SOM7w17
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:
Use of letters (e.g., CC to BB for lower octaves).
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.