science

Pitch interval

Pitch interval 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 Pitch interval rather than just read about it. In short: In musical set theory, there are four kinds of interval: Ordered pitch interval Unordered pitch interval Ordered pitch-class interval Unordered pitch-class interval Pitch intervals Ordered pitch interval The ordered pitch interval is the number of semitones that separates one pitch from another, upward or downward. It is thus more specific than the unordered pitch interval in that it represents the directionality of…

Key takeaways

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

Reference excerpt

In musical set theory, there are four kinds of interval:

Ordered pitch interval Unordered pitch interval Ordered pitch-class interval Unordered pitch-class interval

Pitch intervals

Ordered pitch interval The ordered pitch interval is the number of semitones that separates one pitch from another, upward or downward. It is thus more specific than the unordered pitch interval in that it represents the directionality of the interval. An ordered pitch interval always includes a plus or minus sign. Thus this interval type can describe a melodic as well as a harmonic interval.

Unordered pitch interval The unordered pitch interval does not include directionality information and is thus less specific than the ordered pitch interval. It is still the distance between two pitches measured in semitones, but that distance is not qualified by a positive (+) or negative symbol. (-). An unordered pitch interval can describe a harmonic interval but not a melodic interval. Both types of pitch intervals describe octave information in that they do not treat all octaves as being equivalent. Pitch intervals, both ordered and unordered, may therefore be larger than 12.

Comparison to pitch-class intervals By treating all octaves as being equivalent, pitch-classes contain less information (ex 'C') than pitches (ex: C3). Pitch-class intervals (below) are therefore never larger than 12 semitones.

Pitch-class intervals

In musical set theory, pitch-class intervals do not distinguish between octaves since pitch-classes themselves treat all octaves as being equivalent. There are two kinds of pitch-class intervals:

ordered pitch-class interval (also called a pitch-interval class - PIC) unordered pitch-class interval (also called an 'interval class')

Ordered pitch-class intervals ('pitch interval class; PIC') The ordered pitch-class interval describes the number of ascending semitones from one pitch-class to the next, ordered from lowest to highest. Since pitch-classes have octave equivalence, the ordered pitch -class interval can be computed mathematically as "the absolute value of the difference between the two pitch-classes modulo 12". See Equations, below. A more visual way to do this calculation is to place the pitch-classes on a clockface and measure the difference, always going clockwise (i.e. always ascending).

Unordered pitch-class intervals ('interval class; IC') Unlike the ordered, the unordered pitch-class interval (often called the 'Interval class') does not require the two pitch-classes to be ordered from lowest to highest. Rather, this type of interval measures in semitones whichever interval is smallest. Because of symmetry, the smallest semitone interval between any two pitch-classes can only be an integer between 0 and 6. (hence the seven 'interval classes') The tonal interval names 'minor 2nd' and 'major 7th' both correspond to "interval class 1" for example, this is because both are composed of one semitone and directional order is unimportant when the criteria become to select the smallest interval. Similarly, the 'augmented fourth' and the 'diminished fifth' both correspond to 'interval class 6'. There is no 'interval class 7' therefore, since counting down five semitones can describe the perfect fifth more parsimoniously that counting up seven semitones can. A visual way to do determine an unordered pitch-class interval is to place the pitch-classes on a clockface and measure clockwise or counter-clockwise, whichever distance is smaller.

Equations Using integer notation and modulo 12, ordered pitch interval, ip, may be defined, for any two pitches x and y, as:

ip ⁡ ⟨ x , y ⟩ = y − x {\displaystyle \operatorname {ip} \langle x,y\rangle =y-x}

and:

ip ⁡ ⟨ y , x ⟩ = x − y {\displaystyle \operatorname {ip} \langle y,x\rangle =x-y}

the other way. Ascending intervals are indicated by a positive value, and descending intervals by a negative one. One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, similar to the interval of tonal theory. This may be defined as:

ip ⁡ ( x , y ) = | y − x | {\displaystyle \operatorname {ip} (x,y)=|y-x|}

The interval between pitch-classes may be measured with ordered and unordered pitch-class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. Thus, the ordered pitch-class interval, i⟨x, y⟩, may be defined as:

i ⁡ ⟨ x , y ⟩ = y − x {\displaystyle \operatorname {i} \langle x,y\rangle =y-x} (in modular 12 arithmetic) The unordered pitch-class interval i(x, y) may be defined as

i ( x , y ) = the smaller of i ⟨ x , y ⟩ and i ⟨ y , x ⟩ {\displaystyle i(x,y)={\text{ the smaller of }}i\langle x,y\rangle {\text{ and }}i\langle y,x\rangle }

Illustrations

Worked examples

Example 1 — a first encounter with Pitch interval

Start with the simplest possible case. Write down what Pitch interval 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 Pitch interval 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 Pitch interval 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 Pitch interval

In research: Pitch interval 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 Pitch interval 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: Pitch interval is common in secondary-school and first-year university syllabi. It links to neighbouring topics Intervals (music), Musical set theory, so understanding it makes those chapters shorter.
In everyday life: Look for Pitch interval 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 Pitch interval in 20 minutes

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

Frequently asked questions

What is Pitch interval in simple terms?

In musical set theory, there are four kinds of interval: Ordered pitch interval Unordered pitch interval Ordered pitch-class interval Unordered pitch-class interval Pitch intervals Ordered pitch interval The ordered pitch interval is the number of semitones that separates one pitch from another, up…

Why does Pitch interval 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 Pitch interval?

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 Pitch interval.