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Generic and specific intervals

Generic and specific intervals is a biology topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Generic and specific intervals rather than just read about it. In short: In diatonic set theory, a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members.

Generic and specific intervals — main illustration
Generic and specific intervals — illustration

Key takeaways

  • Generic and specific intervals belongs to biology; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Generic and specific intervals to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Generic and specific intervals from memory before moving on to harder problems.

Reference excerpt

In diatonic set theory, a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members. (Johnson 2003, p.26) A specific interval is the clockwise distance between pitch classes on the chromatic circle (interval class), in other words the number of half steps between notes. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal temperament the largest specific interval is 11. (Johnson 2003, p.26) In the diatonic collection the generic interval is one less than the corresponding diatonic interval:

Adjacent intervals, seconds, are 1 Thirds = 2 Fourths = 3 Fifths = 4 Sixths = 5 Sevenths = 6 The largest generic interval in the diatonic scale being 7 − 1 = 6.

Myhill's property Myhill's property is the quality of musical scales or collections with exactly two specific intervals for every generic interval, and thus also have the properties of cardinality equals variety, structure implies multiplicity, and being a well-formed generated collection. In other words, each generic interval can be made from one of two possible different specific intervals. For example, there are major or minor and perfect or augmented/diminished variants of all the diatonic intervals:

The diatonic and pentatonic collections possess Myhill's property. The concept appears to have been first described by John Clough and Gerald Myerson and named after their associate the mathematician John Myhill. (Johnson 2003, p. 106, 158)

Sources Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.

Further reading Clough, Engebretsen, and Kochavi. "Scales, Sets, and Interval Cycles": 78–84.

Illustrations

Generic and specific intervals: The major scale is maximally even. For example, for every generic interval of a second there are only two possible specific intervals: 1 semitone (a minor second) or 2 semitones (a major second).
The major scale is maximally even. For example, for every generic interval of a second there are only two possible specific intervals: 1 semitone (a minor second) or 2 semitones (a major second).

Worked examples

Example 1 — a first encounter with Generic and specific intervals

Start with the simplest possible case. Write down what Generic and specific intervals claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In biology, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Generic and specific intervals before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Generic and specific intervals ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Generic and specific intervals

In research
Generic and specific intervals appears in biology research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Generic and specific intervals in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Generic and specific intervals is common in secondary-school and first-year university syllabi. It links to neighbouring topics Diatonic set theory, Intervals (music), so understanding it makes those chapters shorter.
In everyday life
Look for Generic and specific intervals outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Generic and specific intervals in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Generic and specific intervals means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Generic and specific intervals out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Generic and specific intervals in simple terms?

In diatonic set theory, a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members.

Why does Generic and specific intervals matter?

Because it connects several biology ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Generic and specific intervals?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Generic and specific intervals.

Tags

  • Diatonic set theory
  • Intervals (music)

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