Understanding Intervals

An Interval represents the distance between two musical pitches. Understanding intervals is fundamental to music theory as they form the basis of scales, chords, and melodies.

What is an Interval?

An interval consists of two components:

Interval Number: The numeric distance between notes (1 = unison, 2 = second, 3 = third, etc.)
Interval Quality: The specific type of interval (Perfect, Major, Minor, Augmented, Diminished)

Interval Qualities

Different intervals have different available qualities:

Unison (1), Fourth (4), Fifth (5), and Octave (8) can be:

- **Diminished**: One semitone smaller than perfect - **Perfect**: The standard interval - **Augmented**: One semitone larger than perfect

// Perfect intervals
var perfectUnison = new Interval(IntervalQuality.Perfect, 1);
var perfectFourth = new Interval(IntervalQuality.Perfect, 4);
var perfectFifth = new Interval(IntervalQuality.Perfect, 5);
var perfectOctave = new Interval(IntervalQuality.Perfect, 8);

// Altered perfect intervals
var augmentedFourth = new Interval(IntervalQuality.Augmented, 4); // Tritone
var diminishedFifth = new Interval(IntervalQuality.Diminished, 5); // Tritone

Second (2), Third (3), Sixth (6), and Seventh (7) can be:

- **Diminished**: One semitone smaller than minor - **Minor**: One semitone smaller than major - **Major**: The standard interval - **Augmented**: One semitone larger than major

// Major intervals
var majorSecond = new Interval(IntervalQuality.Major, 2);
var majorThird = new Interval(IntervalQuality.Major, 3);
var majorSixth = new Interval(IntervalQuality.Major, 6);
var majorSeventh = new Interval(IntervalQuality.Major, 7);

// Minor intervals
var minorSecond = new Interval(IntervalQuality.Minor, 2);
var minorThird = new Interval(IntervalQuality.Minor, 3);
var minorSixth = new Interval(IntervalQuality.Minor, 6);
var minorSeventh = new Interval(IntervalQuality.Minor, 7);

Creating Intervals

Direct Construction

Create intervals by specifying quality and number:

// Common intervals
var majorThird = new Interval(IntervalQuality.Major, 3);
var perfectFifth = new Interval(IntervalQuality.Perfect, 5);
var minorSeventh = new Interval(IntervalQuality.Minor, 7);

// Extended intervals
var majorNinth = new Interval(IntervalQuality.Major, 9);
var perfect11th = new Interval(IntervalQuality.Perfect, 11);
var major13th = new Interval(IntervalQuality.Major, 13);

Calculating Between Notes

Calculate the interval between two notes:

// Create notes
var c4 = new Note(NoteName.C, Alteration.Natural, 4);
var e4 = new Note(NoteName.E, Alteration.Natural, 4);
var g4 = new Note(NoteName.G, Alteration.Natural, 4);

// Calculate intervals
var third = Interval.Between(c4, e4);  // Major Third
var fifth = Interval.Between(c4, g4);  // Perfect Fifth

// Works across octaves
var c5 = new Note(NoteName.C, Alteration.Natural, 5);
var octave = Interval.Between(c4, c5); // Perfect Octave

// Handles altered notes
var eb4 = new Note(NoteName.E, Alteration.Flat, 4);
var minorThird = Interval.Between(c4, eb4); // Minor Third

Interval Properties

Basic Properties

var interval = new Interval(IntervalQuality.Major, 6);

// Access properties
IntervalQuality quality = interval.Quality; // Major
int number = interval.Number;               // 6
int semitones = interval.Semitones;        // 9

// Check interval type
bool isPerfectInterval = (number == 1 || number == 4 || 
                         number == 5 || number == 8);

Semitone Calculation

The library automatically calculates semitones based on quality and number:

Interval	Semitones	Example (from C)
Perfect Unison	0	C → C
Minor Second	1	C → Db
Major Second	2	C → D
Minor Third	3	C → Eb
Major Third	4	C → E
Perfect Fourth	5	C → F
Tritone	6	C → F#/Gb
Perfect Fifth	7	C → G
Minor Sixth	8	C → Ab
Major Sixth	9	C → A
Minor Seventh	10	C → Bb
Major Seventh	11	C → B
Perfect Octave	12	C → C

Using Intervals

Transposing Notes

Use intervals to transpose notes up or down:

var c4 = new Note(NoteName.C, Alteration.Natural, 4);

// Transpose up
var e4 = c4.Transpose(new Interval(IntervalQuality.Major, 3), Direction.Up);
var g4 = c4.Transpose(new Interval(IntervalQuality.Perfect, 5)); // Up is default

// Transpose down
var a3 = c4.Transpose(new Interval(IntervalQuality.Major, 3), Direction.Down);
var f3 = c4.Transpose(new Interval(IntervalQuality.Perfect, 5), Direction.Down);

// Chain transpositions
var d5 = c4
    .Transpose(new Interval(IntervalQuality.Perfect, 5), Direction.Up)  // → G4
    .Transpose(new Interval(IntervalQuality.Perfect, 5), Direction.Up); // → D5

Building Scales

Intervals define scale patterns:

// Major scale intervals from root
var majorScaleIntervals = new[]
{
    new Interval(IntervalQuality.Perfect, 1),  // Root
    new Interval(IntervalQuality.Major, 2),    // Major 2nd
    new Interval(IntervalQuality.Major, 3),    // Major 3rd
    new Interval(IntervalQuality.Perfect, 4),  // Perfect 4th
    new Interval(IntervalQuality.Perfect, 5),  // Perfect 5th
    new Interval(IntervalQuality.Major, 6),    // Major 6th
    new Interval(IntervalQuality.Major, 7),    // Major 7th
    new Interval(IntervalQuality.Perfect, 8)   // Octave
};

// Build C major scale
var c = new Note(NoteName.C, Alteration.Natural, 4);
var cMajorScale = majorScaleIntervals
    .Select(interval => c.Transpose(interval))
    .ToList();

Building Chords

Intervals define chord structures:

// Major triad: Root, Major 3rd, Perfect 5th
var root = new Note(NoteName.C, Alteration.Natural, 4);
var third = root.Transpose(new Interval(IntervalQuality.Major, 3));
var fifth = root.Transpose(new Interval(IntervalQuality.Perfect, 5));

// Minor 7th chord: Root, Minor 3rd, Perfect 5th, Minor 7th
var minorThird = root.Transpose(new Interval(IntervalQuality.Minor, 3));
var minorSeventh = root.Transpose(new Interval(IntervalQuality.Minor, 7));

Interval Inversions

Invert intervals using the Invert() method:

// Basic inversions
var majorThird = new Interval(IntervalQuality.Major, 3);
var minorSixth = majorThird.Invert(); // Minor 6th

var perfectFifth = new Interval(IntervalQuality.Perfect, 5);
var perfectFourth = perfectFifth.Invert(); // Perfect 4th

var minorSecond = new Interval(IntervalQuality.Minor, 2);
var majorSeventh = minorSecond.Invert(); // Major 7th

// Quality inversion rules:
// - Perfect remains Perfect
// - Major becomes Minor
// - Minor becomes Major
// - Augmented becomes Diminished
// - Diminished becomes Augmented

// Number rule: original + inverted = 9
var unison = new Interval(IntervalQuality.Perfect, 1);
var octave = unison.Invert(); // Perfect 8th (1 + 8 = 9)

// Compound intervals are reduced before inversion
var majorNinth = new Interval(IntervalQuality.Major, 9);
var minorSeventh = majorNinth.Invert(); // Minor 7th (not 0!)

// Tritone inverts to tritone
var augFourth = new Interval(IntervalQuality.Augmented, 4);
var dimFifth = augFourth.Invert(); // Diminished 5th

Enharmonic Intervals

Different spellings of the same sonic interval:

// Augmented 4th and Diminished 5th (both 6 semitones)
var augFourth = new Interval(IntervalQuality.Augmented, 4);
var dimFifth = new Interval(IntervalQuality.Diminished, 5);

bool sameSound = augFourth.Semitones == dimFifth.Semitones; // true
bool areEnharmonic = augFourth.IsEnharmonicWith(dimFifth);   // true

// Example from C
var c = new Note(NoteName.C, Alteration.Natural, 4);
var fSharp = c.Transpose(augFourth);  // F#4
var gFlat = c.Transpose(dimFifth);    // Gb4
// Same pitch, different spelling

Common Interval Patterns

Consonant and Dissonant Intervals

// Consonant intervals
var consonantIntervals = new[]
{
    new Interval(IntervalQuality.Perfect, 1),   // Unison
    new Interval(IntervalQuality.Major, 3),     // Major 3rd
    new Interval(IntervalQuality.Minor, 3),     // Minor 3rd
    new Interval(IntervalQuality.Perfect, 4),   // Perfect 4th
    new Interval(IntervalQuality.Perfect, 5),   // Perfect 5th
    new Interval(IntervalQuality.Major, 6),     // Major 6th
    new Interval(IntervalQuality.Minor, 6),     // Minor 6th
    new Interval(IntervalQuality.Perfect, 8)    // Octave
};

// Dissonant intervals
var dissonantIntervals = new[]
{
    new Interval(IntervalQuality.Minor, 2),     // Minor 2nd
    new Interval(IntervalQuality.Major, 2),     // Major 2nd
    new Interval(IntervalQuality.Augmented, 4), // Tritone
    new Interval(IntervalQuality.Diminished, 5),// Tritone
    new Interval(IntervalQuality.Major, 7),     // Major 7th
    new Interval(IntervalQuality.Minor, 7)      // Minor 7th
};

var c = new Note(NoteName.C, Alteration.Natural, 4);
var fifth = new Interval(IntervalQuality.Perfect, 5);

// Move clockwise (adding sharps)
var g = c.Transpose(fifth);                    // G
var d = g.Transpose(fifth);                    // D
var a = d.Transpose(fifth);                    // A

// Move counter-clockwise (adding flats)
var f = c.Transpose(fifth, Direction.Down);    // F
var bb = f.Transpose(fifth, Direction.Down);   // Bb
var eb = bb.Transpose(fifth, Direction.Down);  // Eb

Advanced Topics

Compound Intervals

Intervals larger than an octave:

// Compound intervals (> octave)
var ninth = new Interval(IntervalQuality.Major, 9);      // Octave + 2nd
var eleventh = new Interval(IntervalQuality.Perfect, 11); // Octave + 4th
var thirteenth = new Interval(IntervalQuality.Major, 13); // Octave + 6th

// Reduce to simple interval
int simpleIntervalNumber = ((ninth.Number - 1) % 7) + 1; // 2

Interval Addition

Combining intervals:

// Adding intervals (conceptually)
var majorThird = new Interval(IntervalQuality.Major, 3);    // 4 semitones
var minorThird = new Interval(IntervalQuality.Minor, 3);    // 3 semitones
// Together they make a perfect fifth (7 semitones)

var c = new Note(NoteName.C, Alteration.Natural, 4);
var e = c.Transpose(majorThird);      // E
var g = e.Transpose(minorThird);      // G
// C to G is a perfect fifth

Best Practices

Use appropriate quality: Perfect for 1, 4, 5, 8; Major/Minor for 2, 3, 6, 7
Consider enharmonic context: Aug 4th vs Dim 5th based on musical context
Validate interval creation: Some quality/number combinations are invalid
Think musically: Choose intervals that make sense in the harmonic context

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