On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation

Abstract

For well-formed generalized Pythagorean scales, it is explained how to fill in a bidimensional table, referred to as a scale keyboard, to represent the scale tones, arranged bidimensionally as iterates and cardinals, together with the elementary intervals between them. In the keyboard, generalized diatonic and chromatic intervals are easily identified. Two factor decompositions of the scale tones, which are particular cases of duality, make evident several properties of the sequence of intervals composing the octave, such as the number of repeated adjacent intervals and the composition of the generic step-intervals. The keyboard is associated with two matrix forms. When they are mutually transposed, the keyboard is reversible, as in the 12-tone Pythagorean scale. In this case, the relationship between the two main factor decompositions is given by an involutory matrix.

Keywords:

• generated scale
• closure condition
• Pythagorean tuning
• generated tone system
• well-formed scales
• scale temperament
• generalized diatonic scales
• Myhill's property

2010 Mathematics Subject Classification:

• 05C12
• 11A05
• 54E35

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Scale tone refers to a scale note in the frequency domain $[1,2)$, i.e. the representative of a frequency class (FC) in the octave, while in the ${\log }_2$ space it is a pitch class (PC) in $[0,1)$.

2 We will also refer to one iteration as a fifth, since it corresponds to the perfect fifth musical interval, 5 being the number of notes of the diatonic scale comprising the interval.

3 According to this article, cyclic scales are well-formed scales that can be obtained for any real positive value h by defining the octaves of the fundamental frequency ratio 1 from the monogenous group $\{{\omega }^k,k\in \mathbb{Z}\}$, for any real number $\omega >1$, and not only for $\omega =2$. This is a particular subset of the cyclic scales as defined in Jedrzejewski (2006, 169–172), which are not restricted to well-formed scales. For well-formed scales, the partition of the octave induced by the scale notes has exactly two sizes of scale step and each number of generic intervals occurs in two different sizes, while the other scales have three sizes of scale step and generic interval. In both cases, the generator h can be assigned to specific real numbers, such as the Golden Ratio, e, π, the Euler constant, etc., as Jedrzejewski explains.

4 It can also be expressed as $N=\text{〚}n-k\text{〛}+\text{〚}k\text{〛}+1$, for 0<k<n.

5 This equation is equivalent to the following ones, with n, N coprime: (7) $Nm-n\left(\text{〚}m\text{〛}\right)=1,n(\text{〚}M\text{〛}+1)-NM=1.$(7)

6 If the generator is an FC, i.e. it satisfies 1<h<2, then $\text{〚}1\text{〛}=\lfloor {\log }_{2}h\rfloor =0$ and N is just the cardinal µ of the generator.

7 Since m, M are coprime, we may recall Bézout's lemma stating that there are two integers a and b that satisfy 1 = ma−Mb. This equation admits infinitely many pairs of solutions, although there are two solutions satisfying $|a|<|M|$ and $|b|<|m|$. Furthermore, since m and M are positive, there exists a unique couple of values $(a,b)$ satisfying 0<a<M, $0\le b<m$. The three pairs $(a,b)$, $(a,m)$, $(M,b)$ are also coprime numbers. The values for $(a,b)$ are those given in equation (8(8) $1=(\text{〚}M\text{〛}+1)m-\text{〚}m\text{〛}M.$(8) ). If we multiply the above equation by an integer k>0 and write p = ka, q = kb, we get the Diophantine equation k = mp−Mq, corresponding to the first equation in equation (20(20) $\left(\begin{array}{c}k\\ \text{〚}k\text{〛}\end{array}\right)=\left(\begin{array}{cc}m& -M\\ \text{〚}m\text{〛}& -(\text{〚}M\text{〛}+1)\end{array}\right)\left(\begin{array}{c}p\\ q\end{array}\right)$(20) ). Then, there only exists one pair of values $(p,q)$ satisfying $0<p\le M,0\le q<m$, although such an equation is also valid for the value k = 0 in the trivial case p = q = 0. It is possible to determine directly the values $(p,q)$ by using the appropriate algorithm by working only with the first component in equation (20(20) $\left(\begin{array}{c}k\\ \text{〚}k\text{〛}\end{array}\right)=\left(\begin{array}{cc}m& -M\\ \text{〚}m\text{〛}& -(\text{〚}M\text{〛}+1)\end{array}\right)\left(\begin{array}{c}p\\ q\end{array}\right)$(20) ); however, the linear system provides them directly.

8 From the point of view of the algebraic combinatorics of words, equation (38(38) ${\mathrm{\Phi }}^{+}(p,q)=\{\begin{array}{ll}\mathrm{\Phi }(p+1,q)={\nu }_{k+m};& \mathrm{i}\mathrm{f}p<{\displaystyle \frac{M}{m}}(q+1),k<M\\ \mathrm{\Phi }(p,q+1)={\nu }_{k-M};& \mathrm{i}\mathrm{f}p\ge {\displaystyle \frac{M}{m}}(q+1),k\ge M.\end{array}$(38) ) defines the scale as a Christoffel word of the alphabet $\{U,D\}$ with slope $\frac{m}{M}$ and length n – see for example Noll (2008). The letter in the position j + 1 is U when the h-iteration increases from k to k + m, according to the case k<M; hence, owing to equation (35(35) $k=jmmodn,0\le j<n.$(35) ), $jmmodn<(j+1)mmodn$. Otherwise, the letter in the position j + 1 is D when the h-iteration decreases from k to k−M, according to the case $k\ge M$; therefore $jmmodn>(j+1)mmodn$. This matches the definition of a Christoffel word.

9 The cyclic scales with n = 17, 29, 41, 53 have the same index m and factor U; therefore, they are also represented in this keyboard if the respective rows are eliminated, beginning at the bottom. Then, the last factor should accumulate the ones removed, namely $D^{\prime }=UD$ for n = 41, $D^{\prime }=UUD$ for n = 29, and $D^{\prime }=UUUD$ for n = 17.