This page is a sub-page of our page on Expandable Learning Objects.


The sub-pages of this page are:

Pythagorean Intervals
The Logarithmic Piano
Solfege: An Abstract Key System
Chord Ladder
Chord-Set Inclusion Graph
Harmony and Melody
Instantiations of Solfege in Different Keys
Abstract and Concrete Chord Circles
Transposition = Shift of Base in Music
Interactions of Quarts and Quints
Generating the Quint Circle from the Mathematical Cogwheels
Uniformisation of Intervals
Bengt Ulin om Musik (in Swedish)


Related sources of information:

Music and mathematics at Wikipedia.
Musical notation at Wikipedia.
Music Theory (at the Sibelius Academy).
Modes and Scales in Western Music at Wikipedia.
Musical Set Theory at Wikipedia.
Accidental (music) at Wikipedia.
Piece for triple piano in hyperbolic space by Vi Hart.
This visualisation of a Bach prelude lets you see just how clever the piece really is.
But what is a Fourier series? From heat flow to circle drawings.

In Swedish:

Musikteori på Wikipedia.
Harmoni på Wikipedia.
• Melodi på Wikipedia.
Diatonik på Wikipedia.
Pentatonisk skala på Wikipedia.
Funktionsanalys (musik) på Wikipedia.
Grundfunktionerna (musik) på Wikipedia.
Polyfoni på Wikipedia.
Kontrapunkt på Wikipedia.
Tonfrekvenser på ett liksvävande stämt piano på Musikipedia.


Representation and Reconstruction of a Presentant with respect to a Background

Representation: \, [ \, p_{resentant} \, ]_{B_{ackground}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{ackground}}

Reconstruction: \, \left( \, \left< \, r_{epresentant} \, \right>_{B_{ackground}} \, \right)_{B_{ackground}} \mapsto \,\, p_{resentant}

/////// In Swedish:

Representation och Rekonstruktion av en Presentant med hjälp av en Bakgrund

Representation: \, [ \, p_{resentant} \, ]_{B_{akgrund}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{akgrund}}

Rekonstruktion: \, \left( \, \left< \, r_{epresentant} \, \right>_{B_{akgrund}} \, \right)_{B_{akgrund}} \mapsto \,\, p_{resentant}

/////// Back to English:

Partial representations

Representation and Reconstruction of a \, t_{une} \, with respect to an \, A_{rrangement} \, :


Representation: [ \, t_{une} \, ]_{A_{rr}} \mapsto \left< \,s_{core} \, \right>_{A_{rr}} .

Reconstruction: \left(\left< \, s_{core} \, \right>_{A_{rr}}\right)_{A_{rr}} \mapsto t_{une} .

s_{core} \, is another name for Sheet music. Here we will model it as:

s_{core} = \{ \, o_{rchestration}, c_{horeography} \, \}

where \, o_{rchestration} \, specifies which instruments that are involved in the \, A_{rrangement} \, , and \, c_{horeography} \, specifies how these instruments should interact with each other.

Partial Representation and Partial Reconstruction of a \, t_{une} \, with respect to a \, K_{ey} \, :

The Representation and Reconstruction formulas for Numbers carry over to corresponding partial representation and partial reconstruction formulas for Tunes.

In tonal music, the tonal \, B_{ase} \, , the so-called tonic, of a \, t_{une} \, is given by its \, K_{ey} \, .

In the case of numbers and their digits with respect to a base, knowledge of the digits and the base is enough to uniquely determine the underlying number.

In contrast, in the case of tunes and their chord-sequences with respect to a key, knowledge of the chord-sequence and the key is not enough to uniquely determine the underlying tune. There are a lots of other aspects of a tune (such as melody, rhythm, etc) that are part of its arrangement.

Hence, while a \, n_{umber} \, is totally represented by its \, d_{igits} \, with respect to a certain \, B_{ase} \, , a \, t_{une} \, is only partially represented by its \, c_{hordSequence} \, with respect to a certain \, K_{ey} \,.

Therefore, in the case of tunes and their chord-sequences with respect to different keys, we can only state that:

Partial Representation: [ \, t_{une} \, ]_{A_{rr}} \supset [ \, t_{une} \, ]_{K_{ey}} \mapsto \left< \,c_{hordSequence} \, \right>_{K_{ey}} \, .

Partial Reconstruction: {\left( \left< \, c_{hordSequence} \, \right>_{K_{ey}} \, \right)}_{K_{ey}} \supset t_{une} \, .

This raises the question of which tune that is “the most reasonable one” that is represented by a given chord-sequence. Of course, the answer is highly dependent on the contexts at hand as well as on the individual participants’ interpretations of these contexts.

This opens up a huge area of mathematics, which deals with the most reasonable solutions to systems of equations. Such a solution is called a pseudoinverse for the system.


Don Knuth: Fantasia Apocalyptica.

Fyrklöver i Reaktorn:
Fyrklöver i Reaktorn


Skärmavbild 2018-02-04 kl. 22.52.09


Skärmavbild 2018-02-04 kl. 22.52.30


Lim and colim of melodies, harmonies, and beats:



Steve Fishell explains how pedal steel guitar works
(on YouTube):

Pedal Steel Guitar


The Entertainer (by Scott Joplin) in piano coordinates:

The Entertainer (in piano coordinates)


Numbers are naturally related to Music:
Numbers  are Naturally Related to Music


Geometry of Music (Bill Wesley at TEDxAmericasFinestCity 2011):


The TRUTH Why Modern Music Is Awful (Thoughty2 on YouTube, 2017):


But what is a Fourier Series? – From heat flow to circle drawings
(Steven Strogatz on YouTube):


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