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P2.2. Music Temperaments Evaluation Based on Triads

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P2.2. Music Temperaments Evaluation Based on Triads Music Temperaments Evaluation Based on Triads Tong Meihui Tojo Satoshi JAIST JAIST [email protected] [email protected] ABSTRACT Ogata [4] has proposed the idea of consonance value among chords, calculating the consonance value of each two tones It is impossible for one temperament to achieve optimally in the triads and adding them up, and then drew a 3D dis- both of consonance and modulation. The dissonance level sonance curve of the chords. Also, Cook [5] showed the has been calculated by the ratio of two pitch frequencies, acoustical properties of triads, claiming the perception of however in the current homophonic music, the level should harmony is not simply a sum of inner consonance. In this be measured by chords, especially by triads. In this re- research, we revise the Ogata’s calculation and formalize search, we propose to quantify them as Dissonance Index the dissonance level in a more rigorous way. of Triads (DIT). We select eight well-known temperaments This paper is organized as follows. In the following sec- and calculate seven diatonic chords in 12 keys and compare tion, we show preliminaries including the introduction of the weighted average and standard deviation to quantify various temperaments. Thereafter, we propose our formal- the consonance, and then we visualize our experimental re- ization and show its visualization. Then, we analyze the sults in a two-dimensional chart to compare the trade-offs results, and finally conclude. between consonance and modulation. 2. PRELIMINARIES 1. INTRODUCTION 2.1 Scale and Temperaments Nowadays, 12-tone equal temperament is prevalent and other temperaments has fallen to only historical and math- A set of notes employed in a music piece is, when arranged ematical interests. However, even now in the actual perfor- in a pitch order in an octave, is called a scale. The ratio of mance, string and wind instruments are played in an ad hoc frequency of two pitches is fixed by natural science disci- adjustment of pitches unless accompanied by keyboard in- plines such as physical science, acoustics, and psychology, struments. In this age, those electronic instruments ease among which mathematics plays the most important role, us in using any scale more freely. Then, our motivation in and one fixed series of ratios in a scale gives the notion of this paper is to give quantitative understanding to the dis- temperament. sonance level in various temperament in terms of triads. Pythagoras in ancient Greece discovered that the perfect The difference of temperaments has been often mentioned fifth interval with the frequency ratio of 3:2 as the most by the ratio of two pitch frequencies and such web site as consonant, next to the octave of 2:1, around 550 BC [6]. Pianoteq 1 provides us a very convincing interface to ex- The Sanfen Sunyi-fa by Jing Fang in China (BC77–BC37) perience the difference of temperaments; however, there is considered to have invented the same temperament with were no mathematical formulation to evaluate the conso- Pythagorean tuning [7]. nance and modulation 2 of a chord. Since then, musicologists have been constantly exploring Consonance and dissonance are ambiguous psychologi- how to solve the problem of Pythagorean comma, that is cal notions. The purpose of this research is to explore the the error which slightly exceeds the octave when the 12th mathematical model of Dissonance Index of Triads (DIT). tone is introduced by multiples of 3/2. If we perempto- In 1863, Helmholtz [1] proposed the mathematical model rily regard the 12th locates at the octave, the interval be- of consonance and dissonance in tones in terms of beats tween the 11th and the 12th becomes narrower than the and roughness. In 1965, Plomp and Levelt [2] defined the other fifths. In later years, Pythagoras pitch was amended dissonance curve between two pure tones. Later, the math- to place the narrow fifth, so called the wolf fifth 3 , between ematical formula of the curve has been improved, and Vas- G] and D] where the fifth is rarely used. silakis [3] claimed that the formula he proposed had been The ultimate temperament for consonance is the just in- believed to be most reliable; and thus, we employ it also in tonation, introducing the multiple by 5 in addition to 3/2, this paper, though adding the effect of overtones. where the ratio between the intervals can be expressed all by small integers [8]. However, in contrast, the just intona- 1 https://www.pianoteq.com/ 2 In this paper, the modulation means a key transposition. tion is very clumsy in modulation. The scale evolved into the mean-tone systems [9, 10], well-temperaments by An- dreas Werckmeister (1645–1706) [11] or by Johann Philipp Copyright: c 2019 Tong Meihui et al. This is an open-access article distributed Kirnberger (1721–1783) [12], and since then there exist under the terms of the Creative Commons Attribution 3.0 Unported License, which hundreds or even thousands of music temperaments. permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 3 It is named after the unpleasant sound like the roaring of wolves. C bDD bEEF #FG bAA bBB 37 9 39 81 311 729 3 212 27 215 243 Sanfen Sunyi-fa 1 211 8 214 64 217 512 2 38 16 310 128 256 9 32 81 4 729 3 128 27 16 243 Pythagorean Tuning 1 243 8 27 64 3 512 2 81 16 9 128 16 9 6 5 4 45 3 8 5 16 15 Just Intonation 1 15 8 5 4 3 32 2 5 3 9 8 1 6 3 5 3 2 2 4 1 4 2 4 2 5 2 5 2 5 4 8 5 2 5 Quarter-Comma Meantone 1 5 2 3 4 1 23 5 5 2 1 22 5 4 5 4 5 4 5 2 7 1 6 3 5 4 2 2 4 1 2 4 2 4 5 5 2 5 2 5 4 5 5 2 5 Conventional QC Meantone 1 24 2 3 4 1 23 5 24 2 1 22 5 4 5 4 5 2 p p p p p 28 64 2 32 213 4 2 4 210 8 4 8 128 210 4 2 16 128 4 2 Werckmeister 1 35 81 27 38 3 36 9 81 36 9 81 p 135 9 32 5 4 45 3 128 3 5 16 15 Kirnberger 1 128 8 27 4 3 32 2 81 4 9 8 1 2 3 4 5 6 7 8 9 10 11 Equal Temperament 1 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 Table 1: Ratios of Temperaments in Fractions The equal temperament has been the product of compro- mise, which systematically compensated the Pythagorean comma, defining each half tone to be the 12th root of 2. Then, the temperament perfectly eased the modulation, that is, to enable us to change from one key to another freely, and was applied to the tuning of most modern musical in- struments around the world. But we can never say that the equal temperament is satisfactory because it rejects the original intention of the temperament, viz., the consonance between intervals. Table1 lists the frequency ratios of some typical music temperaments introduced above. 2.2 Helmholtz’s Theory of Beats In physics, the superposition of two simple sinusoidal waves with similar but slightly different frequency will cause pe- Figure 1: Dissonance Curve From a Fixed Tone to Another riodic fluctuation in strength through time. This phenomenon Tone [2] is known to piano tuners as beats. Hermann Helmholtz [1] concluded that dissonance is produced by the beats be- tween two pure tones (without overtones) or between a pair of dissonance, and the interval is from 0 to 1 from top to of partials of two complex sounds. bottom. The lower the vertical value is in this figure, the When the difference in frequency is small, the beats can more dissonant. The horizontal axis is the frequency dif- be easily heard. As the difference is increased to 20-30 ference between higher tone and the base tone, divided by Hz, the beats will create the impression like ‘jarring and the value of the critical bandwidth. As the frequency dif- rough’ described by Helmholtz. Beyond this approximate ference gradually becomes larger, we can observe the re- point, the beats gradually become too rapid to be identified sult of the dissonance value d between the two pure tones and the sensation of roughness disappears. varying. The most dissonant position (d = 1) is said to be about a quarter of the critical band. When the frequency of the tone is too high or too low 2.3 Dissonance Curve to be heard by human ears, the identification of the tones In 1965, Plomp and Levelt confirmed Helmholtz’s hypoth- becomes not that easy. When the horizontal axis of the dis- esis by several experiments [2]. They plotted the disso- sonance curve only takes the frequency difference (without nance curve and proposed the concept of critical band- divided by critical bandwidth), we need to draw many dif- width. Note that though the sound produced by the mu- ferent graphs according to the difference of base tones. sical instruments has a complex timbre this psychologi- cal experiment employed only pure tones with the simplest 2.4 Numerical Calculation of Dissonance in Two spectrum. The combined experimental results is shown in Tones Figure1, and nowadays this result is widely accepted.
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