science

Symmetric scale

Symmetric scale is a science 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 Symmetric scale rather than just read about it. In short: In music theory, a music scale can have certain symmetries, namely translational symmetry and inversional or mirror symmetry. The most prominent examples are scales which equally divides the octave.

Key takeaways

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

Reference excerpt

In music theory, a music scale can have certain symmetries, namely translational symmetry and inversional or mirror symmetry. The most prominent examples are scales which equally divides the octave. The concept and term appears to have been introduced by Joseph Schillinger and further developed by Nicolas Slonimsky as part of his famous Thesaurus of Scales and Melodic Patterns. In twelve-tone equal temperament, the octave can only be equally divided into two, three, four, six, or twelve parts, which consequently may be filled in by adding the same exact interval or sequence of intervals to each resulting note (called "interpolation of notes"). This leads to scales with translational symmetry which include the octatonic scale (also known as the symmetric diminished scale; its mirror image is known as the inverse symmetric diminished scale) and the two-semitone tritone scale:

As explained above, both are composed of repeating sub-units within an octave. This property allows these scales to be transposed to other notes, yet retain exactly the same notes as the original scale (Translational symmetry). This may be seen quite readily with the whole tone scale on C:

{C, D, E, F♯, G♯, A♯, C}

If transposed up a whole tone to D, contains exactly the same notes in a different permutation:

{D, E, F♯, G♯, A♯, C, D} In the case of inversionally symmetrical scales, the inversion of the scale is identical. Thus the intervals between scale degrees are symmetrical if read from the "top" (end) or "bottom" (beginning) of the scale (mirror symmetry). Examples include the Neapolitan Major scale (fourth mode of the Major Locrian scale), the Javanese slendro, the chromatic scale, whole-tone scale, Dorian scale, the Aeolian Dominant scale (fifth mode of the melodic minor), and the double harmonic scale.

Asymmetric scales are "far more common" than symmetric scales and this may be accounted for by the inability of translational symmetric scales to possess the property of uniqueness (containing each interval class a unique number of times) which assists with determining the location of notes in relation to the first note of the scale.

See also Modes of limited transposition Symmetry#In music

References

Further reading Yamaguchi, Masaya. 2006. The Complete Thesaurus of Musical Scales, revised edition. New York: Masaya Music Services. ISBN 0-9676353-0-6. Yamaguchi, Masaya. 2006. Symmetrical Scales for Jazz Improvisation, revised edition. New York: Masaya Music Services. ISBN 0-9676353-2-2. Yamaguchi, Masaya. 2012. Lexicon of Geometric Patterns for Jazz Improvisation. New York: Masaya Music Services. ISBN 0-9676353-3-0.

Illustrations

Worked examples

Example 1 — a first encounter with Symmetric scale

Start with the simplest possible case. Write down what Symmetric scale claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Symmetric scale 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 Symmetric scale 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 Symmetric scale

In research: Symmetric scale appears in science 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 Symmetric scale 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: Symmetric scale is common in secondary-school and first-year university syllabi. It links to neighbouring topics Hexatonic scales, Musical symmetry, so understanding it makes those chapters shorter.
In everyday life: Look for Symmetric scale 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 Symmetric scale in 20 minutes

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

Frequently asked questions

What is Symmetric scale in simple terms?

In music theory, a music scale can have certain symmetries, namely translational symmetry and inversional or mirror symmetry. The most prominent examples are scales which equally divides the octave.

Why does Symmetric scale matter?

Because it connects several science 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 Symmetric scale?

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 Symmetric scale.