TRIANGLES – II

The Tonnetz lattice is a fascinating musical concept and visual representation of harmonic relationships. It is a two-dimensional grid that helps illustrate the interconnections between musical chords and their relations to one another.

With its geometric layout and intricate network of lines, the Tonnetz lattice provides a unique perspective on harmony and offers valuable insights into the structure of music. Whether you are a musician, music theorist, or simply curious about the intricate workings of chords and harmony, exploring the Tonnetz lattice can be an enriching and captivating endeavor.

The Tonnetz lattice was originally developed by Leonhard Euler, a Swiss mathematician and physicist, as a conceptual diagram representing tonal space. This grid-like structure was a way to show the relationships between musical pitches. It was later adopted and extensively used by German music theorist Hugo Riemann in the 19th century.

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Tonal music, also known as harmony-based music, refers to a genre that utilizes a specific pitch center or tonal system as its foundation. In tonal music, melodic and harmonic elements interact to create a sense of tonal stability and resolution. This traditional form of music has been widely embraced and cherished for its ability to evoke emotions and engage listeners on a profound level. Whether it’s the rich harmonies of a symphony or the catchy melodies of a popular song, tonal music continues to be a beloved and significant part of our musical heritage.,

In tonal music, certain pitch combinations and progressions are perceived as more stable or harmonious than others. These relationships correspond to the geometric patterns formed by notes on the Tonnetz lattice. The lattice can help visualize and understand these harmonic relationships.

Here’s how it works:

1. Pitch Relationships: Each point on the grid represents a musical pitch. Adjacent points correspond to specific musical intervals. For example, moving horizontally might correspond to a perfect fifth, moving diagonally up-right might correspond to a major third, and so on.
   
2. Chord Shapes: Chords appear as shapes on the Tonnetz lattice. For instance, major and minor triads appear as equilateral triangles. This can help visualize how different chords are related to each other.
   
3. Chord Progressions: Transformations or movements on the lattice can represent chord progressions in a piece of music. For example, moving along a straight line might correspond to a sequence of perfect fifths (a common progression in tonal music).
   
4. Modulation and Key Relationships: The Tonnetz lattice can also help visualize key relationships and modulation (changing from one key to another). Closely related keys are located near each other on the lattice.
   
5. Isochords: Isochords are chords that share the same shape on the lattice, regardless of their root note or inversion. This concept can be useful for understanding the deep structure of tonal music.

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Isochords are a concept in music theory related to the The Tonnetz lattice. They refer to chords that share the same shape on the Tonnetz lattice, regardless of their root note or inversion.

In the context of the Tonnetz lattice, an isochord is a set of pitches that forms the same geometric pattern, regardless of where it is located on the lattice. For instance, all major and minor triads are isochords of each other because they form the same triangular pattern on the Tonnetz lattice, even though they consist of different pitches.

This is a fairly abstract and mathematical way of thinking about music. However, it can be useful for understanding the deep structure of tonal music and the relationships between different chords.

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In his book “Music: Its Secret Influence throughout the Ages”, Cyril Scott delves into the profound effect of music on human moods and societal values. He cites prominent passages from Plato’s “Republic” to underline his points. According to Plato, music has a powerful influence on a person’s mood and can even shape societal norms and values.

Plato, in his “Republic”, discusses the significance of music in shaping an ideal society. He believed that music is decisive because of its influence on one’s sense of beauty. Furthermore, Plato asserted that musical training is a potent tool as rhythm and harmony find their way into the innermost soul and take strongest hold upon it.

Scott and Plato both acknowledge the expressive character of music. They believed that it resembles certain aspects of the inner life and points to higher, spiritual realities. They also recognized the potential dangers of certain musical scales, indicating that not all music is beneficial or uplifting.

Cyril Scott expressed a negative view towards jazz music, associating it with a decline in sexual morals and self-control. He attributes the spread of jazz to the workings of what he calls “Dark Forces” and accuses the genre of promoting an overemphasis on sexual nature and erotic adventures.

Scott argues that the syncopated rhythm of jazz, devoid of any higher musical content, creates a hyper-excitement of the nerves that weakens self-control. This, he believes, leads to a false exhilaration, a deceptive endurance, and an insatiable desire resulting in damaging moral and physical reactions.

However, Scott was probably referring to the intemperate fast and hard syncopated beat “devoid of any higher musical content,” and not to the “cool jazz” characterized by relaxed tempos and lighter tone. Jazz has evolved into a highly sophisticated form of musical expression that can actually have an uplifting and stimulating effect on its listeners. The smooth, soulful sound of cool jazz, with its complex improvisations, can evoke a feeling of relaxation and contentment in the mind and spirit. It is also capable of conveying feelings of joy, love, sadness or mystery.

Clearly, Scott’s perspective on jazz contrasts sharply with his views on deva-inspired or buddhic music, which he sees as spiritually uplifting and transformative. Buddhi is a Sanskrit term meaning “faculty of wisdom.” This type of music is intended to evoke feelings of oneness and spiritual enlightenment, further highlighting music’s potential as a tool for spiritual growth and understanding.

Cyril Scott also discusses the role of ultra-discordant music in the early 20th century. He suggests that this kind of music was necessary to destroy certain undesirable thought-forms that had persisted since the Middle Ages and were responsible for instances of cruelty, such as the actions of the inquisitors.

Scott argues that only discordant music could touch these coarse vibrations because beautiful music, with its refined vibrations, could not reach them, much like a stagnant pool of slime cannot be affected by the soft light of dawn.

He attributes the task of creating this discordant music to composers like Stravinsky and Schönberg. Despite the jarring nature of their compositions, Scott believes they served an essential purpose in breaking down conventionality and making people more receptive to new ideas.

However, Scott also critiques the enduring influence of ultra-discordant music. He argues that while it was necessary at one point, many composers continue to produce it simply because it has become fashionable, even dismissing non-ultra-discordant contemporary music as “romantic”—a term used derogatorily.

Scott believes that discords are essential to high art, but only an inspired composer knows how to resolve them properly. He criticizes the cacophonous school as a perversion of Deva-music, stating it would be true Deva-music if the composers knew how to resolve their discords in the Devic way.

Lastly, Scott asserts that musical fashions come and go, and it’s those composers who resist them whose works endure, citing Brahms as an example. Accordingly, he states that the Hierarchy has indicated that the ultra-dissonant phase of music will not last, nor was it ever intended to.

Cyril Scott’s writings reveal a complex and nuanced understanding of the power of music to influence human behavior and societal norms.

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Therefore, music may be defined as the art and science of bringing harmony (triangles of consonance) through conflict (squares of dissonance). In doing so, music is assisting in “the effort on Earth today (as seen by the planetary Logos) … to bring about a transformation of the web of the planet and thus slowly change the existing squares into triangles” (Esoteric Astrology by Alice A. Bailey, p. 480). This planetary web is the foundation upon which a future hierarchical democracy will be built.

Dmitri Tymoczko, a renowned music theorist and composer, has contributed significantly to our understanding of music structure. He suggests that tonal music, which is the dominant musical style of Western music from 1600 to 1900, is both self-similar and hierarchical.

1. Self-Similarity: Tymoczko refers to the concept that chord progressions in tonal music often use efficient voice leading to link structurally similar chords. Voice leading is the process of writing smooth melodic lines (or voices) to form harmonies (chords). When chords are structurally similar, they share common tones or have tones that are near each other in pitch, making the voice leading more efficient and smooth. This creates a sense of continuity and coherence in the music.
   
2. Hierarchy: In addition to this, Tymoczko also suggests that modulations in tonal music often use efficient voice leading to link structurally similar scales. Modulation is the process of changing key within a piece of music. When keys are structurally similar, they share many common pitches, making the transition between them smoother. This creates a hierarchical structure in the music, as some keys (and the scales associated with them) can be seen as more closely related than others.
   

These two principles combine to create the rich, complex sound of tonal music, with its characteristic tension and resolution, its sense of movement and rest. Accordingly, these are fundamental aspects of how tonal music is structured and understood (Tymoczko, Dmitri. “The Generalized Tonnetz“. Journal of Music Theory 56, no. 1 (2012): 1-66).

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Thus, the concept of a hierarchical democracy can be enriched by considering correspondences based on the “right relations” between consonance and dissonance in tonal music, as revealed by the “musical science of triangles” portrayed in the Tonnetz grid, and enhanced by Cyril Scott’s hierarchy of musical expressions, and Tymoczko’s concept of hierarchical structures in music.

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https://hierarchicaldemocracy.wordpress.com/2023/08/27/triangles/

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