An Interval Class Vector (ICV) Calculator is a tool used in music theory to analyze the intervallic content of a set of pitches. The Interval Class Vector is a six-element vector that represents the number of occurrences of each interval class within a pitch class set.

## Understanding the Calculator's Purpose and Functionality

The primary purpose of the Interval Class Vector Calculator is to provide a concise and systematic representation of the intervallic relationships within a set of pitches. This tool is essential for music theorists and composers to analyze and understand the structure and characteristics of musical compositions.

## Inputs

To calculate the Interval Class Vector, the following input is required:

**Pitch Class Set**- A set of pitch classes, typically represented as integers from 0 to 11, where 0 is C, 1 is C#, and so on up to 11, which is B.

## Formula

To determine the Interval Class Vector, follow these steps:

**Calculate All Intervals**: Compute the interval between each pair of pitch classes in the set, modulo 12.**Convert to Interval Classes**: For each interval, convert it to its interval class, which is the smallest distance between two pitch classes. This is done using the formula:

[ \text{Interval Class} = \min(\text{Interval}, 12 - \text{Interval}) ]**Count Occurrences**: Count the number of occurrences of each interval class (from 1 to 6).

## Calculation Example

Let's go through an example to understand how the Interval Class Vector Calculator works.

## Example: Calculating the ICV for the Pitch Class Set {0, 4, 7}

**Pitch Class Set**: {0, 4, 7}**Calculate All Intervals**:

- Interval between 0 and 4: (4 - 0 = 4)
- Interval between 0 and 7: (7 - 0 = 7)
- Interval between 4 and 7: (7 - 4 = 3)

**Convert to Interval Classes**:

- Interval 4: Interval Class 4 (since ( \min(4, 12 - 4) = 4 ))
- Interval 7: Interval Class 5 (since ( \min(7, 12 - 7) = 5 ))
- Interval 3: Interval Class 3 (since ( \min(3, 12 - 3) = 3 ))

**Count Occurrences**:

- Interval Class 1: 0 occurrences
- Interval Class 2: 0 occurrences
- Interval Class 3: 1 occurrence
- Interval Class 4: 1 occurrence
- Interval Class 5: 1 occurrence
- Interval Class 6: 0 occurrences

So, the Interval Class Vector for the pitch class set {0, 4, 7} is [0, 0, 1, 1, 1, 0].

## Relevant Information Table

Pitch Class Set | Intervals | Interval Classes | Interval Class Vector |
---|---|---|---|

{0, 4, 7} | 4, 7, 3 | 4, 5, 3 | [0, 0, 1, 1, 1, 0] |

## Conclusion

The Interval Class Vector Calculator is an essential tool for music theorists and composers. It provides a concise representation of the intervallic content of a pitch class set, aiding in the analysis and understanding of musical structures. By using this calculator, one can quickly determine the interval classes present in any set of pitches, making it a valuable resource for music analysis and composition.By utilizing the Interval Class Vector Calculator, music professionals and enthusiasts can gain deeper insights into the compositional makeup of musical works, fostering greater creativity and analytical skills.