The article investigates an extension of the theory of well-formed modes and proposes a model of the major and minor modes in harmonic tonality. The established theory of well-formed modes is well adapted to the description of the medieval diatonic modes. Its core is the conversion of the circle-of-fifths encoding into the circle-of-steps encoding of the seven generic scale degrees. This conversion is a linear automorphisms of the cyclic group of order 7 (known from well-formed scale theory). This automorphism can be lifted to three refined levels of description: (1) to the diatonic Regener transformation on the two-dimensional note-interval system and (2) to a conjugacy class of Sturmian morphisms, corresponding to the filling of the authentically (or plagally) divided octave by the modal species of the fifth and the fourth and (3) to associated lattice path transformations on chains of anchored note intervals. In the present paper the place of the authentic division of the octave into a perfect fifth and a perfect fourth is taken by a triadic division of the octave into a major third, a minor third, and a perfect fourth. The three triadic intervals are filled with patterns formed by three different step intervals. In the consequence, all the above-mentioned constructions on two-dimensional note-interval system have to be replaced by analogous constructions which are based on a three-dimensional system of Euler-notes and intervals. The resulting constructions are mathematically inspired by Arnoux and Ito [2001. “Pisot Substitutions and Rauzy Fractals.” Bulletin of the Belgian Mathematical Society Simon Stevin 8 (2): 181–207] and include the lifting of the diatonic Euler-Regener-transformation to substitutions on words in three letters, their geometrical action as lattice-path transformations and the study of their duals. A constitutive innovation is the music-theoretical interpretation of the eigen-values, eigen-vectors and eigen-co-vectors of the Euler-Regener-transformation and its dual. A modified instance of the classical Euler-Oettingen-Riemann tone net can be defined on the Handschin plane, the kernel of the linear eigen-height form. Finally, comparing the image under the dual lattice-path transformations of the dual of the step-interval trihedron anchored at the Euler-note C₄ and of the dual of the step-interval anti-trihedron targeting the Euler-note C₄, a dynamical system arises on a region of the continuous Handschin plane.

本文研究了格式良好的理论的扩展，并提出了谐波调性中主要和次要模式的模型。建立良好的模式理论很适合于中世纪全音阶模式的描述。其核心是将五分之一圈的编码转换为七个通用比例度的步长圈编码。此转换是7阶循环组的线性自同构（从格式良好的标度理论中知道）。这种自同构可以提升为三个精细的描述级别：（1）二维音符间隔系统上的等张Regener变换，以及（2）Sturmian态的共轭类，对应于用第五和第四和（3）的模态种类填充经真实划分（或后部划分）的八度音阶，以完成锚定音符区间链上相关的晶格路径转换。在本文中，将八度音阶的真实划分为完美的五分之一和完美的四度的位置由八度音阶的三阶划分为大三分，小三分和完美四分。三个三元间隔填充有由三个不同步长间隔形成的图案。结果，必须将上述所有关于二维音符间隔系统的上述构造替换为基于欧拉音符和音符的三维系统的类似构造。由此产生的构造在数学上受到Arnoux和Ito [2001年的启发。比利时数学学会简报，Simon Stevin 8（2）：181–207]，包括将全音阶Euler-Regener变换解除为三个字母中的单词替换，它们作为晶格路径变换的几何作用及其研究双打。本构性的创新是对Euler-Regener变换及其对偶变换的本征值，本征向量和本征余向量的音乐理论解释。可以在Handchin平面上定义经典Euler-Oettingen-Riemann音网的修改实例，Handschin平面是线性本征高度形式的内核。最后，比较锚定在Euler-note C ₄上的阶梯间隔三面体对偶的双晶格路径变换下的图像在针对Euler-note C ₄的阶梯间隔反三面体的对偶中，动力学系统出现在连续的Handchin平面的区域上。