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What are musical intervals?
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What are musical intervals?
Motivation
musikalische Intervalle
Motivation
Pfade musikalischer Intervalle
Syntaktische musikalische Intervalle: Paare von Tasten eines Piano (ohne Ton) und geschriebene Noten
Semantische musikalische Intervalle: Paare von gespielten Noten
System vom musikalischen Stimmen in dem die Frequenzen aller Intervalle auf dem Verhältnis 3:2, der reinen Quinte, beruhen
Pythagoreische Stimmung
Algebraische Theorie des
Measurement
Monoid
Beispiel 1
Task: formulate the properties of a commutative monoid for the additive monoid.
Beispiel 2
Task: formulate the properties of a commutative group for the group above
Beispiel 3
Task: show, that the word monoid is a monoid.
(Iso)morphismen
Morphismen
Beispiel
Task: proof, that the map is an isomorphism
Abstraktes Netzwerk
Abstrktes Netzwerk
We can interpret the set A as a set of arrows and T as a set of paths in T. Then the symbol in the second bullet point means s is a subpath of t. So the point s.
Abstraktes Aktionsnetzwerk
Abstraktes Measurement Setup
Abstraktes Measurement setup
Kongruenz
Konstruktion
Splitting thoerem
Splitting theorem
Universelle Eigenschaft
Musical side
Construction
Interpretation
Remark
Example 1
Example 2
Zusammenfassung
- general algebraic theory of measurement
- specifies to a general theory of musical intervals
- focused on algebraic modeling of syntactic musical intervals - to single out feasible musical intervals
- main mathematical result (Theorem 2) states, that every abstract action domain gives rise to a universal abstract measurement setup
- this means, that for any given framework of syntactic musical intervals, there can be derived a universal concept of semantic musical intervals