mathematics

Millioctave

Millioctave is a mathematics 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 Millioctave rather than just read about it. In short: The millioctave (moct) is a unit of measurement for musical intervals. As is expected from the prefix milli-, a millioctave is defined as 1/1000 of an octave.

Key takeaways

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

Reference excerpt

The millioctave (moct) is a unit of measurement for musical intervals. As is expected from the prefix milli-, a millioctave is defined as 1/1000 of an octave. From this it follows that one millioctave is equal to the ratio 21/1000, the 1000th root of 2, or approximately 1.0006934 (). Given two frequencies a and b, the measurement of the interval between them in millioctaves can be calculated by

n = 1000 log 2 ⁡ ( a b ) ≈ 3322 log 10 ⁡ ( a b ) {\displaystyle n=1000\log _{2}\left({\frac {a}{b}}\right)\approx 3322\log _{10}\left({\frac {a}{b}}\right)}

Likewise, if you know a note b and the number n of millioctaves in the interval, then the other note a may be calculated by:

a = b × 2 n 1000 {\displaystyle a=b\times 2^{\frac {n}{1000}}}

Like the more common cent, the millioctave is a linear measure of intervals, and thus the size of intervals can be calculated by adding their millioctave values, instead of multiplication, which is necessary for calculations of frequencies. A millioctave is exactly 1.2 cents.

History and use The millioctave was introduced by the German physicist Arthur von Oettingen in his book Das duale Harmoniesystem (1913). The invention goes back to John Herschel, who proposed a division of the octave into 1000 parts, which was published (with appropriate credit to Herschel) in George Biddell Airy's book on musical acoustics. Compared to the cent, the millioctave has not been as popular because it is not aligned with just intervals. It is however occasionally used by authors who wish to avoid the close association between the cent and twelve-tone equal temperament. Some considers that the millioctave introduces as well a bias for the less familiar 10-tone equal temperament however this bias is common in the decimal system.

See also Cent (music) Savart Musical tuning Logarithm Degree (angle) Chiliagon

Notes

External links Logarithmic Interval Measures

Worked examples

Example 1 — a first encounter with Millioctave

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

In research: Millioctave appears in mathematics 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 Millioctave 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: Millioctave is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1000 (number), 1913 introductions, Equal temperaments, so understanding it makes those chapters shorter.
In everyday life: Look for Millioctave 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 Millioctave in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Millioctave 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 Millioctave out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Millioctave in simple terms?

The millioctave (moct) is a unit of measurement for musical intervals. As is expected from the prefix milli-, a millioctave is defined as 1/1000 of an octave.

Why does Millioctave matter?

Because it connects several mathematics 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 Millioctave?

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 Millioctave.