science

Scale of harmonics

Scale of harmonics 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 Scale of harmonics rather than just read about it. In short: The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale.

Key takeaways

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

Reference excerpt

The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale. Most fret positions appearing on Non-Western string instruments (lutes) are equal to positions of this scale. Unexpectedly, these fret positions are actually the corresponding undertones of the overtones from the harmonic series. The distance from the nut to the fret is an integer number lower than the distance from the fret to the bridge (see: superparticular number).

Origin On the guqin, the left end of the dotted scale is a mirror image of the right end. The instrument is played with flageolet tones (harmonics) as well as pressing the strings on the wood. The flageolets appear on the harmonic positions of the overtone series, therefore these positions are marked as the musical scale of this instrument. The flageolet positions also represent the harmonic consonant relation of the pressed string part with the open string, similar to the calculations Pythagoras did on his monochord. The guqin has one anomaly in its scale. The guqin scale represents the first six harmonics and the eighth harmonic. The seventh harmonic is left out. However this tone is still consonant related to the open string (otherwise it would not be a harmonic) and has a lesser consonant relation to all other harmonic positions. This is the main reason all the ratios of the sevenths family (7:1, 7:2, 7:3, 7:4, 7:5 and 7:6) also often are not present in other musical scales like for instance the just intoned major and minor scale or the major scale in the Pythagorean tuning.

A Vietnamese monochord, called the đàn bầu, also functions with the scale of harmonics. On this instrument only the right half (from the view of the musician) of the scale is present up to the limit of the first seven overtones (see 7-limit). The dots are on the string lengths 1⁄2, 1⁄3, 1⁄4, 1⁄5, 1⁄6, 1⁄7 of the whole string length. The reason for this half scale is because the left half creates the same tones as the right half when played as a flageolet tone and therefore the extra dots on the left half are useless for how this instrument is played. The scale of harmonics was, together with the book of Helmholtz an inspiration for Harry Partch to switch to just intonation and alternate tuning systems to create more consonant music than possible with the equal temperament. Partch's tone selection otonality from his utonality and otonality concept are the complement pitches of the overtones. For instance: the frequency ratio 5:4 is equal to 4⁄5 of the string length and 4⁄5 is the complement of 1⁄5, the position of the fifth harmonic (and the fourth overtone). The Norwegian composer Eivind Groven also wrote a thesis on the scale of harmonics, claiming this to be the oldest usable scale, frequent in Norwegian folk music, and seemingly in other folk musical traditions as well. Groven used the seljefløyte as basis for his research. The flute uses only the upper harmonic scale. The scale is also present on the Moodswinger. Although this functions quite differently to a Guqin, oddly enough the scale occurs on this instrument while it is not played in a just intonation tuning but a regular equal temperament.

See also Acoustic scale Arithmetic progression Harmonic spectrum Otonality and Utonality

References

Further reading Partch, Harry (1979). Genesis Of A Music: An Account Of A Creative Work, Its Roots, And Its Fulfillments (Second Edition). ISBN 0-306-80106-X.

External links

"3rd Bridge Helix", PerfectSoundForever. Article about the overtoning positions and their relation to musical scales.

Illustrations

Worked examples

Example 1 — a first encounter with Scale of harmonics

Start with the simplest possible case. Write down what Scale of harmonics 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 Scale of harmonics 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 Scale of harmonics 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 Scale of harmonics

In research: Scale of harmonics 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 Scale of harmonics 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: Scale of harmonics is common in secondary-school and first-year university syllabi. It links to neighbouring topics Acoustics, Hemitonic scales, Hexatonic scales, so understanding it makes those chapters shorter.
In everyday life: Look for Scale of harmonics 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 Scale of harmonics in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Scale of harmonics 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.
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Frequently asked questions

What is Scale of harmonics in simple terms?

Why does Scale of harmonics 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 Scale of harmonics?

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 Scale of harmonics.