Harmony as an underlying ingredient in the numerical search for suitable equal temperaments

Melo, L.F.

doi:10.1590/1806-9126-RBEF-2023-0270

L.F. Melo ^** Correspondence email address: lucasfelipe.bmo@gmail.com

Universidade Federal de Minas Gerais, Instituto de Ciências Exatas, Departamento de Física, Belo Horizonte, MG, Brasil.

https://orcid.org/0009-0001-5415-8212

* Correspondence email address: lucasfelipe.bmo@gmail.com

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Figure 1
Representation of harmonics as vibrations on a string in terms of either distance or time.

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Figure 2
Krenek diagrams for major and minor pentatonic (5 notes per octave) and heptatonic (7 notes per octave) scales. The addressed heptatonic minor scale is also known as natural minor scale.

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Figure 3
Piano octaves of 12-TET and 19-TET. In colors we show the corresponding representations of the C major (left) and minor (right) chords.

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Figure 2
Krenek diagrams for major and minor pentatonic (5 notes per octave) and heptatonic (7 notes per octave) scales. The addressed heptatonic minor scale is also known as natural minor scale.

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Figure 3
Piano octaves of 12-TET and 19-TET. In colors we show the corresponding representations of the C major (left) and minor (right) chords.

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Figure 4
(a) Plots of Eq. (³ 3. Finding N-TETs Using M-element Sets of Pure Intervals 3.1. Method In this section, we built a method for finding which N-TETs, letting N be other than 12, suitably fit a given M-element subset of the chromatic just scale. Firstly, we set the error we are willing to take by addressing specific partitions of the semi-tone which are called commas. For example, in 12-TET each semi-tone has 4.5 commas, yielding a total of 54 such partitions per octave [15]. N-TETs have different densities of commas per interval, μ=μ⁢(N), while maintaining the total number of such partitions within the octave. We choose to use the minimum of those partitions as a measure of frequency resolution6: (2) Δ f ( μ ) = ( 2 1 N μ − 1 ) f 0 , which is slightly lower than the Pythagorean comma [14]. Throughout this paper, without loss of generality, f0 is chosen to be the frequency of the C4 which is approximately 261.63 Hz, so Δ⁢f⁢(μ)≈3.38 Hz. Note in Tab. 1 that some frequency differences between the 12-TET and just intonation are higher than (2) yielding out-of-tune notes, but it stands as a good fit in average. We now design a measure for quantifying the average performance of an N-TET in producing a certain target set of pure intervals. Consider an ordered set of frequencies corresponding to just intervals 𝒫M={𝒫kM}k=0M with M≤12, where 𝒫k+1M>𝒫kM⁢⁢∀k. Using definitions (1) and (2) we address the following quantity: (3) β M ( N ) = 1 Δ f ∑ k = 0 M min ℓ ≤ N ( ℱ ℓ N − ρ k M ) 2 M . Then if βM⁢(N)<1 frequency deviations are less than Δf in average, indicating that the corresponding N-TET properly fits the target set, whereas the (N-M) extra intervals are labelled as intermediate notes. Note that if ||ℱℓ+1N-𝒫kM|-|ℱℓN-𝒫kM||<Δ⁢f for a certain ℓ∈[0,N], an evident ambiguity would take place when labelling which frequency corresponds to the target interval and a potentially useful intermediate note would be lost. Therefore, we sum the two differences in (3) as a measure of a non-desirable ambiguity. For being an unique set, the chromatic just scale will be denoted by 𝒫12. With the intent of testing different target sets 𝒫M with M<12 we use subscripts as useful labels: Major and minor scales will be denoted by 𝒫⁢m⁢a⁢j⁢o⁢rM and 𝒫⁢m⁢i⁢n⁢o⁢rM respectively, whereas uniform scales will be denoted by 𝒫⁢u⁢n⁢i⁢fM. The corresponding parameter (3) of each addressed set will carry the same label (like β⁢m⁢a⁢j⁢o⁢rM and so on). The test whether βM⁢(N)<1 or βM⁢(N)>1 is read as it is a good fit or it is not a good fit respectively. For comparing within range N∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] the average similarity between the results when addressing 𝒫12 and 𝒫M as target sets, we design the following quantity: (4) ℱ ( ρ M ) = 1 − ∑ k | Θ ( β M ( k ) − 1 ) − Θ ( β 12 ( k ) − 1 ) | N max − N min , where k∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] and Θ is the Heaviside function. Equation (4) then computes the coincidences between βM⁢(N)<1 and β12⁢(N)<1, and returns a value between 0 and 1. Evidently, ℱ⁢(𝒫12)=1. For that, we call Eq. (4) fidelity. It allows us to compare which target sets most faithfully reproduce the results of the chromatic just scale. 3.2. Numerical results We start by addressing the simplest divisions of the chromatic universe, which constitute the chromatic just scale 𝒫12, built only by semi-tone steps, and the whole-tone scale 𝒫⁢u⁢n⁢i⁢f6, built only by tone steps (see Tab. 2). The corresponding values of parameter (3) are shown in Fig. 4(a) within range N∈[9,54]. By addressing the chromatic just scale as target set we find TETs with 12, 19, 22, 24, 31, 34, 41 and 53 intervals as suitable and non-ambiguous7. As for the parameter β⁢u⁢n⁢i⁢f6, corresponding to the uniform hexatonic (see Fig. 4(b)), the correct identifications of the proper N-TETs are not reproduced. This means that some configurations can reasonably adjust the whole-tone intervals, like the 18-TET for example, while making other intervals audibly out-of-tune. Figure 4 (a) Plots of Eq. (3) within range N∈[9,54] when addressing as target sets the chromatic just scale, built only by semi-tone steps, and the whole-tone scale, built only by tone steps, yielding β12 and β⁢u⁢n⁢i⁢f6 respectively. (b) Krenek diagram of the whole-tone scale. In Figs. 5(a) and 5(b) (Figs. 5(c) and 5(d)) we plot parameter (3) for major and minor pentatonic (heptatonic) scales, respectively (see Fig. 2). Note that, when compared with the results achieved by addressing the chromatic just scale, major scales present more faithful answers about suitable N-TETs than the minor counterparts; the similarity increases with the scales sizes as expected. Note that the 53-TET is the most accurate fit for all addressed subsets of the chromatic universe. Figure 5 Comparisons between values of (3) for the chromatic scale (gray triangles) and: (a) the major pentatonic (purple circles), (b) the minor pentatonic (orange circles), (c) the major heptatonic (blue circles) and (d) the natural minor heptatonic (red circles). We computed fidelities (4) within range N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 by taking major, minor and uniform scales of different sizes as targets (see Fig. 2 and Tab. 2). The corresponding plot in Fig. 6 shows the difference between the convergences for each harmonic set: Major scales provide more faithful answers about proper N-TETs than minor and uniform ones of corresponding sizes, the latter being the most disadvantageous reference set. For example, the fidelities achieved when addressing the major pentatonic (M = 5) and the minor heptatonic (M = 7) as targets are comparable, both standing above the fidelity achieved when addressing the whole-tone scale (M = 6). Note in Fig. 6 that as we increase the scales sizes, thus addressing more information about pure intervals, we get more faithful identifications of the proper N-TETs for fitting the chromatic just scale with exception of the uniform case: The decreasing fidelity from M = 3 to M = 4 is legitimate because the uniform triad is not a subset of the uniform tetratonic. Figure 6 Comparison between fidelities for N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 when taking major, minor and uniform scales as target sets. Moreover, two observations suggest that the major and minor thirds are the most relevant ingredients for the convergence: (a) The main difference between major and minor scales of any size is the third interval, indicating that the extra major/minor sevenths are not crucial for the convergence, and (b) The uniform tetratonic, which produces the lower fidelity in Fig. 6, is the only uniform set which contains the minor third instead of the major one (see Tab. 2). The reason for a major thirds preference when setting a suitable equal-temperament, if not just a mathematical issue involving prime numbers, is quite mysterious but it seems to dialog with the fact that lower integer fractions correspond to the most appealing harmonies: The major and minor thirds are the nearest consonances from the tonic, moreover the 5/4 ratio is made of lower integers than the 6/5 ratio. The observation that strategic harmonic subsets of the chromatic universe can store sufficient information for setting proper equal-temperaments for the whole chromatic just scale indicates that extensions of the 5-limit just intonation to more than twelve tones, which exhibits notable difficulty in the construction of more and more fractions between powers of prime numbers, may not be necessary. Our results suggest that suitable fits for major scales are preferable in advance. Evidently, the average faithfulness will depend on the range of interest: Note in Fig. 5(a) that if we wish to look for suitable N-TETs within range N∈[9,42] then the set of pure intervals corresponding to a major pentatonic would be sufficient for finding the proper solutions, i.e. ℱ⁢(𝒫⁢m⁢a⁢j⁢o⁢r5)=1. ) within range

N \in [9, 54]

when addressing as target sets the chromatic just scale, built only by semi-tone steps, and the whole-tone scale, built only by tone steps, yielding β¹² and

β_{u n i f}^{6}

respectively. (b) Krenek diagram of the whole-tone scale.

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Figure 5
Comparisons between values of (³ 3. Finding N-TETs Using M-element Sets of Pure Intervals 3.1. Method In this section, we built a method for finding which N-TETs, letting N be other than 12, suitably fit a given M-element subset of the chromatic just scale. Firstly, we set the error we are willing to take by addressing specific partitions of the semi-tone which are called commas. For example, in 12-TET each semi-tone has 4.5 commas, yielding a total of 54 such partitions per octave [15]. N-TETs have different densities of commas per interval, μ=μ⁢(N), while maintaining the total number of such partitions within the octave. We choose to use the minimum of those partitions as a measure of frequency resolution6: (2) Δ f ( μ ) = ( 2 1 N μ − 1 ) f 0 , which is slightly lower than the Pythagorean comma [14]. Throughout this paper, without loss of generality, f0 is chosen to be the frequency of the C4 which is approximately 261.63 Hz, so Δ⁢f⁢(μ)≈3.38 Hz. Note in Tab. 1 that some frequency differences between the 12-TET and just intonation are higher than (2) yielding out-of-tune notes, but it stands as a good fit in average. We now design a measure for quantifying the average performance of an N-TET in producing a certain target set of pure intervals. Consider an ordered set of frequencies corresponding to just intervals 𝒫M={𝒫kM}k=0M with M≤12, where 𝒫k+1M>𝒫kM⁢⁢∀k. Using definitions (1) and (2) we address the following quantity: (3) β M ( N ) = 1 Δ f ∑ k = 0 M min ℓ ≤ N ( ℱ ℓ N − ρ k M ) 2 M . Then if βM⁢(N)<1 frequency deviations are less than Δf in average, indicating that the corresponding N-TET properly fits the target set, whereas the (N-M) extra intervals are labelled as intermediate notes. Note that if ||ℱℓ+1N-𝒫kM|-|ℱℓN-𝒫kM||<Δ⁢f for a certain ℓ∈[0,N], an evident ambiguity would take place when labelling which frequency corresponds to the target interval and a potentially useful intermediate note would be lost. Therefore, we sum the two differences in (3) as a measure of a non-desirable ambiguity. For being an unique set, the chromatic just scale will be denoted by 𝒫12. With the intent of testing different target sets 𝒫M with M<12 we use subscripts as useful labels: Major and minor scales will be denoted by 𝒫⁢m⁢a⁢j⁢o⁢rM and 𝒫⁢m⁢i⁢n⁢o⁢rM respectively, whereas uniform scales will be denoted by 𝒫⁢u⁢n⁢i⁢fM. The corresponding parameter (3) of each addressed set will carry the same label (like β⁢m⁢a⁢j⁢o⁢rM and so on). The test whether βM⁢(N)<1 or βM⁢(N)>1 is read as it is a good fit or it is not a good fit respectively. For comparing within range N∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] the average similarity between the results when addressing 𝒫12 and 𝒫M as target sets, we design the following quantity: (4) ℱ ( ρ M ) = 1 − ∑ k | Θ ( β M ( k ) − 1 ) − Θ ( β 12 ( k ) − 1 ) | N max − N min , where k∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] and Θ is the Heaviside function. Equation (4) then computes the coincidences between βM⁢(N)<1 and β12⁢(N)<1, and returns a value between 0 and 1. Evidently, ℱ⁢(𝒫12)=1. For that, we call Eq. (4) fidelity. It allows us to compare which target sets most faithfully reproduce the results of the chromatic just scale. 3.2. Numerical results We start by addressing the simplest divisions of the chromatic universe, which constitute the chromatic just scale 𝒫12, built only by semi-tone steps, and the whole-tone scale 𝒫⁢u⁢n⁢i⁢f6, built only by tone steps (see Tab. 2). The corresponding values of parameter (3) are shown in Fig. 4(a) within range N∈[9,54]. By addressing the chromatic just scale as target set we find TETs with 12, 19, 22, 24, 31, 34, 41 and 53 intervals as suitable and non-ambiguous7. As for the parameter β⁢u⁢n⁢i⁢f6, corresponding to the uniform hexatonic (see Fig. 4(b)), the correct identifications of the proper N-TETs are not reproduced. This means that some configurations can reasonably adjust the whole-tone intervals, like the 18-TET for example, while making other intervals audibly out-of-tune. Figure 4 (a) Plots of Eq. (3) within range N∈[9,54] when addressing as target sets the chromatic just scale, built only by semi-tone steps, and the whole-tone scale, built only by tone steps, yielding β12 and β⁢u⁢n⁢i⁢f6 respectively. (b) Krenek diagram of the whole-tone scale. In Figs. 5(a) and 5(b) (Figs. 5(c) and 5(d)) we plot parameter (3) for major and minor pentatonic (heptatonic) scales, respectively (see Fig. 2). Note that, when compared with the results achieved by addressing the chromatic just scale, major scales present more faithful answers about suitable N-TETs than the minor counterparts; the similarity increases with the scales sizes as expected. Note that the 53-TET is the most accurate fit for all addressed subsets of the chromatic universe. Figure 5 Comparisons between values of (3) for the chromatic scale (gray triangles) and: (a) the major pentatonic (purple circles), (b) the minor pentatonic (orange circles), (c) the major heptatonic (blue circles) and (d) the natural minor heptatonic (red circles). We computed fidelities (4) within range N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 by taking major, minor and uniform scales of different sizes as targets (see Fig. 2 and Tab. 2). The corresponding plot in Fig. 6 shows the difference between the convergences for each harmonic set: Major scales provide more faithful answers about proper N-TETs than minor and uniform ones of corresponding sizes, the latter being the most disadvantageous reference set. For example, the fidelities achieved when addressing the major pentatonic (M = 5) and the minor heptatonic (M = 7) as targets are comparable, both standing above the fidelity achieved when addressing the whole-tone scale (M = 6). Note in Fig. 6 that as we increase the scales sizes, thus addressing more information about pure intervals, we get more faithful identifications of the proper N-TETs for fitting the chromatic just scale with exception of the uniform case: The decreasing fidelity from M = 3 to M = 4 is legitimate because the uniform triad is not a subset of the uniform tetratonic. Figure 6 Comparison between fidelities for N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 when taking major, minor and uniform scales as target sets. Moreover, two observations suggest that the major and minor thirds are the most relevant ingredients for the convergence: (a) The main difference between major and minor scales of any size is the third interval, indicating that the extra major/minor sevenths are not crucial for the convergence, and (b) The uniform tetratonic, which produces the lower fidelity in Fig. 6, is the only uniform set which contains the minor third instead of the major one (see Tab. 2). The reason for a major thirds preference when setting a suitable equal-temperament, if not just a mathematical issue involving prime numbers, is quite mysterious but it seems to dialog with the fact that lower integer fractions correspond to the most appealing harmonies: The major and minor thirds are the nearest consonances from the tonic, moreover the 5/4 ratio is made of lower integers than the 6/5 ratio. The observation that strategic harmonic subsets of the chromatic universe can store sufficient information for setting proper equal-temperaments for the whole chromatic just scale indicates that extensions of the 5-limit just intonation to more than twelve tones, which exhibits notable difficulty in the construction of more and more fractions between powers of prime numbers, may not be necessary. Our results suggest that suitable fits for major scales are preferable in advance. Evidently, the average faithfulness will depend on the range of interest: Note in Fig. 5(a) that if we wish to look for suitable N-TETs within range N∈[9,42] then the set of pure intervals corresponding to a major pentatonic would be sufficient for finding the proper solutions, i.e. ℱ⁢(𝒫⁢m⁢a⁢j⁢o⁢r5)=1. ) for the chromatic scale (gray triangles) and: (a) the major pentatonic (purple circles), (b) the minor pentatonic (orange circles), (c) the major heptatonic (blue circles) and (d) the natural minor heptatonic (red circles).

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Figure 6
Comparison between fidelities for

N_{m i n} = 9

and

N_{m a x} = 54

when taking major, minor and uniform scales as target sets.

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Table 1
Pure intervals alongside their 12-TET counterpart frequency ratios [¹¹[11] J. Fonville, Perspectives of New Music 29, 106 (1991).].

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Table 2
Major, minor and uniform sets of sizes 3, 4 and 6. The uniform hexatonic is also called whole-tone scale.

(1)

(2)

(3)

(4)

Figure 4 (a) Plots of Eq. (³ 3. Finding N-TETs Using M-element Sets of Pure Intervals 3.1. Method In this section, we built a method for finding which N-TETs, letting N be other than 12, suitably fit a given M-element subset of the chromatic just scale. Firstly, we set the error we are willing to take by addressing specific partitions of the semi-tone which are called commas. For example, in 12-TET each semi-tone has 4.5 commas, yielding a total of 54 such partitions per octave [15]. N-TETs have different densities of commas per interval, μ=μ⁢(N), while maintaining the total number of such partitions within the octave. We choose to use the minimum of those partitions as a measure of frequency resolution6: (2) Δ f ( μ ) = ( 2 1 N μ − 1 ) f 0 , which is slightly lower than the Pythagorean comma [14]. Throughout this paper, without loss of generality, f0 is chosen to be the frequency of the C4 which is approximately 261.63 Hz, so Δ⁢f⁢(μ)≈3.38 Hz. Note in Tab. 1 that some frequency differences between the 12-TET and just intonation are higher than (2) yielding out-of-tune notes, but it stands as a good fit in average. We now design a measure for quantifying the average performance of an N-TET in producing a certain target set of pure intervals. Consider an ordered set of frequencies corresponding to just intervals 𝒫M={𝒫kM}k=0M with M≤12, where 𝒫k+1M>𝒫kM⁢⁢∀k. Using definitions (1) and (2) we address the following quantity: (3) β M ( N ) = 1 Δ f ∑ k = 0 M min ℓ ≤ N ( ℱ ℓ N − ρ k M ) 2 M . Then if βM⁢(N)<1 frequency deviations are less than Δf in average, indicating that the corresponding N-TET properly fits the target set, whereas the (N-M) extra intervals are labelled as intermediate notes. Note that if ||ℱℓ+1N-𝒫kM|-|ℱℓN-𝒫kM||<Δ⁢f for a certain ℓ∈[0,N], an evident ambiguity would take place when labelling which frequency corresponds to the target interval and a potentially useful intermediate note would be lost. Therefore, we sum the two differences in (3) as a measure of a non-desirable ambiguity. For being an unique set, the chromatic just scale will be denoted by 𝒫12. With the intent of testing different target sets 𝒫M with M<12 we use subscripts as useful labels: Major and minor scales will be denoted by 𝒫⁢m⁢a⁢j⁢o⁢rM and 𝒫⁢m⁢i⁢n⁢o⁢rM respectively, whereas uniform scales will be denoted by 𝒫⁢u⁢n⁢i⁢fM. The corresponding parameter (3) of each addressed set will carry the same label (like β⁢m⁢a⁢j⁢o⁢rM and so on). The test whether βM⁢(N)<1 or βM⁢(N)>1 is read as it is a good fit or it is not a good fit respectively. For comparing within range N∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] the average similarity between the results when addressing 𝒫12 and 𝒫M as target sets, we design the following quantity: (4) ℱ ( ρ M ) = 1 − ∑ k | Θ ( β M ( k ) − 1 ) − Θ ( β 12 ( k ) − 1 ) | N max − N min , where k∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] and Θ is the Heaviside function. Equation (4) then computes the coincidences between βM⁢(N)<1 and β12⁢(N)<1, and returns a value between 0 and 1. Evidently, ℱ⁢(𝒫12)=1. For that, we call Eq. (4) fidelity. It allows us to compare which target sets most faithfully reproduce the results of the chromatic just scale. 3.2. Numerical results We start by addressing the simplest divisions of the chromatic universe, which constitute the chromatic just scale 𝒫12, built only by semi-tone steps, and the whole-tone scale 𝒫⁢u⁢n⁢i⁢f6, built only by tone steps (see Tab. 2). The corresponding values of parameter (3) are shown in Fig. 4(a) within range N∈[9,54]. By addressing the chromatic just scale as target set we find TETs with 12, 19, 22, 24, 31, 34, 41 and 53 intervals as suitable and non-ambiguous7. As for the parameter β⁢u⁢n⁢i⁢f6, corresponding to the uniform hexatonic (see Fig. 4(b)), the correct identifications of the proper N-TETs are not reproduced. This means that some configurations can reasonably adjust the whole-tone intervals, like the 18-TET for example, while making other intervals audibly out-of-tune. Figure 4 (a) Plots of Eq. (3) within range N∈[9,54] when addressing as target sets the chromatic just scale, built only by semi-tone steps, and the whole-tone scale, built only by tone steps, yielding β12 and β⁢u⁢n⁢i⁢f6 respectively. (b) Krenek diagram of the whole-tone scale. In Figs. 5(a) and 5(b) (Figs. 5(c) and 5(d)) we plot parameter (3) for major and minor pentatonic (heptatonic) scales, respectively (see Fig. 2). Note that, when compared with the results achieved by addressing the chromatic just scale, major scales present more faithful answers about suitable N-TETs than the minor counterparts; the similarity increases with the scales sizes as expected. Note that the 53-TET is the most accurate fit for all addressed subsets of the chromatic universe. Figure 5 Comparisons between values of (3) for the chromatic scale (gray triangles) and: (a) the major pentatonic (purple circles), (b) the minor pentatonic (orange circles), (c) the major heptatonic (blue circles) and (d) the natural minor heptatonic (red circles). We computed fidelities (4) within range N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 by taking major, minor and uniform scales of different sizes as targets (see Fig. 2 and Tab. 2). The corresponding plot in Fig. 6 shows the difference between the convergences for each harmonic set: Major scales provide more faithful answers about proper N-TETs than minor and uniform ones of corresponding sizes, the latter being the most disadvantageous reference set. For example, the fidelities achieved when addressing the major pentatonic (M = 5) and the minor heptatonic (M = 7) as targets are comparable, both standing above the fidelity achieved when addressing the whole-tone scale (M = 6). Note in Fig. 6 that as we increase the scales sizes, thus addressing more information about pure intervals, we get more faithful identifications of the proper N-TETs for fitting the chromatic just scale with exception of the uniform case: The decreasing fidelity from M = 3 to M = 4 is legitimate because the uniform triad is not a subset of the uniform tetratonic. Figure 6 Comparison between fidelities for N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 when taking major, minor and uniform scales as target sets. Moreover, two observations suggest that the major and minor thirds are the most relevant ingredients for the convergence: (a) The main difference between major and minor scales of any size is the third interval, indicating that the extra major/minor sevenths are not crucial for the convergence, and (b) The uniform tetratonic, which produces the lower fidelity in Fig. 6, is the only uniform set which contains the minor third instead of the major one (see Tab. 2). The reason for a major thirds preference when setting a suitable equal-temperament, if not just a mathematical issue involving prime numbers, is quite mysterious but it seems to dialog with the fact that lower integer fractions correspond to the most appealing harmonies: The major and minor thirds are the nearest consonances from the tonic, moreover the 5/4 ratio is made of lower integers than the 6/5 ratio. The observation that strategic harmonic subsets of the chromatic universe can store sufficient information for setting proper equal-temperaments for the whole chromatic just scale indicates that extensions of the 5-limit just intonation to more than twelve tones, which exhibits notable difficulty in the construction of more and more fractions between powers of prime numbers, may not be necessary. Our results suggest that suitable fits for major scales are preferable in advance. Evidently, the average faithfulness will depend on the range of interest: Note in Fig. 5(a) that if we wish to look for suitable N-TETs within range N∈[9,42] then the set of pure intervals corresponding to a major pentatonic would be sufficient for finding the proper solutions, i.e. ℱ⁢(𝒫⁢m⁢a⁢j⁢o⁢r5)=1. ) within range $N \in [9, 54]$ when addressing as target sets the chromatic just scale, built only by semi-tone steps, and the whole-tone scale, built only by tone steps, yielding β¹² and $β_{u n i f}^{6}$ respectively. (b) Krenek diagram of the whole-tone scale.

Figure 5 Comparisons between values of (³ 3. Finding N-TETs Using M-element Sets of Pure Intervals 3.1. Method In this section, we built a method for finding which N-TETs, letting N be other than 12, suitably fit a given M-element subset of the chromatic just scale. Firstly, we set the error we are willing to take by addressing specific partitions of the semi-tone which are called commas. For example, in 12-TET each semi-tone has 4.5 commas, yielding a total of 54 such partitions per octave [15]. N-TETs have different densities of commas per interval, μ=μ⁢(N), while maintaining the total number of such partitions within the octave. We choose to use the minimum of those partitions as a measure of frequency resolution6: (2) Δ f ( μ ) = ( 2 1 N μ − 1 ) f 0 , which is slightly lower than the Pythagorean comma [14]. Throughout this paper, without loss of generality, f0 is chosen to be the frequency of the C4 which is approximately 261.63 Hz, so Δ⁢f⁢(μ)≈3.38 Hz. Note in Tab. 1 that some frequency differences between the 12-TET and just intonation are higher than (2) yielding out-of-tune notes, but it stands as a good fit in average. We now design a measure for quantifying the average performance of an N-TET in producing a certain target set of pure intervals. Consider an ordered set of frequencies corresponding to just intervals 𝒫M={𝒫kM}k=0M with M≤12, where 𝒫k+1M>𝒫kM⁢⁢∀k. Using definitions (1) and (2) we address the following quantity: (3) β M ( N ) = 1 Δ f ∑ k = 0 M min ℓ ≤ N ( ℱ ℓ N − ρ k M ) 2 M . Then if βM⁢(N)<1 frequency deviations are less than Δf in average, indicating that the corresponding N-TET properly fits the target set, whereas the (N-M) extra intervals are labelled as intermediate notes. Note that if ||ℱℓ+1N-𝒫kM|-|ℱℓN-𝒫kM||<Δ⁢f for a certain ℓ∈[0,N], an evident ambiguity would take place when labelling which frequency corresponds to the target interval and a potentially useful intermediate note would be lost. Therefore, we sum the two differences in (3) as a measure of a non-desirable ambiguity. For being an unique set, the chromatic just scale will be denoted by 𝒫12. With the intent of testing different target sets 𝒫M with M<12 we use subscripts as useful labels: Major and minor scales will be denoted by 𝒫⁢m⁢a⁢j⁢o⁢rM and 𝒫⁢m⁢i⁢n⁢o⁢rM respectively, whereas uniform scales will be denoted by 𝒫⁢u⁢n⁢i⁢fM. The corresponding parameter (3) of each addressed set will carry the same label (like β⁢m⁢a⁢j⁢o⁢rM and so on). The test whether βM⁢(N)<1 or βM⁢(N)>1 is read as it is a good fit or it is not a good fit respectively. For comparing within range N∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] the average similarity between the results when addressing 𝒫12 and 𝒫M as target sets, we design the following quantity: (4) ℱ ( ρ M ) = 1 − ∑ k | Θ ( β M ( k ) − 1 ) − Θ ( β 12 ( k ) − 1 ) | N max − N min , where k∈[N⁢m⁢i⁢n,N⁢m⁢a⁢x] and Θ is the Heaviside function. Equation (4) then computes the coincidences between βM⁢(N)<1 and β12⁢(N)<1, and returns a value between 0 and 1. Evidently, ℱ⁢(𝒫12)=1. For that, we call Eq. (4) fidelity. It allows us to compare which target sets most faithfully reproduce the results of the chromatic just scale. 3.2. Numerical results We start by addressing the simplest divisions of the chromatic universe, which constitute the chromatic just scale 𝒫12, built only by semi-tone steps, and the whole-tone scale 𝒫⁢u⁢n⁢i⁢f6, built only by tone steps (see Tab. 2). The corresponding values of parameter (3) are shown in Fig. 4(a) within range N∈[9,54]. By addressing the chromatic just scale as target set we find TETs with 12, 19, 22, 24, 31, 34, 41 and 53 intervals as suitable and non-ambiguous7. As for the parameter β⁢u⁢n⁢i⁢f6, corresponding to the uniform hexatonic (see Fig. 4(b)), the correct identifications of the proper N-TETs are not reproduced. This means that some configurations can reasonably adjust the whole-tone intervals, like the 18-TET for example, while making other intervals audibly out-of-tune. Figure 4 (a) Plots of Eq. (3) within range N∈[9,54] when addressing as target sets the chromatic just scale, built only by semi-tone steps, and the whole-tone scale, built only by tone steps, yielding β12 and β⁢u⁢n⁢i⁢f6 respectively. (b) Krenek diagram of the whole-tone scale. In Figs. 5(a) and 5(b) (Figs. 5(c) and 5(d)) we plot parameter (3) for major and minor pentatonic (heptatonic) scales, respectively (see Fig. 2). Note that, when compared with the results achieved by addressing the chromatic just scale, major scales present more faithful answers about suitable N-TETs than the minor counterparts; the similarity increases with the scales sizes as expected. Note that the 53-TET is the most accurate fit for all addressed subsets of the chromatic universe. Figure 5 Comparisons between values of (3) for the chromatic scale (gray triangles) and: (a) the major pentatonic (purple circles), (b) the minor pentatonic (orange circles), (c) the major heptatonic (blue circles) and (d) the natural minor heptatonic (red circles). We computed fidelities (4) within range N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 by taking major, minor and uniform scales of different sizes as targets (see Fig. 2 and Tab. 2). The corresponding plot in Fig. 6 shows the difference between the convergences for each harmonic set: Major scales provide more faithful answers about proper N-TETs than minor and uniform ones of corresponding sizes, the latter being the most disadvantageous reference set. For example, the fidelities achieved when addressing the major pentatonic (M = 5) and the minor heptatonic (M = 7) as targets are comparable, both standing above the fidelity achieved when addressing the whole-tone scale (M = 6). Note in Fig. 6 that as we increase the scales sizes, thus addressing more information about pure intervals, we get more faithful identifications of the proper N-TETs for fitting the chromatic just scale with exception of the uniform case: The decreasing fidelity from M = 3 to M = 4 is legitimate because the uniform triad is not a subset of the uniform tetratonic. Figure 6 Comparison between fidelities for N⁢m⁢i⁢n=9 and N⁢m⁢a⁢x=54 when taking major, minor and uniform scales as target sets. Moreover, two observations suggest that the major and minor thirds are the most relevant ingredients for the convergence: (a) The main difference between major and minor scales of any size is the third interval, indicating that the extra major/minor sevenths are not crucial for the convergence, and (b) The uniform tetratonic, which produces the lower fidelity in Fig. 6, is the only uniform set which contains the minor third instead of the major one (see Tab. 2). The reason for a major thirds preference when setting a suitable equal-temperament, if not just a mathematical issue involving prime numbers, is quite mysterious but it seems to dialog with the fact that lower integer fractions correspond to the most appealing harmonies: The major and minor thirds are the nearest consonances from the tonic, moreover the 5/4 ratio is made of lower integers than the 6/5 ratio. The observation that strategic harmonic subsets of the chromatic universe can store sufficient information for setting proper equal-temperaments for the whole chromatic just scale indicates that extensions of the 5-limit just intonation to more than twelve tones, which exhibits notable difficulty in the construction of more and more fractions between powers of prime numbers, may not be necessary. Our results suggest that suitable fits for major scales are preferable in advance. Evidently, the average faithfulness will depend on the range of interest: Note in Fig. 5(a) that if we wish to look for suitable N-TETs within range N∈[9,42] then the set of pure intervals corresponding to a major pentatonic would be sufficient for finding the proper solutions, i.e. ℱ⁢(𝒫⁢m⁢a⁢j⁢o⁢r5)=1. ) for the chromatic scale (gray triangles) and: (a) the major pentatonic (purple circles), (b) the minor pentatonic (orange circles), (c) the major heptatonic (blue circles) and (d) the natural minor heptatonic (red circles).

	Label	Just intonation	12-TET	Difference (Hz)
Unison	C	1	1	0
Minor Second	C#/D $♭$	16/15	$2^{1 / 12}$	$-$ 4.66
Major Second	D	9/8	$2^{2 / 12}$	+0.66
Minor Third	D#/E $♭$	6/5	$2^{3 / 12}$	+2.82
Major Third	E	5/4	$2^{4 / 12}$	$-$ 2.60
Fourth	F	4/3	$2^{5 / 12}$	$-$ 0.39
Diminished Fifth	F#/G $♭$	45/32	$2^{6 / 12}$	$-$ 2.08
Major Fifth	G	3/2	$2^{7 / 12}$	+0.44
Minor Sixth	G#/A $♭$	8/5	$2^{8 / 12}$	+3.30
Major Sixth	A	5/3	$2^{9 / 12}$	$-$ 3.96
Minor Seventh	A#/B $♭$	9/5	$2^{10 / 12}$	+4.76
Major Seventh	B	15/8	$2^{11 / 12}$	$-$ 3.34
Perfect octave	C	2	2	0

Size	3	4	6
Major	C, E, G	C, E, G, A	C, D, E, F, G, A
Minor	C, E $♭$ , G	C, E $♭$ , G, B $♭$	C, D, E $♭$ , F,G, B $♭$
Uniform	C, E, G#	C, E $♭$ , G $♭$ , A	C, D, E, G $♭$ , A $♭$ , B $♭$

[1] * Correspondence email address: lucasfelipe.bmo@gmail.com

Brasil

Brasil

Harmony as an underlying ingredient in the numerical search for suitable equal temperaments

Harmonia como um ingrediente subjacente na busca numérica por temperamentos iguais apropriados