SHOSYS ACADEMY 9.1 LESSON: Conventions Of Labeling Intervals
Kelvin Sholar
1 Introduction To The Blog Series
This series of lessons and tests incorporates an easy music appreciation curriculum for adult beginners who are remote learning, or are self-taught. Lessons are posted on Mondays while Tests are posted on Saturdays. For more in depth and private guidance, I offer personal instruction by Zoom (Personal Meeting ID 8522954569) – for 1 dollar a minute. Time schedules range from a minimum of 30 minutes to a maximum of 60 minutes. Email me at [email protected] to set up personal instruction. I accept payments and cash gifts by Cash App ($KelvinSholar), Zelle ([email protected]) or Paypal (paypal.me/kelvinsholar).
2 Revisiting The Tree Of Knowledge
In Lesson 8, we learned about conventions of tone labeling. In this Lesson, we will learn about conventions of labeling degrees and intervals. This knowledge resides in the Ways branch (1.20) of the Tree of Knowledge (1.00), at the third leaf from the left (1.21) – Conventions.
2.1 Conventions Of Labeling Degrees
Sonology is the study of tonal structures in and of themselves, and the study of how tonal structures interact in time; while harmony is the study of simultaneous tonal structures and melody is the study of sequential tonal structures. When we study how tonal structures interact in time, we are partially concerned with rhythm because rhythm is they study of structures in time.
Conventions of labeling harmonies began with labeling tones, which was the subject of lesson 8. However, here we will define yet another way that tones were labeled with numerals in early Western music. Traditionally, (in England and America), the first seven ascending tones of the A minor scale are represented by the first seven letters of the Roman alphabet: A B C D E F G. Tones were numbered with Roman numerals (called degrees), in order to label them as ordered positions of a scale, in a given direction (Loy 16). For example, the seven tones of the ascending A minor scale are labeled as ordered positions in an upwards direction:
- A=I=Tonic
- B=II=Supertonic
- C=III=Mediant
- D=IV=Subdominant
- E=V=Dominant
- F=VI=Submediant
- G=VII=Subtonic
2.2 Conventions Of Labeling Intervals
Intervals are distances between tones (Loy 14). We measure intervals in units (like degrees or semitones) in order to quantify said distances. The first thing we should notice about intervals is that they are transposition equivalent. This means that an interval is a harmonic length which is independent of the particular tones involved.
Intervals can be defined numerically in terms of direction and magnitude. In terms of direction we will use terms like “evolution” to describe ascension upwards, or “involution” to describe descent downwards (Hanson 17). In terms of magnitude, we can describe intervals as big or small in distance – regardless of direction. For example, B is the second of A when ascending, because it is the second scale tone upwards in the sequence A B C D E F G. But, B is in the seventh tone of the A minor scale when descending downwards from A in the sequence A G F E D C B.
Diatonic intervals demonstrate the directional equivalence of intervals. For example, a major second upwards (e.g. A B) is the same interval as a minor seventh downwards (B A) when direction, (i.e. evolution or involution), is unimportant (Hanson 27). Related to directional equivalence is inversion equivalence: A B is the same interval as B A; though B A is the first upwards inversion of A B.
As the equal temperament tuning system gradually became standard, (and the twelve tones gradually replaced seven tones), intervals had to be altered. For example, some intervals were made smaller, (as “major” intervals, like the second, third, sixth and seventh, became “minor” intervals), other intervals were made smaller or larger, (as “perfect” intervals, like the fifth or fourth, became “diminished” or smaller, and “augmented” or larger).
Intervals which are defined as kinds of low numbered degrees within the octave, regardless of direction, (e.g. minor second, augmented fourth, diminished fourth, perfect fifth or major third), are also able to be defined as high numbered degrees beyond the octave (e.g. minor ninth, sharp eleventh or flat thirteenth), regardless of direction.
Notice that the traditional definition of western intervals require three things: a tonal reference point of the pair (e.g. the tone A of the pair B A), a scale reference (e.g. the A minor scale), and directed scale degrees (B is second up or seventh down from A in the A minor scale).
3 Bibliography
Bloom, B. S.; Engelhart, M. D.; Furst, E. J.; Hill, W. H.; Krathwohl, D. R. Taxonomy Of Educational Objectives: The Classification Of Educational Goals. Handbook I: Cognitive Domain. New York: David McKay Company, 1956
Loy, Gareth. Musimathics The Mathematical Foundations of Music: Volume 1. Cambridge, Massachusetts: The MIT Press, 2006
Hanson, Howard. Harmonic Materials Of Modern Music, New York: Appleton-Century-Crofts, 1960