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Page 1: The Complete Pythagorean C Major Scale

• Key Concept: Ratios are derived from the Circle of Fifths.

• Inclusion of F allows obtaining ratios for all notes in the C major scale.

• Note Representation:

  • C: 1 (base note)

  • Ratios for each note:

    • D: 9/8 --> This value is derived from the interval between C and D, which contributes a whole tone.

    • E: 81/64 from D to E (a whole tone)

    • F: 4/3 --> Next whole tone interval from E to F (perfect fourth)

    • G: 3/2 --> From F to G (perfect fifth)

    • A: 27/16 from G to A (whole tone)

    • B: 243/128 from A to B (whole tone)

• Notes and their ratios help understand the structure and intervals of the scale more clearly.

Page 2: Names of Notes in Different Octaves

• Distinctions in note names for varying octaves:

  1. Use of letters (e.g., CC to BB for lower octaves).

  2. C1, C2 for clarity across octaves.

• Example of Notation:

  • C2, C3, C4, etc., indicate increasing octaves, facilitating easier identification.

Page 3: Tones and Semitones in the Pythagorean Scale

• Construction relies solely on intervals of a fifth, revealing:

  • Whole tones have a ratio of 9/8.

  • Semitones represented as 256/243.

• Notable feature: The Pythagorean semitone refers to a different mathematical relationship than modern pianos.

Page 4: The First Five Harmonics of a Vibrating String

• Fundamental frequency described as the first harmonic (f).

• Higher harmonics follow:

  • Second Harmonic: 2f (e.g., 528 Hz, C5).

  • Third harmonic: 3f (e.g., 792 Hz, G5).

• Relationships between harmonics create the full range of sounds through the vibrating string.

Page 5: The Fifth Harmonic of a String

• Definition: Touching a vibrating string at one-fifth point divides the string into five equal sections.

• Result: Each section vibrates at a frequency five times that of the original frequency.

• Visualization of nodes and anti-nodes helps understand the harmonic structure.

Page 6: Deriving the Interval of a Third from the Fifth Harmonic

• Calculating frequencies from the fifth harmonic (1320 Hz) derives alternative notes.

• E described as 330 Hz, showcasing an alternative designation indicating the Just third.

• Ratios: 81/64 emphasize frequency relationships

Page 7: Using Both the Fifth and Third to Build the Just Scale

• Concept of constructing Just scale utilizing simple ratios from fundamental intervals:

  • Thirds derived from varying harmonics create true thirds.

  • Relationship of fifths and thirds clarifies Just scale complexity versus Pythagorean origins.

Page 8: The Intervals of the Sixth and Seventh in the Just Scale

• Intervals created by multiplying frequencies:

  • F to G (7:8 ratio).

  • Logical extensions derive notes like A and B from initial constructs of C.

Page 9: Intervals in the Complete C Major Just Scale

• Each note derived from middle C through simple multiplicands with clean ratios:

  • Example notes from C major scale, understanding the purity of relationships.

Page 10: Just Scale and Intonation

• The Just scale utilizes octave, third, and fifth, highlighting proportional intervals derived from physical properties of strings.

• Tuning based on these properties leads to pure intonation perceived by some critics and theorists.

Page 11: Problems with Just Intonation

• Challenges in Just intonation arise with intervals not yielding expected harmonious ratios:

  • Example: D to A interval does not fulfill expectations.

Page 12: Two Kinds of Whole Tone

• Distinct whole tone intervals result from different sections of the scale, yielding contrasting ratios further complicating Just scale dynamics.

Page 13: Comparison of the Just and Pythagorean Scales

• Distinctions emphasizing simple ratios in Just scale versus the more complex fifths and imperfect thirds in Pythagorean scaling.

Page 14: Frequencies in the Just and Pythagorean Scales

• Frequencies in Just scale yield whole number outcomes based on simple multiplicative ratios in comparison to Pythagorean scale's irrational intervals for certain notes.

Page 15: Fifths and Thirds in the Pythagorean Scale

• Analysis reveals that fifths meet exact ratios while thirds skew higher due to structure derivation from fifths.

Page 16: The Pythagorean Third as Four Fifths

• The concept of stacking fifths to create the third, illustrating the fundamental relationships within the scale and harmonic structures.

Page 17: Reducing the Pythagorean Third

• Adjusting the E to generate a Just third, demonstrating the manipulation of harmonic ratios to achieve an intended outcome.

Page 18: Reducing the Pythagorean Third (Continued)

• Further breakdown of ratios manipulating frequency distributions to refine harmonic relationships.

Page 19: The Fourth as the Inversion of the Fifth

• Exploring the mathematical relationship between fourths and fifths, reinforcing their emotional and musical roles.

Page 20: Major Third and Minor Third

• Introduces distinctions between major and minor thirds through frequency ratios compelling usage.

Page 21: The Triad Chord

• Defines triad structures based around specific notes within scales, deepening harmonic understanding of music composition.

Page 22: Ratios of the Triad Chord

• Ratios describing the triad provide insight into harmonic relationships, enhancing foundational music theory.