An alternative approach to generalized Pythagorean scales. Generation and properties derived in the frequency domain

Abstract

Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The refinement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales avoiding continued fractions. New proofs of the main properties are obtained as a consequence of the generating procedure. When the scale tones are generated from the two elementary factors associated with the generic widths of the step intervals we get the partition of the octave leading to the fundamental Bézout's identity relating several characteristic scale indices. This relationship is generalized to prove a new relationship expressing the partition that the frequency ratios associated with the two sizes composing the different step-intervals induce to a specific set of octaves.

Keywords:

• well-formed scales
• equal temperament
• Pythagorean tuning
• Pythagorean comma
• Bézout's identity
• continued fractions

2010 Mathematics Subject Classification:

• 05C12
• 11A05
• 54E35

Acknowledgments

I would like to thank the Editors Emmanuel Amiot, Jason Yust and, in particular, Thomas Fiore and Clifton Callender for valuable remarks and comments. Clifton Callender pointed out the existence of a similar algorithm (Carey 1998) for obtaining well-formed scales. I would also like to thank several anonymous referees for their thorough revision of this paper or earlier versions thereof, as well as pointing out crosslinks, helpful discussions, hints, and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This is an instance of a maximally even set, where each generic interval is either a single integer or two consecutive integers (Clough and Douthett 1991).

2 In a more formal context, we consider a GIS (Lewin 1987; Kolman 2004) $\pi :\mathrm{\Omega }\times \mathrm{\Omega }\to {\mathrm{\Omega }}_0$ and a canonical GIS $\pi :{\mathrm{\Omega }}_0\times {\mathrm{\Omega }}_0\to {\mathrm{\Omega }}_0$, with the α-projection function defined as ${\pi }_{\alpha }(\cdot )\equiv \pi (\alpha ,\cdot )$.

3 Two FCs α and β other than 1 are complementary if they satisfy $\alpha \text{β}=2$. This definition is not valid for the FC 1.

4 For any $x\in \mathbb{R}$ and any positive $N\in \mathbb{Z}$, there exists integers p and q such that $1\le q\le N$ and $|qx-p|<1/N$.

5 The shortest distance between two FCs α, β is measured in the unit circle $S_0$ as $d(\alpha ,\text{β})=min(|{\log }_2(\alpha /\text{β})|,1-|{\log }_2(\alpha /\text{β}\left)|\right)$. For a FC ${\pi }_{\alpha }^k\in (1,2)$ we have $d({\pi }_{\alpha }^k,1)=min({\log }_2{\pi }_{\alpha }^k,1-{\log }_2{\pi }_{\alpha }^k)$, while for its complementary FC, ${\pi }_{\alpha }^{-k}=2/{\pi }_{\alpha }^k$, we also have $d({\pi }_{\alpha }^{-k},1)=min(1-{\log }_2{\pi }_{\alpha }^k,{\log }_2{\pi }_{\alpha }^k)$. Thus, $d({\pi }_{\alpha }^k,1)=d({\pi }_{\alpha }^{-k},1)$.

6 In our approach, the scale remains composed of tones corresponding to positive powers of 3, and the 12-tone Pythagorean scale is formed by successive fifths of 1; while for these authors a natural scale is also composed by tones expressed as negative powers of 3, so that the Pythagorean scale is formed by successive fifths of $\frac{256}{243}$.

7 It is worth remembering that ${\mathbb{Z}}_n$ has the same structure of commutative ring than the integers for the addition and product. In the ring ${\mathbb{Z}}_n$ of integers modulo n, 1 is the neutral element for the product. The invertible elements form the multiplicative group of units $U\left({\mathbb{Z}}_n\right)$. Therefore, $a\in {\mathbb{Z}}_n$ has an inverse if and only if there exists $b\in {\mathbb{Z}}_n$ such that ab = 1, hence $a,b\in U\left({\mathbb{Z}}_n\right)$. Similarly, two integers $x,y\in \mathbb{Z}$, such that $xy\equiv 1\left(\mathrm{m}\mathrm{o}\mathrm{d}n\right)$, belong to two classes of $U\left({\mathbb{Z}}_n\right)$ that are mutually inverse, or belong to the class of 1. This condition can be written using Bézout's identity. This leads to the property that the generators of $({\mathbb{Z}}_n,+)$, which are coprime with n, are just the elements of $U\left({\mathbb{Z}}_n\right)$.

8 If $x,y\in \mathbb{R}$ we have $\lfloor x\rfloor +\lfloor y\rfloor \le \lfloor x+y\rfloor \le \lfloor x\rfloor +\lfloor y\rfloor +1$. If $h\in (1,2)$, then $\text{⟦}1\text{⟧}=\lfloor {\log }_{2}h\rfloor =0$ and the value $\text{⟦}p+1\text{⟧}$ matches either $\text{⟦}p\text{⟧}$ or $\text{⟦}p\text{⟧}+1$.

9 The Stern-Brocot tree is a full binary tree where the nodes are labeled in such a way that each positive rational number occurs exactly once. Vertically, it provides the usual ordering of the rationals. For $p_1/q_1$ and $p_2/q_2$, the mediant is the fraction $\left(p_1+p_2\right)/\left(q_1+q_2\right)$ whose parents they are. Every row consists of the fractions that are mediants of elements of previous rows. Positive irrational numbers can be associated with a unique infinite pathway down the tree and the nodes which are passed by on such a finite or infinite path are called semi-convergents of the corresponding rational or irrational number.

10 The scale closure is an intrinsic parameter of the scale. However, comparing between scales according to the closeness of the closure to 1 (its logarithm closer to 0) is actually equivalent to evaluating the best comma as defined in Douthett and Krantz (2007).

11 It will be notated as $¢\left({\gamma }_n\right)=1200{\log }_2{\gamma }_n$, corresponding to the width of the spectrum associated with the frequency ratios $2/{\nu }_M$ and ${\nu }_m$ (e.g. Douthett and Krantz 2007).

12 Since $\frac{1}{\sqrt{2}}<{\gamma }_n<\sqrt{2}$, then $-\frac{1}{2}<n{\log }_{2}h-N<\frac{1}{2}$, and $N<n{\log }_{2}h+\frac{1}{2}<N+1$. Hence, by taking integer parts, we get $N\le \lfloor n{\log }_{2}h+\frac{1}{2}\rfloor <N+1$, from where $N=\lfloor n{\log }_{2}h+\frac{1}{2}\rfloor$. Usually, the chromatic length N is defined in this way. However, in the current paper such a relationship is not of particular interest. Instead, we are interested in the dependency of the scale properties on the indices m, M, and their respective octaves.

13 We know that, if k<n then $\text{⟦}k\text{⟧}\le \text{⟦}n\text{⟧}$. Nevertheless, if 1<h<2, in case (i) we have ${\nu }_n<{\nu }_{n-1}$ and ${\nu }_n=\left(h/2\right){\nu }_{n-1}$, therefore $\text{⟦}n\text{⟧}=\text{⟦}n-1\text{⟧}+1$; hence, $\text{⟦}k\text{⟧}\le \text{⟦}n-1\text{⟧}=\text{⟦}n\text{⟧}-1$. Instead, in case (ii), ${\nu }_{n-1}<{\nu }_n$ and $h{\nu }_{n-1}={\nu }_n$, therefore $\text{⟦}n-1\text{⟧}=\text{⟦}n\text{⟧}$; hence, $\text{⟦}k\text{⟧}\le \text{⟦}n-1\text{⟧}=\text{⟦}n\text{⟧}$. Thus, in both cases, $\text{⟦}k\text{⟧}\le N-1$ for $0\le k\le n-1$.

14 According to equations (30(30) ${\nu }_{-k}=\frac{2}{{\nu }_k}={\pi }_h^{-k}={\pi }_{\frac{1}{h}}^k,k\in \mathbb{Z}$(30) ) and (32(32) ${\nu }_{n-k}{\nu }_k=2{\gamma }_n;k=1,\dots ,n-1.$(32) ), the complementary tones of the scale (other than the fundamental) are given by ${\nu }_{-k}={\gamma }_n^{-1}{\nu }_{n-k}$, which do not match the tones of the scale $E_n^h$. The inverse scale $E_n^{1/h}$, built by starting at the fundamental by negative iterations of the h-interval, does not provide the same scale as $E_n^h$. However, since the tones of $E_n^h$ and $E_n^{1/h}$ differ by excess or by default in one comma, they belong to the same classes of projection functions. It happened in a similar way with Hellegouarch's (1999) natural scales, that were a particular choice of representatives of the abstract scale ${\overline{E}}_n^h$.

15 The terms “good” and “best” approximation, used in recent books such as Loya (2017), are equivalent to best approximation “of the first kind” and “of the second kind”, respectively, used in Khinchin (1964).

16 For the cyclic scale $E_n^h$, with extreme tones of indices m and M, the value n provides a convergent or semi-convergent $N/n$ of ${\log }_{2}h$ according to one of these situations: $\text{⟦}m\text{⟧}/m<N/n<{\log }_{2}h<\left(\text{⟦}M\text{⟧}+1\right)/M$ or $\text{⟦}m\text{⟧}/m<{\log }_{2}h<N/n<\left(\text{⟦}M\text{⟧}+1\right)/M$, depending on whether ${\nu }_n$ is a new minimum or maximum of the sequence $S_n^h$. We shall see in Section 6.2 that equations (45(45) $\begin{array}{rl}& Nm-n\text{⟦}m\text{⟧}=1\end{array}$(45) ) and (46(46) $\begin{array}{rl}& n(\text{⟦}M\text{⟧}+1)-NM=1\end{array}$(46) ) are satisfied. Therefore, we meet a situation such as $a^{\prime }/a<{\log }_{2}h<b^{\prime }/b$, together with the Bézout's identity corresponding to pairs of coprime numbers, $a{b}^{\prime }-b{a}^{\prime }=1$, which leads to a new improvement $\left(a^{\prime }+b^{\prime }\right)/\left(a+b\right)$ of the approximation. This situation is common to the continued fractions approach, the Farey sums, the structure of the Stern-Brocot tree, and its dual, the Raney tree (Berstel and de Luca 1997; Raney 1973), also known as Calkin-Wilf tree (Calkin and Wilf 2000; Gibbons, Lester, and Bird 2006). Thus, for two consecutive approximations, by using the extended Euclidean algorithm (Appendix 4) it is possible from one to determine the other.

17 According to equation (37(37) $\left(\begin{array}{cc}{m}^{+}& M^{+}\\ \text{⟦}{m}^{+}\text{⟧}& \text{⟦}{M}^{+}\text{⟧}+1\end{array}\right)=\left(\begin{array}{cc}m& M\\ \text{⟦}m\text{⟧}& \text{⟦}M\text{⟧}+1\end{array}\right)\left(\begin{array}{cc}1& \delta \\ 1-\delta & 1\end{array}\right).$(37) ), the subsequent matrices of indices are obtained multiplying by a matrix in one of the forms $L=\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$ or $R=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$. These are equivalent to the $2\times 2$ matrices representing the branches of the Stern-Brocot tree (Noll 2006, 2007; Jedrzejewski 2009, 2008), providing the strings of L's and R's that encode the subsequent nodes of the fractions $N/n$. This is equivalent to the retrograde Euclidean algorithm along the path to the root from that node in the Raney tree. On the other hand, the fractions $m/M$ follow a similar path in the Raney tree, with the same strings as $N/n$.

18 One consequence of the fundamental identity is that all the following pairs are coprime: $(m,M)$; $(\text{⟦}M\text{⟧}+1,M)$; $(\text{⟦}M\text{⟧}+1,\text{⟦}m\text{⟧})$; $(m,\text{⟦}m\text{⟧})$; $(m,n)$; $(M,n)$; $(n,N)$; $(\text{⟦}m\text{⟧},N)$; $(\text{⟦}M\text{⟧}+1,N)$.

19 The procedure to determine cyclic scales and their refinements is closely related to the concept of mechanical or Sturmian words used in the new approaches to the theory of well-formed scales and modes (Noll 2008; Clampitt and Noll 2011; Noll 2015) based on methods of combinatorics on words (e.g. Lothaire 2002). A cyclic scale can be defined as a Christoffel word of the alphabet $\{U,D\}$ with slope $m/M$ and length n. For cyclic scales, the first step of the octave after 1 must be U, and the last step before 2 must be D, although generic scales do not need to satisfy such a requirement. Given a scale whose factors satisfy $U^M{D}^m=2$, if U<D then, owing to Myhill's property, it can be refined by factorizing $D=U{D}^{\prime }$, otherwise by factorizing $U=U^{\prime }D$, and so on. The refinement of cyclic scales correspond to the binary tree of Christoffel words.

20 For $i\ge 2$, these variables are equivalent to duplicating equation (A5(A5) $d_{i+1}\equiv r_i=d_{i-1}-q_i{d}_i,i\ge 1$(A5) ), dividing it by a and b, and defining $x_i=d_i/a$, $y_i=d_i/b$. As $x_1=y_0=0$, the new variables are integers.