Common and distinct quantitative characteristics of Chinese and Western music in terms of modes, scales, degrees and melody variations

Figures & data

Figure 1. The scale formation of the Western major mode, based on 12-TET.

Figure 2. The three scale formations of the Western minor modes, based on 12-TET. The descending melodic minor scale is consistent with the natural scale.

Figure 3. The scale-degree formation of five Chinese pentatonic modes in a Tonggong system with the first degree as the tonal head (tonic) and Gong as the tone master.

Figure 4. The scale formation of three Chinese heptatonic scales, based on 12-TET. Each of the three heptatonic scales has five modes with the tonal heads (tonics) of Gong, Shang, Jue, Zhi, and Yu.

Figure 5. The Chinese Xiazhi scale, the Western C major scale and the Western A natural minor scale, based on 12-TET.

The Chinese Xiazhi scale consists of five pentatonic degrees and an added-tone pair. It has five modes with tonal heads (tonics) corresponding to the pentatonic degrees in the Tonggong system. The Western natural scale has two forms: the natural major and natural minor scales, which are relative scales. The inner relationships among the degrees in the Chinese Xiazhi scale and the Western natural scale are different.

Figure 6. The mode distribution of the four Chinese musical genres: Chinese Folk Song (CFS), Chinese Traditional Instrumental Music (CTIM), Chinese Opera (CO) and Chinese Popular Music (CPM).

Figure 7. The mode and key distribution of Western major/minor music.

The proportions of the Western major and minor modes are 59.56% and 40.44%. The profiles of the twelve keys of the major modes are 14.7% (C), 10.3% (D), 8.1% (G), 6.6% (E), 5.2% (F), 4.4% (D# /E b), 3.7% (A # /B b), 2.2% (C #/D b, 1.5% (G #/ A b), 1.5% (A), 0.7% (B), and 0.7% (F # / G b); those of the minor modes are 9.6% (G), 9.6% (D), 4.4% (A), 3.7% (C), 2.9% (F), 2.2% (E), 2.2% (C \# / D b), 2.2% (A # / B b), 1.5% (B), 1.5% (D # /E b), and 0.7% (F # / G b).

Figure 8. Proportions of twelve degrees of Chinese pentatonic music and Western heptatonic music.

Figure 9. The three heptatonic scale distributions of Chinese Folk Song (CFS), Chinese Traditional Instrumental Music (CTIM), Chinese Opera (CO) and Chinese Popular Music (CPM).

The three heptatonic scale distributions of Chinese Folk Song (CFS), Chinese Traditional Instrumental Music (CTIM), Chinese Opera (CO) and Chinese Popular Music (CPM). In each musical genre, the proportion of the Xiazhi scale music is the highest, the proportion of the Zhengsheng scale is sharply reduced, and that of the Qingsheng scale music is the lowest. (Sometimes two different heptatonic scales are used in a musical piece, so the total proportions may exceed 100%.).

Table 1. The 3-section temperament recorded in Diyunpian of Guanzi.

Producing order	Tone	Calculation
1	Gong (Do/ 1)	$1\times 3^4=9\times 9=81$
2	Zhi (Sol/ 5)	$81\times 4/3=108$
3	Shang (Re/ 2)	$108\times 2/3=72$
4	Yu (La/ 6)	$72\times 4/3=96$
5	Jue (Mi/ 3)	$96\times 2/3=64$

Table 2. The 3-section temperament recorded in Yinlüpian, Lüshichunqiu of Shiji.

Producing order	Tone	Calculation
1	Gong (Do/ 1)	$1\times 3^4=9\times 9=81$
2	Zhi (Sol/ 5)	$81\times 2/3=54$
3	Shang (Re/ 2)	$54\times 4/3=72$
4	Yu (La/ 6)	$72\times 2/3=48$
5	Jue (Mi/ 3)	$48\times 4/3=64$

Figure 10. Two columns of melody variation CCDFs measured by the three interval-dividing metrics, with Chinese pentatonic music in the left column and Western music in the right column. Case A, B, and C are the CCDFs of the melody variations of a folk song, Midu Shan’ge, from Yunnan, China measured by three interval-dividing metrics; Case D, E, and F are the results of Beethoven, Turkish March in $B^b$ Major, Op. 113, No. 4 measured by the same metrics.

Case A, B, and C in the figure are the CCDFs of the melody variations measured by three interval-dividing metrics of a folk song from Yunnan province in China – Midu Shan’ge; Case D, E, and F are the results of Beethoven, Turkish March in B b Major, Op. 113, No. 4. Case A and D in the figure are the results measured by the five-tone interval-dividing metric, Case B and E measured by the seven-tone interval-dividing, and Case C and F measured by the twelve-tone interval-dividing. They all observe the power law except the Chinese folk song measured by the twelve-tone interval-dividing metric (Case C), which demonstrates a ladder pattern at the tail of the distribution.

Figure 11. The adjusted-$R^2$ of the CCDFs of Chinese pentatonic music and Western heptatonic music measured by the three interval-dividing metrics. Chinese pentatonic music includes four genres: Chinese Folk Song (CFS), Chinese Traditional Instrumental Music (CTIM), Chinese Opera (CO) and Chinese Popular Music (CPM). Western heptatonic music includes the eight genres/periods: Baroque, Classicism (Class.), Romanticism (Romant.), nineteenth Century Opera (19thCO), Nationalism (Natl.), Impressionism (Impr.), Neoclassicism (Neoclass.), and Popular Music (Pop).

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Adjusted- R2s of Chinese Folk Songs (CFS) measured by the three interval-dividing metrics are respectively 0.95, 0.93, and 0.76; Chinese Traditional Instrumental Music (CTIM) 0.98, 0.97, and 0.83; Chinese Opera (CO) 0.98, 0.95, and 0.80, and Chinese Popular Music (CPM) 0.97, 0.95, and 0.81. Adjusted- R2s of Baroque measured by the three interval-dividing metrics are respectively 0.97, 0.92, and 0.90, Classicism (Class.) 0.96, 0.92, and 0.90, Romanticism (Romant.) 0.95, 0.92, and 0.88, nineteenth Century Opera (19thCO) 0.94, 0.91, and 0.88, Nationalism (Natl.) 0.98, 0.96, and 0.93, Impressionism (Impr.) 0.90, 0.88, and 0.84, Neoclassicism (Neoclass.) 0.95, 0.91, and 0.83, and Popular Music (Pop) 0.95, 0.90, and 0.85.

Figure 12. The parameters $c$ and $D$ of the three interval-dividing metrics for the four Chinese musical genres (CFS, CTIM, CO, and CPM) and the eight Western musical genres/periods (Baroque to Pop).

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Subfigure A is the result of parameter c. The parameter c of the four Chinese musical genres and the eight Western musical genres/periods measured by the five-tone interval-dividing metric are respectively 1.02, 1.01, 1.01, 1.01, 1.01, 1.02, 1.02, 1.03, 1.03, 1.02, 1.04, and 1.04; the seven-tone interval-dividing metric 1.02, 1.02, 1.03, 1.03, 1.05, 1.05, 1.05, 1.05, 1.06, 1.03, 1.07, 1.05, and 1.06; and the twelve-tone interval-dividing metric 1.14, 1.13, 1.15, 1.13, 1.09, 1.09, 1.10, 1.10, 1.08, 1.13, 1.14, and 1.12. Subfigure B is the result of parameter D. The parameter D of the four Chinese musical genres and the eight Western musical genres/periods measured by the five-tone interval-dividing metric are respectively 1.02, 1.01, 1.01, 0.41, 1.13, 0.99, 0.89, 0.96, 0.81, 0.60, 0.87, and 0.60; the seven-tone interval-dividing metric 1.20, 0.92, 1.38, 0.47, 1.03, 0.94, 0.86, 0.95, 0.79, 0.61, 0.89, and 0.59; and the twelve-tone interval-dividing metric 0.87, 0.80, 0.97, 0.59, 0.90, 0.83, 0.81, 0.84, 0.80, 0.59, 0.90, and 0.56.