Acoustic analysis of musical intervals in modern Byzantine Chant scales

Byzantine Chant (BC) or Byzantine Music (BM) has been exclusively a monophonic vocal music (Wellesz, 1961) performed by chanters called psaltes. The term “echos” used in BM refers to the specialized meaning of the term “mode” (Arnold, 1983) denoting the musical scale being applied in a melody with a definite “tonic” or main note and specific music phrases, as well as its microintervallic structure or genus. There are three musical genera in BC: Diatonic (DI), Chromatic (CR)—subdivided in Malakon (CM) and Skliron (CS)—and Enharmonic (EN). The basic difference among these genera lies in their musical intervals. In modern BM, especially during the last $200years$⁠, two main octave divisions have been proposed; first by Chrysanthos of Madytos (1832) and later by the Patriarchal Music Committee (1883). These approaches raised a number of discussions among those studying the BM. On the other hand, there are open issues related to the intervallic system on which a specific performer actually sings, along with the associated accuracy during his/her performance (Gabrielsson, 2003). Similar issues have been studied in other musical cultures (Sundberg, 1987) and types of music (Prame, 1997; Loosen, 1993; Ely, 1992; Fyk, 1995; Greer, 1970; Rakowski, 1990). Moreover, a corresponding system of music performance rules has been found to exist with regard to western music (Friberg et al., 2006).

To the authors’ knowledge there is no published systematic work to verify the results proposed by the PMC. The present study deals with the relative length of the musical intervals among notes in performed musical scales, considered as a whole. This approach was based on the finding that musical context may have a significant influence on intonation (Frances, 1958), and skilled musicians developed expertise for playing and perceiving typical musical sequences as a whole (Krumhansl, 1979). According to BM theory, a tetrachord is an interval of fourth (3:2) and there is a trend during the performance forcing the intervals inside the tetrachord to vary with respect to the direction of the note sequence performance (Chrysanthos from Madytos, 1832). The notes at the edges of the tetrachord remain almost unchanged, while the others inside it are moved upward or downward depending on the musical scale direction during the performance (PMC, 1883). This phenomenon was recently reported (Tsiappoutas et al., 2004, 2006), but the conclusions were based on the performance of a two psaltes and two pieces of BM (not on musical scales). In this paper, the microintervallic structure of BM was studied as a function of two parameters: (a) musical genus and (b) ascending or descending musical scale direction.

The PMC (1883) proposed for the BC a 72-tone equal temperament system, with steps named “moria,” each representing a frequency ratio of $21∕72$⁠, or 16.667 cents. PMC’s values concern seven intervals: $(9∕8)=12.23m∕203.9c$¹ for all genera, $(9∕8)×(80∕81)2=9.65m∕160.9c$ for DI, $(27∕25)=7.99m∕133.2c$ for DI and CM, $(9∕8)×(25∕24)×(80∕81)2=13.89m∕231.5c$ for CM, $(9∕8)×(24∕25)×(80∕81)2=5.41m∕90.2c$ for CS and EN, as well as $(9∕8)×(27∕25)=20.23m∕337.1c$ and $(25∕24)=4.24m∕70.6c$⁠, for CS. Although the above-mentioned rather complex ratios, do not involve small enough numbers, in the sequel, they will be considered to be “justly” tuned intervals. All these values were approximated by PCM, according the 72-tone equal temperament system, into 12, 10, 8, 14, 6, 20, and 4 moria through truncating or rounding. Thus, the proposed PMC scales for the genera (along with their corresponding tetrachords) are DI: 12-10-8-12-12-10-8 (12-10-8), CM: 8-14-8-12-8-14-8 (8-14-8), CS: 6-20-4-12-6-20-4 (6-20-4), and EN: 12-12-6-12-12-12-6 (12-12-6), where between the two tetrachords of each scale a disjunctive interval of 12 moria exists.

A total of 13 male subjects, average age $50.8years$⁠, range 40–60, participated, following a systematic selection procedure. They were all famous psaltes, very experienced in BC performance, healthy, and without voice problems. They were asked to chant ascending and descending BC scales for the single Greek vowel /a/ in all genera, in a comfortable intensity, after a few warming-up exercises. The fundamental frequency range spanned from 132 up to $264Hz$⁠. The pressure signal, captured in a studio by a condenser microphone (K2 RODE) at a constant distance of $30cm$⁠, and the electroglottograph (EGG) signal were digitally recorded (console AWE16G) at a rate of $44.1kHz$⁠, $16bits$⁠. For each subject, a musical scale per genus was recorded five times, resulting in a corpus of 520 data scales. These recordings are part of the tagged DAMASKINOS prototype Acoustic Corpus of Byzantine Ecclesiastic Chant Voices (Chryssochoidis et al., 2007). Four characteristic examples are presented in multimedia (Mm.) files.

Three pitch detection algorithms (Rabiner and Schafer, 1978), i.e., the autocorrelation, the harmonic product spectrum (with eight replicas), and the cepstrum (with a lifter length corresponding to the range $50–500Hz$⁠), implemented in MATLAB, were applied. A nonoverlapping $30ms$ Hamming window was used to the pressure signal of each data scale. The autocorrelation algorithm was also applied to the EGG signal. The time domain autocorrelation method makes use of the filtering technique, leading to enhanced performance, while the other two frequency domain methods differ considerably in the way they exploit the spectral characteristics of a signal. Out of these four pitch methods, the one with the smooth pitch track and consisting of almost constant notes (which, however, include a vibrato), without octave’s jumps and other errors, was selected for further analysis. Initially, the specification of the on- and offset of each note was based on the first derivative of a smoothed pitch track, in which the maxima and minima correspond to an increase and a decrease of the pitch, respectively. Only the extremes, over or under a threshold, were taken into further consideration. Then, the main “body” of a note was specified to be at a certain pitch value if the rate and the extent of its vibrato were kept within certain limits (6.5 undulations per s and $±30$ cents vibrato extent) (Sundberg, 1979). In order to reduce the influence of the transition parts at the boundaries of a note, a statistical analysis over all the frequencies inside each note was performed, so that the values closed to on- and offset of a note, which were larger than the median value of note’s frequencies, plus or minus a standard deviation, were discarded. Since humans perceive the mean pitch in a logarithmic scale (Shonle and Horan, 1980), the use of moria and cents in pitch measurements is justified. Moria are used here for historical reasons. Notes with gliding tones (glissando) exceeding the threshold of 0.16 semitones (16c) per s, for $1s$ tone duration were discarded, since their pitches would be somewhat problematic (d’Alessandro et al., 1998).

The seven intervals of each eight-note scale were computed, relative to the starting note, for both ascending and descending scale directions. Thus, the total number of intervals for all the data scales was 3640. Whenever a pitch value was missed at some point of the scale, the next one was taken as the new starting point. The intervals of a tetrachord, along with the disjunctive one, were represented as, first (1st), second (2nd), third (3rd), and fourth (4th), respectively, for the genera DI, CM, CS, and EN. The following analysis was based on the assumption that the corresponding interval lengths of two tetrachords are identical, from a performance point of view. Each measured interval was compared with its counterpart of the PMC scale by calculating the difference $Δ=∣measured∣−∣PMC∣$ with an accuracy of one decimal point, namely 0.1m/1.6c.

The $Δ$ values for all five trials of all psaltes were statistically analyzed. The intraperformers’ variability was assumed to be comparable to the variability of each one of the data samples, since the mean value of all $Δ$ for one performer did not differ from the values of the other performers by more than 3.3c. The average value of the standard deviation for all the performers was 31.6c for both ascending and descending scales. The one-way analysis of variance demonstrated closed relation among the performers for all intervals: $F=0.72$⁠, $p=0.7053$⁠, in descending scales and $F=0.54$⁠, $p=0.8869$⁠, in ascending scales.

All the mean $Δ$ values do not exceed the range of $−40c$ (min) to 33.3c (max). For both scale directions, the corresponding measures were $−30c$ and 28.3c, respectively. The standard deviation varies between 21.6c and 50c. The average standard deviation for all the intervals was 30c, which was found to be comparable to the average value of the standard deviation for all the performers (31.6c).

Searching for possible trends in $Δ$ values, with regard to the scale direction, the corresponding t-tests showed that for the intervals 2ndCM, 3rdCM, 3rdDI, 4thDI, 2ndEN, and 3rdEN (1st set) there is no statistically significant difference (95% confidence interval) between descending and ascending scales: $0.529⩽p⩽0.991$⁠, while for the 3rdCS, 2ndDI, and 4thEN (2nd set), there is a small difference: $0.099⩽p⩽0.426$⁠. For the rest of the intervals, namely 1stCM, 4thCM, 1stCS, 2ndCS, 4thCS, 1stDI, and 1stEN (3rd set), it appears that the $Δ$ values differ between scale directions significantly $(p<0.023)$⁠, except for the 1stEN $(p=0.087)$⁠. The corresponding differences between the means of $Δ$ values for the sets of intervals (defined as $ΔM$⁠), were measured to be smaller than 2.4c, for the first group of intervals, 4.4c, 4.9c, 4.2c, for the second one, and between 7.3c and 15.8c for the third one. The apparent trend in these findings is that the 1st interval of the tetrachord for all genera increases in the ascending with respect to the descending scale, while the 4th interval decreases for the two chromatic genera between the two scale directions (Fig. 1).

The t-test for the 1st, 2nd, and 3rd interval between the CM and CS genera showed that there are statistically significant (95%) differences in both scale directions. The $p$ values were about zero $(p=0.000)$ and the corresponding $ΔM$ values were $−13.5c$⁠, 57.2c, $−45.1c$ for the ascending scale, and $−14.5c$⁠, 64.5c, $−49.8c$ for the descending one. The opposite signs in $ΔM$ values between the adjacent intervals in the tetrachord indicate the opposite changes in length between them. In other words, when an interval that belongs to one chromatic genus increases, the corresponding interval of the other genus decreases and visce versa (Fig. 1). The $ΔM$ value for the 4th interval, between the two chromatic genera, was also found to be significant $(p=0.003)$ in ascending direction of the scale $(−13c)$⁠, but not in descending $(p=0.117)$ with value $−7c$⁠. Between the DI and EN, for the 2nd and 3rd intervals, the $ΔM$ values were computed to be 16.1c and $−25.6c$ for the ascending scale, and 19.9c and $−23.3c$ for the descending scale. Thus, they exhibit an explicit trend $(p=0.000)$ to decrease for the 2ndEN interval relative to 2ndDI, and inverse, to increase for the 3rdEN relative to 3rdDI interval for both directions of the scale. The $ΔM$ values, for the 1st and 4th intervals, between the diatonic and enharmonic genera, were found statistically insignificant $(0.269<p<0.754)$ and were computed to be smaller than 4.3c.

The less audible music interval, for PMC’s musician experts, was considered to be 1m or 16.7c (PMC, 1883). This seems to differ from findings in other styles of singing for tones with vibrato, while it is in line for nonvibrato tones (Sundberg, 1987). Provided that two intervals in a scale differ at least 2m or 33.4c, any deviation more than 1m should be audible by musician experts (Burns, 1999).

By considering that the interval values proposed by the PMC are approximations of their “just” counterparts and by examining their relations with the corresponding $Δ$ values, the following remarks could be made: Although the interval of major second has been approximated by 12 moria for all the genera, this was found true only for the disjunctive interval and this can be also inferred from the proceedings of the PMC. Also, the two intervals of 8 moria (in DI and CM) and 6 moria (in EN and CS) have been considered to be identical. As it mentioned earller, the difference between the justly tuned intervals and the equal temperament ones proposed by the PMC are 0.23m/3.8c, $−0.35m∕−5.8c$⁠, $−0.01m$/0.2c, $−0.11m∕−1.8c$⁠, $−0.59m$/9.8c, 0.23m/3.8c, and 0.24m/4c for the intervals 12, 10, 8, 14, 6, 20, and 4 moria, respectively. The corresponding mean $Δ$ values inside the confidence interval at the level of 95% were found $2.2±2.1c$⁠, $6.7±2.9c$⁠, $−13.8±1.9c$⁠, $26.3±5.9c$⁠, $7.1±2.3c$⁠, $−34.4±4.2c$⁠, and $30.6±3.3c$ [Fig. 2(a)]. Each one of the previous values does not belong in the confidence interval of the corresponding mean $Δ$ value, showing an explicit deviation from that of the PMC’s. The only exception is for the case of the disjunctive interval of 12 moria.

The statistically significant $ΔM$ values between the CM and CS genera for the 1st, 2nd, and 3rd intervals and the ones between the DI and EN genera for 2nd and 3rd intervals indicate a strong interaction among them. The $Δ$ values for adjacent intervals (without overlapping of their confidence intervals) have opposite signs between the two chromatic or diatonic genera (Fig. 1), except for the 1st CM-CS in the descending scale, and each absolute $ΔM$ value between the two genera, for an interval, does not exceed the absolute difference of its PMC values, for the two genera. For example, for the 1st interval between the two chromatic genera, the absolute $ΔM$ value is 0.8m/13.3c in ascending scale, which does not exceed the absolute difference of its PMC values for the two genera: $∣8−6∣=2m(33.3c)$⁠. This finding also holds true for all the above-mentioned intervals: (⁠$0<1st$ $(13.3c,15c)<8−6(33.3c)$⁠, $0<2nd$ $(56.7c,65c)<20−14(100c)$⁠, $0<3rd$ $(45c,50c)<8−4(66.7c)$⁠) (CM-CS), and (⁠$0<2nd$ $(16.7c,25c)<12−10(33.3c)$⁠, $0<3rd$ $(20c,23.3c)>8−6(33.3c)$⁠) (DI-EN). An explanation for this could be that, while a chanter is trying to perform a very small or very large interval, as for example a chromatic one, he does not attain that (in the meaning of average values). The result is the production of an interval with intermediate length between its PMC values for the two genera. This phenomenon could be ascribed to an interaction between “relative” genera. Thus, we call this phenomenon of pitch alteration on the corresponding notes between two musical scales, intra-genera interaction.

The second factor that causes intervallic changes is the direction of the musical scale sequence. The influence of this factor can be apparent from $ΔM$ values, for each interval, between the ascending and descending scale. The effect of intervallic change, due to the scale direction, has been mentioned in the literature on BC. However, what we know about it comes from the performers’ experience. The trend for an interval to be modified, due to the scale direction change, may be based on the psychological effect of the attraction to the target note (Tsiappoutas et al., 2004). As we mentioned before, in BM the scale is organized in the form of tetrachords or pentachords (a tetrachord including a disjunctive tone). That means as a performer chants the notes inside a tetrachord or pentachord starting from the tonal (starting note) and moving to the last note and then comes back he/she tends to drift them out of position. The most significant result of this effect is the alteration of the 1st interval, of which the main trend is to increase in length in ascending scale compared to one in descending scale, for all genera. When performing scales in the form of tetrachords, there is a tendency to modify the 3rd interval, as in the case of CS (small difference) or DI (very small difference), while in the case of EN there is a negligible modification (indicating a rigid interval). Also, the 3rd interval should appear an inverse change with respect to the 1st interval, (5c, $−10c$⁠)(CS), (17c, $−6.7c$⁠)(DI), (⁠$−6.7$⁠, 0.0c)EN, although there is a statistically insignificant modification. When performing in the form of pentachords, which is not rare in chanting a scale, the modification of the 4th interval is the main effect, as in the case of CM, CS, and EN genera. Such a phenomenon of a note’s pitch change caused by a kind of attraction to the nearest note in scale has recently been reported as the attraction effect (Tsiappoutas et al., 2006) but also it has been reported for several years in BM literature (Panagiotopoulos, 1981).

Provided that the $Δ$ values were negative and positive ones, an expected mean of $Δ$ values for all intervals inside an octave should be zero valued, however this is not the case. There exists a deviation from zero showing, of course, a trend existing in the performance and called octave stretch (Sundberg, 1972). Octave stretch concerns pure octave intervals, as well as sums of successive intervals of seconds inside an octave scale. Average values of all subjects’ octave stretches with their corresponding confidence intervals were computed for each musical genus and scale direction [Fig. 2(b)]. This computation revealed that averages of octave stretches among different genera of same scale direction did not differ more than 10c in ascending and 21.7c in descending scale, but between the directions of scale there was an important difference [Fig. 2(b)] 46.7c as the maximum value. The average octave stretch for all genera and scale directions was computed to be 0.7c.

The former estimated octave stretch could explain the small alteration of the 3rd interval due to the scale direction for all genera. It can be suggested that by scaling down the music scale, especially in descending order, any change in the interval length could be masked by that stretch effect.

In order to have an estimation of the contribution for each of the above-mentioned factors of alteration of intervals, a score was fabricated from the sum of the absolute $Δ$ values for each contributor divided by the number of values. Thus, the score value was measured to be 20.8c per interval, concerning the intra-genera interaction, and 7.4c per interval, due to the scale direction. Under the assumption that the two contributors are independent of each other, the smallest contribution should be due to the direction of the musical sequence.

All the above-mentioned measurements concern sung notes extracted from a scale, namely each tone was sung in the context of a preceding or subsequent tone in that scale. It is known that a sung tone in such a context changes its pitch tending to come close to the next tone toward the direction of scale (Fyk, 1997). This results in an alteration of the music interval between the two successive tones and this is verified from our data.

A comparative analysis was conducted between experimental values of BC intervals during performance of musical scales and the ones of PMC’s scales. Differences in interval lengths were measured from 0.1c up to 39.3c, while the biggest differences were found for the chromatic genera. All interval deviations are either positive or negative depending on the intra-genera interaction, the scale direction, and octave stretch effects. The corresponding intervals of the two chromatic genera deviate from PCS values in reverse way between them, which can be ascribed to the effect of the intra-genera interaction. Thus, DI and CM interact with EN and CS, respectively. Differences in interval sizes between the ascending and descending musical scales, during the performance, were measured. Thus, inside a pentachord or tetrachord, the first interval length increase and the fourth or third interval length decrease, in ascending scales, and, inversely, in the descending ones. Between the two effects of the intervals alteration, the one of the intra-genera interaction was observed to have the biggest influence in intervals deviation.

This work was cofunded by European Community Funds and Greek National resources under the project “AOIDOS” of the EPEAEK program.