The Chromelodeon Scale: A Psychoacoustical Model of Roughness Versus Harry Partch’s One Footed Bridge Alexandre Torres Porres Music Department, State University of São Paulo (USP), Brazil [email protected]

ABSTRACT Under a Psychoacoustical point of view, and because of the partials’ alignment, it is the spectrum of sounds that determines the consonance of given musical intervals. For example, harmonic spectrums do align their components in harmonic musical intervals. On this paper, I adopt a computer software to analyze the spectrum of the Harmonium built by Harry Partch (the Chromelodeon). Its results show which are the most consonant steps in the span of an octave. I use this data to compare to a similar graph designed by Harry Partch, his One Footed Bridge. A major concern behind this research is to question at what extent musical intervals related to higher harmonics do provide a perceptual significant consonance perception. Another consequence is to promote a revision of Harry Partch’s theoretical work.

I.

INTRODUCTION

A Software – based on a Roughness Model developed during a Master’s research, and still in development on a current doctorate program at USP – was used to measure this perceptual dimension of consonance. Given a sound spectrum, this tool enables the derivation of a scale with consonant steps for this sound, or, in other words, intervals that promote a significant alignment of partials. The term “significant” was adopted by me to illustrate an actual perceptual difference of a particular musical interval’s consonance compared to its neighboring areas. Just Intonation is based on small integer ratios such as, for example, the Fifth [3:2], with 702 cents (one hundredths of the Tempered Semitone). These musical intervals correspond to the relations between the Harmonic Series’ terms (3rd e 2nd on the previous example) and promote consonance in the case of harmonic spectrums (as it is the case of most musical instruments). The smaller the integer values on a ratio, that corresponds to a musical interval, the more they also correspond to smaller terms in the Harmonic Series, and the more consonant they are, for more significant is provided on harmonic musical instruments. Because of this fact, Tuning Systems, in Western Music before the 20th century, never trespassed the relations over the 5th harmonic. But what is the deal when intervals correspond to more distant terms in the Harmonic Series (like [11:9], that corresponds to the relation between the 9th and the 11th harmonic)? The answer, obviously, depends on the spectrum, or, in more detail, on how much energy there is for such distant harmonics, in order for them to provide a perceptually significant alignment, or consonance as a result of the lack of Beatings and Roughness sensations. Such intervals can be found on Harry Partch’s Just Intonation system (1974), which includes relations up to the 11th harmonic. Partch, in a graph he named One Fotted

Bridge (Partch, 1974:155), represented the result of a consonance measurement for all intervals on his system, by having as a basis the Chromelodeon (a Harmonium built by him). This was a perceptual test that did not count on a particular population of individuals, and was carried only under the personal and subjective criteria of Harry Partch’s judgment, where he concluded that the consonance obtained by intervals related up to the 11th harmonic are relevantly significant in the context of harmonic musical instruments, or at least the Chromelodeon. This is because some musical instruments, even though harmonic, like the Recorder, posses a quite simple spectral content, and barely enables the alignment up to the 3rd harmonic. Another point is that more complex spectrums, even with some energy on higher harmonics, do not necessarily promote consonance in intervals corresponding to the same harmonics. In this paper, I investigated the case of the Chromelodeon, by analyzing its spectrum via the Roughness Model Software. The results are compared to One Footed Bridge. Based in Psychoacoustical studies, the Roughness Model’s results present a more detailed analysis like, for instance, a consonance ranking of the most significant alignments, besides giving data for the investigation and discussion of how relevant is the consonance of just intervals that are distant in the Harmonic Series – bearing the example of the Harmonium-like instruments (free reed musical instruments). On the next section I present a bit about the psychoacoustics background. Further on, a brief description of Harry Partch system is given on the third section. After informing some main issues about objectives and methodology of research on the fourth section, I present and discuss the results found by this research. On the sixth and last section of this paper, I draw some final and pertinent considerations.

II. PSYCHOACOUSTICS A. Roughness Model When Partials are a bit unaligned, Beatings and Roughness Sensations are evoked by resulting Amplitude Fluctuations (Vassilakis, 2001), which influence and compromise the perception of Consonance/Tuning. Plomp & Levelt (1965) pointed out a tendency of maximum Roughness perception (a peak) at the interval corresponding to one fourth of the critical bandwidth (Figure 1). The Roughness model, previously developed during a master’s research at Unicamp (State University of Campinas, São Paulo), is mainly based on Plomp & Levelt. It has some enhancements and updates as a mixture of more recent Psychoacoustic Roughness Models based on Plomp e Levelt. Like the work of Sethares (2005), Parncutt (1993), Barlow

(1980), and Vassilakis (2001). For a more detailed description, please refer to Porres & Manzolli (2007a). The Model has two equations that approximate Figure 1: Sethares’ (2005) and Parncutt’s (1993). The first is less accurate, but it permits, with not that much of a compromise, a bigger contrast when drawing Roughness graphs. This is better to detect maximum and minimum points of Roughness, which are the essential data to derive scales of consonant steps by the software, as pursued on this paper. More information on how the software works can be found at Porres & Manzolli (2007b).

draw a dissonance curve of the Harmonium. The results are considered in order to discuss how relevant the consonance of musical intervals found in Harry Partch’s just intonation system are, especially those related to higher harmonics. Before that, on the next section, I present a brief description of Harry Partch’s Just Intonation system, and his One Footed Bridge Graph.

Figure 1. Plomp & Levelt’s Roughness curve for pure tones. The vertical axis is the Consonance/Dissonance. Horizontal axis is the frequency difference in the Critical Band Scale (the Bark Scale).

B. Scale and Spectrum To account the Roughness of complex spectrums – or the Roughness juxtaposed complex spectrums in musical intervals – the Roughness values of every combination of spectral components (pair of pure tones) are summed up. Given a sound spectrum, we can duplicate it, maintain one of them at a fixed pitch, and vary the other ascendantly in pitch up to a specified musical interval. By calculation the Roughness in this specified range, we can draw a Roughness/Dissonance Curve (as shown in Figure 2). Minimum points (a so called “valley” on the graph) represent the consonance as a result of the alignment of partials, and can be considered as a Scale Step of such spectrum (Sethares, 2005). Figure 2 is the Dissonance Curve of a slightly inharmonic saw tooth waveform. Because of this small distortion of the spectrum, the alignment of partials (valleys) occur in corresponding slightly inharmonic musical intervals. Please note the valley just above the octave (12 Semitones). Vertical lines, on this Figure, not only depict the consonant steps (valleys), but also the maximum values of Roughness, or dissonant steps (also defined as “peaks”). A good contrast between peaks and valleys does indicate a greater relevance concerning the perception of Consonance. On the next graph (Figure 2), one can easily spot both the fifth and the octave (each of them slightly sharp because of this spectrum’s distortion) as the most relevant consonant steps. On this paper, the Roughness model is applied to measure and

Figure 2. Saw tooth curve. Vertical axis is the musical intervals in Semitones (upwards). Horizontal axis is the Roughness Perception (from left to right).

III. HARRY PARTCH’S SYSTEM A. The 43 Tone Just Intonation System Partch carried out tuning perception experiments with his Harmonium-like instrument, the Chromelodeon. He stated that he could perfectly tune the Chromelodeon by ear, which means eliminating the sensations of Beatings and Roughness. Up to now, this activity is not surprising at all, since every musician is able to do that with some training. The only thing is that Partch did tune his Chromelodeon in a Just Intonation

system that comprises 43 tones in an octave, and the musical intervals are related up to the 11th harmonic! His 43-tone system is based on two scales of six notes: the O-tonality e U-tonality. “O” stands for “Overtone Series”, while “U” for “Undertone Series”. The last is a theoretical concept of an inverted Harmonic Series. Both of Partch’s “tonalities” are described on the table below.

The harmonic Series, or “Overtone Series” are multiples of a fundamental tone [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ..., ∞]. On the other hand, we can invert it to an “Undertone Series” [1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, ..., ∞]. Here it is possible to maintain the conception of the relationship between terms of a “series” and its “fundamental”, and thus find the inverted relations of the O-tonality scale.

Cents

Interval Name

Interval Ratio

(Sub) Harmonic

Tonality

0

Unison

[1:1]

1st

O-Tonality

Partch’s tonalities were transposed by him to generate a system of 43-tone per octave. The system contains 7 complete O-tonalities & 7 complete U-tonalities (plus others that are incomplete). Partch separates the 43 tones of his system in 4 categories, which are: Power (Unison, Fifth, Fourth and Octave), Suspense (Tritones), Emotion (Thirds and Sixths) & Approximation (Seconds and Sevenths). The 43 tones, as well as their categories, are marked on One Footed Bridge.

204

Major Second

[9:8]

9th

O-Tonality

B. One Footed Bridge

386

Major third

[5:4]

5th

O-Tonality

551

Augmented Fourth

[11:8]

11th

O-Tonality

702

Fifth

[3:2]

3rd

O-Tonality

[7:4]

Minor Seventh

[7:4]

7th

O-Tonality

0

Unison

[1:1]

1st

U-Tonality

231

Major Second

[8:7]

7th

U-Tonality

498

Fourth

[4:3]

4th

U-Tonality

649

Diminished Fifth

[16:11]

11th

U-Tonality

814

Minor Sixth

[8:5]

5th

U-Tonality

996

Minor Seventh

[16:9]

9th

U-Tonality

Table 1. Intervals and Respective (Sub) Harmonics of Partch’s “Tonalities”

The Octave is missing, but it is implied in the next cycle of the scale starting from the Unison. The Unison/Octave is the only interval that belongs to both tonalities. Apart from it, the O-tonality intervals are formed by the ratios between the first term of the series (or, more specifically, its octave relations 2nd, 4th, and 8th) and the, 3rd, 5th, 7th, 9th and 11th harmonics (fourth column from left to right). According to Harry Partch, this is an expansion of the tone material that was until then still locked in the ratios up to the 5th harmonic. The U-tonality is a simple inversion (where the Fifth inverts to the Fourth, the Major Third to the Minor Sixth, and so on). These inverted intervals are not possible to be found on the Harmonic Series, if you wish to maintain the conception of a ratio between one particular harmonic and its fundamental tone of the series.

Figure 3. One Footed Bridge

The One Footed Bridge is a mirrored graph, it is the consonance estimative of Harry Partch of his Just Intonation System. In a similar fashion to Figure 2, the One Footed Bridge represents musical intervals on its vertical axis. The

left half of the mirrored graph, upwards, is the register from the unison to 600 cents. The other half, downwards, goes from 600 cents up to the octave. The horizontal axis is the consonance measurement. Because of its mirrored like shape, we can see how every musical interval has an inversion on this system. Not only that, but we notice also that both an interval and its inversion do have the exact same value of consonance.

V. RESULTS: THE CHROMELODEON SCALE

IV. OBJECTIVES AND METHODOLOGY This study investigates the consonance of Harry Partch’s tuning system. This goal, in a broader sense, concerns every Just Intonation system related to distant terms in the harmonic Series. The research aims to demonstrate and discuss how significant is the consonance of such intervals on musical instruments, having as an example the analysis of the Chromelodeon’s spectrum, or the Chromelodeon Scale of consonant steps. A sample consisting of a digital audio signal of the Chromelodeon (at about 194Hz) was extracted from A Quarter-Saw Section Of Motivations And Intonations (Partch, 2006) – a didactic recording in character that enabled the editing of a musical tone. A FFT analysis of this sample was used to find the first more prominent partials of the Chromelodeon Spectrum (Figure 4). These harmonics allowed a reasonably good reconstruction of the timbre. From this list of frequencies and relative amplitudes, the Roughness Model was able to generate results that were compared to One Footed Bridge. Obviously, the results obtained only reflects a rather restrict portion of the instruments spectrum. In order to analyze a wider range of the Chromelodeon, it would be necessary to have more samples of musical tones. Nevertheless, the range around 200Hz to 400Hz is a good example concerning the musical practice. 1,00

0,75

0,50

0,25

19 4 38 8 58 1 77 5 96 9 11 63 13 57 15 50 17 44 19 38 21 32 23 26 25 19 27 13 29 07 31 01

0,00

Figure 4. Spectrum of the Chromelodeon. Vertical axis is the relative amplitude, horizontal axis is the value in Hertz.

In favor of peak and valley detection, I chose Sethares’ approximation of Plomp & Levelt’s curve. Parcutt’s approximation is considered to be a better one, but if I had used such a more accurate parameter analysis in my software, I would miss some of the results, which are those that represent not much significant alignment of partials. This, at some extent, means that I am distorting a bit the results, in order to have them. The consequence of this procedure is taken into account on the next sections of this paper.

Figure 5. The Chromelodeon Scale. Vertical axis is the interval in cents upwards. Horizontal axis is the Roughness perception in relative percentage, from right to left. The most prominent valleys are pointed by arrows.

All of the Consonants steps (valleys) found by the Software belong to Harry Partch’s system. The first column on table 2 is the ranking, in ascending order, of Roughness Perception. The second column brings the value of the Roughness values in relative percentage. The third column shows the musical intervals in their ratio format, while the fourth gives their values in cents. The fifth column is the Partch’s category of intervals, and, finally, the sixth column tells whether the interval belongs to an O-tonality (marked as “O”), a U-tonality (marked as “U”), both of them (marked as “O/U”), or neither (marked as “X”).

Table 2. Ranking of Valleys from the Chromelodeon Scale

Ranking

Roughness%

Ratio

Cents

Category

T

1

28.30%

[2:1]

1200

Power

O/U

2

34.21%

[1:1]

0

Power

O/U

3

40.81%

[5:3]

884

Emotion

X

4

42.11%

[11:6]

1049

Approx.

X

5

42.30%

[3:2]

702

Power

O

6

45.05%

[7:4]

969

Approx.

O

7

45.82%

[11:7]

783

Emotion

X

8

46.00%

[16:9]

996

Approx.

U

9

46.12%

[10:7]

617

Suspense

X

10

46.18%

[7:5]

583

Suspense

X

11

46.25%

[4:3]

498

Power

U

12

46.97%

[8:5]

814

Emotion

U

13

47.11%

[11:8]

551

Suspense

O

14

47.59%

[12:7]

933

Emotion

X

15

50.49%

[5:4]

386

Emotion

O

16

58.11%

[6:5]

316

Emotion

X

17

58.29%

[7:6]

267

Emotion

X

18

78.01%

[11:10]

165

Approx.

X

Only the interval [9:8], the Major Second, does not appear among the O-tonality intervals. This is not much of an issue, since it can definitely be considered a stable interval of juxtaposed Fifths [3:2], and easily tuned that way. Not only that, but [9:8] was not expected as a valley in the consonance plot, since it is still in an area strongly influenced by the critical band. Its inversion, the interval of [16:9] (the U-tonality’s Minor Seventh) was returned as a valley, but, as it will be better-informed bellow, does not really count. In the same way, [8:7] was not found as a valley among intervals from the U-tonality. One can argue that this is not as a stable interval as [9:8], because it is not a juxtaposition of Fifths. But it is better to consider the idea behind Partch’s architecture, and analyze this interval in connection to its inversion, the interval of [7:4], the one it is formed between [8:7] and the octave [2:1], meaning that this was the way in which Partch tuned it. Besides both [9:8] and [8:7], the only interval from Partch’s tonalities that wasn’t pointed out as a valley is the Tritone formed by the ratio of [16:11]. Outside the critical bandwidth influence as the previous exceptions (Major Seconds), the only explanation for its absence is the not at all significant alignment of partials it promotes. In order to appear as a significant valley, the spectrum of the Chromelodeon should have much more energy on its 16th harmonic, since it is too weak as shown on Figure 4. However, some valleys returned by the analysis do not belong to neither of Partch’s tonalities, one of which – the

Major Sixth [5:3] – is particularly very significant. And by the visual inspection on Figure 5, we can see that most valleys are not that significant, not to mention that Sethares’ approximation was used in order to enhance their significance. What brings me to the fact that the model actually did return a couple of completely insignificant valleys, which did not correspond to any Just Interval and were discarded for being considered a mere chance feature, usually expected. Nevertheless, some alignments, even though not reasonably significant, were kept for the sake of research. This is the case of [16:9] (996 cents) and [12:7] (933 cents), which are not pointed out by arrows on Figure 5. Somewhat reasonably significant valleys are not free from discussion, as I have been pointing out how Sethares’ approximation does favor them. As an example, let me draw the attention to the intervals from the category of Suspense (Tritones). Regarding the Roughness Curve, they are in a particularly constant range bit, in which a small deviation to a more arbitrary interval, like the Tempered Tritone (600 cents), cannot be considered as a significant dissonant contrast.

VI. FINAL CONSIDERATIONS Helmholtz stated “... after all, I do not know that it was necessary to sacrifice correctness of intonation to the convenience of musical instruments” (Helmholtz, 1877:327). One goal of this paper is to shed some light to a plausible answer regarding Helmholtz’s questioning. Partch’s answer is clear: it was an “unnecessary sacrifice”. Based on this paradigm, Partch did build several instruments, and developed his Just Intonation system. One of much information Partch has left us is his One Footed Bridge. The result on Figure 5 returns more consistent data than in One Footed Bridge. Partch considered as equivalent the perception of both intervals and their inversions, what is impossible according to psychoacoustics. Not too keen on psychoacoustical experiments of his time (such as Plomp & Levelt’s), Partch had little knowledge about Psychoacoustical Models of Roughness that could have led him to a common end. Recently, these models have been quite developed, and are now commonly implemented on computer softwares for digital signal analysis. Their results have been widely accepted, as in the work of Sethares (2005). By confronting One Footed Bridge to a more recent psychoacoustical research, this paper revisits Partch’s estimation of consonance for his tuning system, and delivers a more detailed revision. One must also consider debating Partch’s theoretical premiss, that does not consider the minor triad as an inversion of the major triad, and do not depart from actual perceptual principles, as in the concept of an “Undertone Series”. More than perceptual, his choice of musical intervals related to the Overtone and Undertone Series is arbitrary. Had he actually pursued proper perceptually consonant steps, he wouldn’t have let aside the third most significant valley presented here, the Major Sixth [5:3], on his O-tonality. Neither its inversion (the 16th valley in the ranking), the Minor Third [6:5], on his U-tonality. This would even make them both 07-tone scales, and is not a complete distortion of his system, because it actually contains these intervals, but as scale steps from other tonalities.

As to the consonance significance of musical intervals corresponding to higher harmonics, although rather small, it exists. But it suffers from serious practical problems regarding musical instruments, mainly because it requires a tuning system with more than 12 tones per octave, which are totally incompatible to equal temperament. In fact, equal temperament defies and sacrifices correctness of intonation for musical intervals corresponding to the 5th harmonic (Thirds and Sixths). This is what Helmholtz was actually questioning about. In the end, even the more significant perceptual value of Thirds and Sixths were not able to prevail over practical elements (convenience of musical instruments). Perhaps it is because the “convenience” that sacrifices intonation does favor other musical aspects of our tradition, and not only a mere performance issue. If this picture will ever change, only time will tell. At least Partch has given the first steps.

ACKNOWLEDGMENT Clarence Barlow for sharing information regarding Roughness modeling, as well as Vassilakis and Sethares. Jônatas Manzolli for being my friendly supervisor during my master’s research, in which I wrote this paper. Fernando Iazzeta and Marcelo Queiroz for having me in the doctorate program at USP.

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