harmonic

frequency with “C” every three partials and so forth. This shared-frequency (or multiple) can be easily

determined using prime factorization and equivalent denominators and from this information, one can

predict all the shared harmonic frequencies between all pitches and the periodicity of their occurrences.

From this graph, one can infer that less distant ratios have shared harmonic frequencies that occur more

often and at lower frequencies than more distant ratios.

Example 5. Pitch space in Shri Camel, I ; ordered by distance.

partials1

1“C”

3

2“G”

4

3“F”

5

3“A”

7

4“Bb”

7

6“Eb”

9

8“D”

1 100Hz 150Hz 133.333Hz 166.666Hz 175Hz 116.666Hz 112.5Hz

2 200Hz 300HzC 266.666Hz 333.333Hz 350Hz 233.333Hz 225Hz

3 300Hz 450Hz 400HzC 500HzC 525Hz 350HzBb 337.5Hz

4 400Hz 600HzC 533.333Hz 666.666HzF 700HzC 466.666Hz 450HzG

5 500Hz 750Hz 666.666Hz 833.333Hz 875Hz 583.333Hz 562.5Hz

6 600Hz 900HzC 800HzC 1000HzC 1050HzG 700HzCBb 675Hz

7 700Hz 1050Hz 933.333Hz 1166.666Hz 1225Hz 816.66Hz 787.5Hz

8 800Hz 1200HzC 1066.66Hz 1333.333HzF 1400HzC 933.333Hz 900HzCG

9 900Hz 1350Hz 1200HzC 1500HzC 1575Hz 1050HzBb 1012.5Hz

10 1000Hz 1500HzC 1333.333Hz 1666.666Hz 1750Hz 1166.666Hz 1125Hz

11 1100Hz 1650Hz 1466.666Hz 1833.333Hz 1925Hz 1283.333Hz 1237.5Hz

12 1200Hz 1800HzC 1600HzC 2000HzCF 2100HzCG 1400HzCBb 1350HzG

Riley uses simple ratios to deduce his scale, which dilutes any sense of tonal expectation and



instead focus on the relationships of distance between tones. Of course, the music is more sophisticated

than simply a linear presentation of these relationships, as the melodic passages involve delicate

ornamentation and constant pitch-axis pivots. Before we delve any deeper, let us reproach the opening

musical material with a closer look at the relative distances in the passage using integral sums.

Returning to the music from Ex.1, the excerpt in Ex.6 from the Riley is a reduction that

removes the very quick ornaments and combines the two voices into one line in order to suggest that

one voice picks up where another left off. Let's begin with the most obvious linear observations: After

entering a world of 1/1, 2/1 and 1/2, Riley immediately moves away from these smaller distances to

3/2, the next closest interval and pitch axis of the first subsection, which turns around 7/4 and 4/3. As

the next section enters, 9/8 and 4/3 carry over from the previous 2nd

voice, which leads into a pivoted

motion on 9/8 (9/8 4/3 7/6 9/8) which leads into a turn on 7/4 and 3/2 again. The pivot on a 9/8 axis

repeats and a repeat of the turn is setup, but falls back onto 1/1 to return to the droning octaves yet

again. A closing passage that turns around an axis on 5/3 eventually falls back onto the octaves as well,

and the piece moves forward to the next section.

Example 6. Distance relations in non-ornamented pitch reduction of Shri Camel, I.

What is interesting to note here is how a cycle of 3/2's plays a role in the large-scale progression

(1/1 to 3/2 to 4/3 to 9/8) and how integral sums play a role in creating a pull-and-release plot of

distance relations in each subsection. Each subsection seems to deviate further and further away from

1/1, with some voices returning to 1/1 while others continue further away, along a cycle of 3/2s. One



could approach these axes as being always related in distance to only 1/1, which would be logical given

the constant 1/1 octave drones underneath the melodic passage. However, if one where to examine each

subsection as being a self-contained collection, then one could approach how each of these subsections

relate to one another in terms of linear motion rather than deriving a relation between 1/1 and whatever

axis is present. In this sense, it is clear how each subsection could be viewed as both their respective

ratio in the scale as also their ratio in terms of what has come directly before.

With regards to how integral sums work in this passage, the first subsection, that which

embellishes an axis on 3/2, moves from distance 5 to 11 to 7 to 5, where the first distance not the 3/2

axis is also the furthest away. In the next section, the grouped set (9/8 4/3 7/6 9/8) which embellishes

9/8, moves from distance 17 to 7 to 13 to 17, which moves away from the axis exactly as the first

except in the opposite direction. In the last subsection, a turn below and above 5/3, which rounds out

using distance motion from above and from below.

Certainly, this introductory material is later expanded and transformed. Riley begins to organize

his pitches in ways that continually shift axes and the distance relations within. This integral sum-

approach of this music helps to describe some of the relations in Riley's continual reshaped melodic

phrases through acoustical science, deployed as integral sums. Through this rather simple pitch space,

there have already been proven to be many complex relationships occurring at the same time. In spaces

much larger than 7 pitches, the relationships become exponentially more complex. In the next two

sections, we will discuss limit and series tuning theory and how these mathematical functions develop

into interesting musical relationships in 43-tone and Alcestis Scales.

Limit Tuning and the 43-tone Scale

In ratio tuning, there are an infinite amount of possibilities of mathematical functions to

generate intervals. In fact, there is a book on the thousands of tetrachords possible between 1/1 and 4/3

(so in reality, this is a book on the thousands of the possible two pitches between these two ratios).

Indeed, in regards to frequency, since it can be written as a float number, and there are an infinite



amounts of gradation between two integers in a float system, so there are an infinite possible number of

frequencies between 100.0Hz and 101.0Hz. Of course, the human ear has a limit of which it can

actually distinguish between frequencies. Usually this limit is rather low and unless the listener is

incredibly alert to subtlety of pitch, it is close-to impossible to hear the difference between, say,

640/324 and 641/324 (Though not impossible, given the listener can hear the pitches as single

frequency sinewaves coming from a single speaker in a completely “dead” room). So, the amount of

complexity of tones (and resultantly, the integral sum sizes) must stay below a certain threshold if we

expect audibility of the results. Using limit tuning theory, one can proscribe the maximum amount of

complexity in a given collection and ensure audibly-measurable distance can be determined.

Harry Partch, though not the creator of limit tuning theory, is certainly the most well-known

proponent. In his manifesto, Genesis of a Music, he discusses at length the possibilities and properties

of limit theory tuning and how they might apply to expanding musical vocabulary. Creating a scale

using “limits” requires the strict use of a mathematical function:

In a given x-limit tuning, where x is any odd number, the numerator or denominator of an included

ratio (r ) may be any number between 1 and x. If that number is < x, then it may also be multiplied by 2,

where .

For example, to list all the possible ratios in 3-limit tuning, first we shall list all the possible numbers

that we can use to create ratios, which would be 1, 2, 3, and 4. Notice that 6 is not possible because we

cannot include the double of x. No other numbers can exist either because we have used all numbers up

to 3 and we have exhausted all possible doubles (2, the double of 1, and 4, the double of 2). In this 3-

limit, the only ratios then that can be created are: (Note that ratios that are larger than 2 or

smaller than 1 are not included in this tuning.)

What results is a collection of octaves and pure fifths and fourths and nothing else. If we

measure these in terms of distance, we find that we have very small sums that result in very little

1

1 ,

2

1 ,

3

2 ,

4

3

1!r !2



relative distance. Subsequently, all of the pitches include many frequencies from each other's harmonic

series in their own series. Obviously, what this produces is a great amount of unity and very little

diversity and thus for any music that relies on contrasting unity with diversity, this would be a very

difficult limit to create substantial music in.

Noting example 7, if we expand the limit to the next odd number, 5, we find our collection has

expanded much more and our palette includes a few more distant ratios. In example 8, we find that in

the 7-limit scale, even more complex ratios exist. Note that whatever ratios from lower limits are

included in higher limit collections and that the new ratios are always going to include both more

simple and more complex ratios than were included previously.

Example 7. Distance in the 5-limit Scale

distance order 1

1

2

1

3

2

4

3

5

3

5

4

6

5

8

5

integral sum 2 3 5 7 8 9 11 13

Example 8. Distance in the 7-limit Scale

distance order 1

1

2

1

3

2

4

3

5

3

5

4

6

5 ,

7

4

7

5

8

5 ,

7

6

8

7

10

7

12

7

integral sum 2 3 5 7 8 9 11 12 13 15 17 19

For Partch, the 11-limit scale had what he considered to be the appropriate amounts of both

simple and complex ratios, and for this reason, he used it as the basis of his 43-tone scale. By

generating the 11-limit scale, one creates a total of 29 notes, distributed in two bell curves that meet

around the middle of the octave (See Ex.9). Partch felt the need to add additional tones (of increasing

ratio complexity) to “fill in” the gaps left behind in this distribution. By adding the additional 14 tones

in various gaps of the scale, Partch creates the 43-tone scale out of limit theory and structures the scale

by completing the 11-limit scale with additional ratios, allowing for a wide range of ratio complexities



and distance. In Ex.10, we find the scale order and distance order charted for the 43-tone scale.

Example 10. Distance in Partch's 43-tone Scale

scale order 1

1

81

80

33

32

21

20

16

15

12

11

11

10

10

9

9

8

8

7

7

6

integral sum 2 161 65 41 31 23 21 19 17 15 13

scale order 32

27

6

5

11

9

5

4

14

11

9

7

21

16

4

3

27

20

11

8

7

5

integral sum 59 11 20 9 25 16 37 7 47 19 12

scale order 10

7

16

11

40

27

3

2

32

21

14

9

11

7

8

5

18

11

5

3

27

16

integral sum 17 27 67 5 53 23 18 13 29 8 43

scale order 12

7

7

4

16

9

9

5

20

11

11

6

15

8

40

21

64

33

160

81

2

1

integral sum 19 11 25 14 31 17 23 61 97 241 3

distanceorder

1

1

2

1

3

2

4

3

5

3

5

4

6

5 ,

7

4

7

5

7

6 ,

8

5

integralsum

2 3 5 7 8 9 11 12 13

distanceorder

9

5

8

7

9

7

9

8 ,

10

7 ,

11

6

11

7

10

9 ,

11

8 ,

12

7

11

9

11

10

12

11 ,

14

9 ,

15

8

integralsum

14 15 16 17 18 19 20 21 23

distanceorder

14

11 ,

16

9

16

11

18

11

16

15 ,

20

11

21

16

21

20

27

16

27

20

32

21

integralsum

25 27 29 31 37 41 43 47 53

distanceorder

32

27

40

21

33

32

40

27

64

33

81

80

160

81

integralsum

59 61 65 67 97 161 241



For the analysis portion of the

discussion on limit theory and the 43-tone

scale, we will look at the opening passage

from The Wayward: IV. Barstow - Eight

Hitchhiker Inscriptions From A Highway

Railing At Barstow, California. This piece,

for chromelodeon (a 43-tone tuned organ), the diamond marimba (an instrument based on the 11-limit

scale), bass marimba (a variant of the diamond marimba), kithara (a free-standing 43-tone harp),

narrator and chorus is a piece that highlights many of the peculiarities of Partch that make his music so

unique. Ex.11 is a short excerpt from the beginning of the fourth movement, which is a reading of a

series of writings by traveling hobos.

Example 11. Opening of The Wayward: IV. Barstow.

0

25

50

75

100

125

150

175

200

225

250

Distance in the 43-tone Scale

Scale Degree

I n t e g r a l S u m



In the passage indicated, two instruments built by Partch, the Chromelodeon and the Diamond

Marimba play a short interlude before each inscription is read by the narrator. This rather short section,

however, provides some very interesting insight into how Partch approaches distance in his own music.

From his writings on limit theory, we already know he is thinking consciously about what ratios he

chooses and most of his instruments only play subsections of the 43-tone scale (for practical and

aesthetic reasons). But it is quite clear after approaching this short section using integral sums that

Partch's music has very specific structural uses for the simple and complex ratios in the 43-tone scale.

Assuming, as with the Riley, that we can attach to a specific subsection a particular ratio axis, the

Partch example shifts ratio axes every beat for six beats. In each of these axes, a specific ratio is played

on both instruments simultaneously, emphasizing that ratio over the others. In all cases, the diamond

marimba starts its motif with that ratio axis and moves downward in register, moving from simple to

complex ratios in each beat and also over the course of the excerpt from an average of small integral

sums to much larger sums by the end of the passage. It should also be said that every axis is an

included ratio of the previous beat, which makes sense since the marimba is moving down sequentially.

The chromelodeon continually uses simple ratios on the downbeat dotted eighth and on the sixteenth

note, uses much more complex ratios. At the end of the passage, both instruments land on a pure “C”

major triad, where the narrator begins reading the numbered inscription. The axis for each beat is

circled in the example and shows movement from 3/2 to 6/5 to 1/1 to 5/3 to 10/7 and on the last beat,

the chromelodeon steps up in thirds unexpectedly while the diamond marimba repeats the same motif

again (It actually cannot sequence any further mainly because the diamond marimba has already played

its lowest note possible). What is interesting about these ratios axis is to compare their sums. Listing in

order, we find the passage moves from sum 5 – 11 – 2 – 8 - 17. Between the 5 - 11 and 2 – 8 is 6 sums

and between 11 – 2 and 8 – 17 is 9 sums. In effect, by shifting a certain amount in sum value is the

same as in the Riley where the ratios are shifting away from a ratio axis. By making the ratios more or

less complex by this 6 – 9 – 6 – 9 pattern, Partch enables a type of modulation in his 43-tone system

that instead of shifting pitches up or down in absolute frequency, shifts the complexity up or down,



resulting in higher or lower integral sums. This idea would need to be explored further, but as of yet,

the author has been unable to acquire any scores by Partch other than Barstow.

Series Tuning and the Alcestis Scale

As was said earlier, there are a infinite amount of possible mathematical functions to generate

various scales, but in this paper, I wish to highlight mainly those that deal with ratios and specifically

relative distance. Because of this, the systems shown here usually contain a fair amount of both simple

and complex ratios and analysis yields the distance relationships inherent between these ratios. Similar

to limit tuning theory is series tuning theory. This theory works by creating a series of ratios by taking

all consecutive number pairs in a given series (up to a predetermined limit) to form the numerator and

denominator of a ratio in the system. In order to discuss how these scales are generated, one must first

look at how series work, using the fibonacci series as an example, which is the basis for the creation of

the Alcestis scale seen in the next section.

A series is merely an array of numbers where the list is built by performing the same operation

on every number in the array. For example, an additive series such as where An = n + 10, starting with

n = 0, you would get the series: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ... and so on. Another example

would be a multiplicative series such as where Mn = n * 2, starting with n = 0, you would get the

series: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18 ... and so on. A self-operation series is one that operates on a

number in the series using others numbers in the series. For example, in the Fibonacci series,

Fn = Fn-1 + Fn, if n > 1. given that F0 = 0, F1 = 1

This results in the series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 ... and so on. What is interesting in this

series is that every two consecutive pairs of numbers form a ratio that as you proceed further and

further in the series, more closely approximate (but never reach) ! (Phi), which is approximately

1.61803399.... This number is sometimes called the Golden Ratio and is a number invented by the



ancient Greeks to illustrate: , where .

The Alcestis scale is generating using this series in two ways. By forming the two consecutive

numbers pairs as numerator, denominator, such as: and also by forming the pairs

as denominator, numerator, such as: . There two ordered lists of ratios combine

together to form a scale that moves away both above and below the starting point with the same amount

of distance. This scale creates a sense of unity, symmetry and above all, the arbitrary limit in the series

halts after ratio sets (5/3 8/5 13/8) or (3/5 5/8 8/13). Both sets create a subtle orbiting around a pitch of

which is a distance of Phi away from the fundamental. Of course, the ratios formed from the fibonacci

series only more and more closely approximates this number, but never actually reach the pure Phi and

to continue with more ratios would only more and more closely approximate a pitch of which cannot

truly exist. In this sense, the scale can be seen as creating a sense of infinity in either direction.

Ex.11 The scales used in Alcestis' departure from Lemon Island .

In my work for viola and harp, Alcestis' departure from Lemon Island , each instrument uses a

particular version of the Alcestis scale. The viola uses exactly what has been stated in the previous

paragraph, where 1/1 approximates middle C. The harp uses two portions of two Alcestis scales so that

middle C is roughly Phi of each's fundamental. The two scales overlap in a way so that where one

1

1

,2

1

,3

2

,5

3

,8

5

,13

8

--------#a

-----#b

1

1 ,

1

2 ,

2

3 ,

3

5 ,

5

8 ,

8

13

a+b

a =

a

b



instrument is playing 1/1, the other instrument is more and more closely approximating that pitch but

never reaches it. Ex.11 shows this very clearly. Integral sum distance works very well with this system;

as a ratio attempts to closer and closer approximate Phi, it becomes logarithmically more complex and

therefore, integral sums span a very range even with just a few pitches in the Alcestis scale.

Conclusion

This paper's main purpose has been to bring to attention some of the properties of ratio-based

microtonal scale generation and the relationships of the ratios included in these sets. Integral sums,

while they do not fully account for all the tone relationships possible in microtonal scale music, do shed

some light on the functional and structural importance of these tones. Through further analysis, one

may be able to open new avenues of exploration in many musics that have yet to be researched beyond

their basic properties and hopefully bring new and interesting ideas on microtonal scales to the

forefront of theoretical discussion.

References

Allen, Andrew. Alcestis' departure from Lemon Island . 2008.

Partch, Harry. Barstow : eight hitchhiker inscriptions from a highway railing at Barstow, California

(1968 version). Madison, Wis.: Published for the American Musicological Society by A-R

Editions. 2000.

Partch, Harry. Genesis of a Music. Madison, Wis.: University of Wisconsin Press. 1949.

Riley, Terry. Shri Camel . Sony, 1990.

andrew s. allen - distance in microtonal ratio-based scales

Documents