The Temperament Ontology

Working Draft — 18 Dec 2009

This version:
http://purl.org/ontology/temperament/draft1-20091218/
Latest version:
http://purl.org/ontology/temperament/
Last Update: 18. Dec. 2009.
Date: 18. Dec. 2009.
Editors:
Gyorgy Fazekas, Centre for Digital Music Queen Mary University of London
Authors:
Gyorgy Fazekas, Dan Tidhar, Centre for Digital Music Queen Mary University of London
Contributors:
See acknowledgements

Abstract

The temperament ontology can be used to give a detailed description of the tuning of an instrument. It is developed in the OMRAS2 project, and intended to be used in conjunction with the Music Ontology, for example, when describing the tuning that was used in a particular harpsichord recording.

Status of this Document

This is a work in progress! This document is changing on a daily if not hourly basis. Comments are very welcome, please send them to gyorgy.fazekas@elec.qmul.ac.uk. Thank you.

Table of Contents

  1. Introduction
    1. About instrument tuning
    2. An open-ended temperament description model
    3. The many ways of describing temperaments
      1. The Circle of Fifths
      2. Deviation from Equal Temperament
  2. Temperament ontology at a glance
  3. Temperament ontology overview
    1. Example
  4. Cross-reference for Temperament classes and properties

Appendixes

  1. Normative References
  2. Changes in this version (Non-Normative)
  3. Acknowledgements (Non-Normative)

1 Introduction

The temperament ontology aims to describe instrument tuning systems and their particularities. It may also be used to characterise a (potentially unknown) temperament that was used when tuning an instrument for a particular performance or recording.

At this stage the ontology is far from exhaustive. We primarily deal with temperaments in western tonal music with an emphasis on tuning keyboard instruments. One important objective is to be able to describe arbitrary temperaments, or express the results of automatic temperament extraction from audio recordings.

1.1 About instrument tuning

Tuning an instrument consists of choosing the frequency values and spacing (or ratio) of pitches that are used. Pure (just) intervals of pitches correspond to whole number ratios of their frequencies, however these ratios are not compatible with each other as they arranged in scales (the way octaves are divided into discrete pitch classes) in western music. For example, it is not possible to fit twelve pure fifths (3:2)^12 into seven octaves (2:1)^7. The difference is called the Pythagorean or Ditonic comma (23.5 cents). This difference has to be tempered out ---that is, some (or all) fifths has to be mistuned slightly in order to fit them. There are many tuning systems. Most commonly, they differ in the way they compromise pure intervals to solve this problem.

1.2 An open-ended temperament description model

There is no mutual agreement in the literature on the description or classification of temperaments. Therefore, in this ontology we do not impose a hierarchy between types of temperaments. We define an opaque top-level temperament concept. Subclasses of this concept can be used in describing individual temperaments, if necessary, using multiple class memberships. Since there is more than one way to associate tuning systems with their properties, we treat temperament descriptions as concepts as well, and use reification to keep the model open and extensible.

Using the Circle of Fifths to describe a temperament.

Figure 1. Overview of the temperament ontology.

1.3 The many ways of describing temperaments

Temperaments can be characterised in lots of different ways. The most common methods are using either the circle of fifths or give the pitch deviations from equal temperament. We define these descriptions as concepts in the ontology, however, other descriptions may be used and defined in the future. (For example, one might find it convenient to express the same information using the circle of fourths.)

1.3.1 The Circle of Fifths

The circle of fifths has several uses in music theory. It shows the harmonic relationships of the twelve major and minor keys. It can also be seen as a circle of the corresponding pitch class intervals, such as (C-G), (G-D), (D-A) etc... If we go around the circle using pure fifth intervals, it wouldn't close. What remains is the Pythagorean comma. Hence, it is often used to describe temperaments by showing how the comma is distributed among the intervals to close the circle. (Note that there are several types of commas related to different tuning problems. These are defined in the ontology.)

...

1.3.2 Deviations from Equal Temperament

In equal temperament an octave is divided into twelve equal intervals. As a result only octaves are pure. All other intervals are impure, and the deviation from pure is different in case of each interval. Since equal temperament has become very common, other temperaments are often described by the frequency deviations (in cents) of each pitch class from the corresponding pitch class in equal temperament.

2. Temperament ontology at a glance

An alphabetical index of Temperament terms, by class (concepts) and by property (relationships, attributes), are given below. All the terms are hyperlinked to their detailed description for quick reference.

Classes: CircleOfFifths, Comma, DeviationFromFifth, DeviationsFromEqual, Interval, IntervalDeviation, MeantoneTemperament, PitchClass, PitchClassDeviation, PitchClassInterval, RegularTemperament, RestrictedTemperament, Temperament, TemperamentDescription, WellTemperament,

Properties: comma, constraint, description, deviation, frequency_ratio, interval, lower, pitch_class, pitch_class_deviation, upper, value,

Instances: A, Ab, As, B, Bb, Bendeler, Bs, C, Cb, Cs, D, Db, Diaschisma, DidymusComma, DitonicComma, Ds, E, Eb, Es, F, Fb, FifthCommaMeantone, Fs, G, Gb, Gs, KirnbergerIII, Neidhardt, PythagoreanComma, QuarterCommaMeantone, Schisma, SixthCommaMeantone, Stanhope, SyntonicComma, Valotti, WerckmeisterIII, YoungII,

...

3. Temperament ontology overview

3.1. Example

Here is a very basic example describing a pitch class interval in the Valotti temperament using the circle of fifths.

		@prefix rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#>.
		@prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#>.
		@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .
		@prefix tm: <http://purl.org/ontology/temperament/> .
		@prefix pc: <http://purl.org/ontology/temperament/pitchclass/> .

		:ValottiTemperament a tm:WellTemperament;
			tm:description :ValottiDescription1 .

		:ValottiDescription1 a tm:CircleOfFifths;
			tm:interval [
				a tm:FifthInterval ;
				tm:deviation [
					a tm:IntervalDeviation ;
					tm:comma tm:PythagoreanComma ; 
					tm:value "-0.16667"^^xsd:float ;
					rdfs:label "-1/6"
					] ;
				tm:lower pc:C ;
				tm:upper pc:G ] ;
			tm:interval [ # the rest of the intervals on the circle of fifths 
				] .
      

A graphical representation of a similar description is shown in figure 2. We explicitly name the pitch classes involved in each interval on the circle of fifths. The IntervalDeviation concept describes the amount of deviation from a pure interval in terms of a specific type of comma and a corresponding value (a fraction of that comma). We can safely assume that a fifth is pure, unless the deviation is given.

Using the Circle of Fifths to describe a temperament.

Figure 2. Describing temperaments using the circle of fifths.

4. Cross-reference for Temperament classes and properties

...

Class: tm:CircleOfFifths

URI: http://purl.org/ontology/temperament/CircleOfFifths

Circle Of Fifths - We use the circle of fifth to describe a temperament. Each alteration of a fifth can be expressed in a number of ways, for example, we may express interval deviations in a particular comma.

sub-class-of:
tm:TemperamentDescription

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Class: tm:Comma

URI: http://purl.org/ontology/temperament/Comma

Comma - A comma is a very small interval between two enharmonic notes tuned in different ways. (Source: Wikipedia)

in-range-of:
tm:comma

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Class: tm:DeviationFromFifth

URI: http://purl.org/ontology/temperament/DeviationFromFifth

Deviation from Fifth - Represent the deviation from a pure fifth interval.

sub-class-of:
tm:IntervalDeviation

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Class: tm:DeviationsFromEqual

URI: http://purl.org/ontology/temperament/DeviationsFromEqual

Deviations From Equal - A type of temperament description which describes a tuning in terms of the deviation of each note (in cents) from the equal temperament.

sub-class-of:
tm:TemperamentDescription
in-domain-of:
tm:pitch_class_deviation

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Class: tm:Interval

URI: http://purl.org/ontology/temperament/Interval

Interval - Describes the relationship between the pitches of two notes or pitch classes.

in-domain-of:
tm:deviation
tm:lower
tm:upper
in-range-of:
tm:interval

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Class: tm:IntervalDeviation

URI: http://purl.org/ontology/temperament/IntervalDeviation

Interval Deviation (from pure interval) - Represents any type of alteration of a pure interval.

in-domain-of:
tm:comma
tm:value
in-range-of:
tm:deviation

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Class: tm:MeantoneTemperament

URI: http://purl.org/ontology/temperament/MeantoneTemperament

Meantone Temperament - In a meantone temperament, each fifth is narrowed by the same amount relative to its width in Just Intonation.

sub-class-of:
tm:Temperament

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Class: tm:PitchClass

URI: http://purl.org/ontology/temperament/PitchClass

Pitch Class - The set of all pitches that are a whole number of octaves apart.

in-range-of:
tm:lower
tm:upper
tm:pitch_class

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Class: tm:PitchClassDeviation

URI: http://purl.org/ontology/temperament/PitchClassDeviation

Pitch Class Deviation - Deviation in cents of a pitch class from its value in equal temperament.

in-domain-of:
tm:value
tm:pitch_class
in-range-of:
tm:pitch_class_deviation

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Class: tm:PitchClassInterval

URI: http://purl.org/ontology/temperament/PitchClassInterval

Pitch Class Interval - Describes the relationship between two pitch classes.

sub-class-of:
tm:Interval

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Class: tm:RegularTemperament

URI: http://purl.org/ontology/temperament/RegularTemperament

Regular Temperament - Regular temperament is a class of temperaments where each interval is altered by the same amount.

sub-class-of:
tm:Temperament

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Class: tm:RestrictedTemperament

URI: http://purl.org/ontology/temperament/RestrictedTemperament

Restricted Temperament - In restricted temperaments playing some keys results in poor intonation.

sub-class-of:
tm:Temperament

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Class: tm:Temperament

URI: http://purl.org/ontology/temperament/Temperament

Temperament - A temperament specifies a system of instrument tuning. Tempering is a way of altering the size of intervals, such that they no longer based on pure rational intervals.

in-domain-of:
tm:description

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Class: tm:TemperamentDescription

URI: http://purl.org/ontology/temperament/TemperamentDescription

Temperament Description - A super class of temperament descriptions using, for example, deviation from equal temperament, or the circle of fifths.

in-domain-of:
tm:constraint
tm:interval
in-range-of:
tm:description

[back to top]

Class: tm:WellTemperament

URI: http://purl.org/ontology/temperament/WellTemperament

Well Temperament - Well temperament represents a system of tuning where 5ths are tempered by irregular amounts, such that some 5ths are tempered by an equal amount, while others remain pure.

sub-class-of:
tm:Temperament

[back to top]

Property: tm:comma

URI: http://purl.org/ontology/temperament/comma

comma - Links an interval deviation with a type of comma.

OWL Type:
ObjectProperty
Domain:
tm:IntervalDeviation
Range:
tm:Comma

[back to top]

Property: tm:constraint

URI: http://purl.org/ontology/temperament/constraint

mathematical constraint - A mathematical constrain on the temperament (This can be described in a named graph or and N3 graph literal).

OWL Type:
ObjectProperty
Domain:
tm:TemperamentDescription

[back to top]

Property: tm:description

URI: http://purl.org/ontology/temperament/description

temperament description - Associates a temperament with its detailed description.

OWL Type:
ObjectProperty
Domain:
tm:Temperament
Range:
tm:TemperamentDescription

[back to top]

Property: tm:deviation

URI: http://purl.org/ontology/temperament/deviation

interval deviation - Links an interval with a particular deviation from pure that characterises the interval.

OWL Type:
ObjectProperty
Domain:
tm:Interval
Range:
tm:IntervalDeviation

[back to top]

Property: tm:frequency_ratio

URI: http://purl.org/ontology/temperament/frequency_ratio

frequency ratio - The frequency ratio of the two pitch classes in the interval. This may be a single value or an expression (given for example in an N3 graph literal).

OWL Type:
ObjectProperty
Domain:
tm:intarval

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Property: tm:interval

URI: http://purl.org/ontology/temperament/interval

interval - Associates a specific interval on the circle of fifths.

OWL Type:
ObjectProperty
Domain:
tm:TemperamentDescription
Range:
tm:Interval

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Property: tm:lower

URI: http://purl.org/ontology/temperament/lower

lower pitch class - Links an interval a pitch calss which is lower on a scale.

OWL Type:
ObjectProperty
Domain:
tm:Interval
Range:
tm:PitchClass

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Property: tm:pitch_class

URI: http://purl.org/ontology/temperament/pitch_class

pitch class - The pitch class of a particular pitch class deviation. This can be used to express deviations from equal temperament.

OWL Type:
ObjectProperty
Domain:
tm:PitchClassDeviation
Range:
tm:PitchClass

[back to top]

Property: tm:pitch_class_deviation

URI: http://purl.org/ontology/temperament/pitch_class_deviation

pitch class deviation - The deviation of a particular pitch class in the described temperament from its equivalent in equal temperament.

OWL Type:
ObjectProperty
Domain:
tm:DeviationsFromEqual
Range:
tm:PitchClassDeviation

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Property: tm:upper

URI: http://purl.org/ontology/temperament/upper

upper pitch class - Links an interval with a pitch class which is higher on a scale compared to the other pitch class of this interval.

OWL Type:
ObjectProperty
Domain:
tm:Interval
Range:
tm:PitchClass

[back to top]

Property: tm:value

URI: http://purl.org/ontology/temperament/value

deviation value - Links an interval deviation to its float value (in cents).

OWL Type:
DatatypeProperty
Domain:
tm:IntervalDeviation
tm:PitchClassDeviation
Range:
xsd:float

[back to top]

Instance: A

URI: http://purl.org/ontology/temperament/pitchclass/A

A - pitch class A

RDF Type:
tm:PitchClass

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Instance: Ab

URI: http://purl.org/ontology/temperament/pitchclass/Ab

Ab - pitch class Ab

RDF Type:
tm:PitchClass

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Instance: As

URI: http://purl.org/ontology/temperament/pitchclass/As

As - pitch class As

RDF Type:
tm:PitchClass

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Instance: B

URI: http://purl.org/ontology/temperament/pitchclass/B

B - pitch class B

RDF Type:
tm:PitchClass

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Instance: Bb

URI: http://purl.org/ontology/temperament/pitchclass/Bb

Bb - pitch class Bb

RDF Type:
tm:PitchClass

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Instance: Bendeler

URI: http://purl.org/ontology/temperament/Bendeler

Bendeler -

RDF Type:
tm:WellTemperament

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Instance: Bs

URI: http://purl.org/ontology/temperament/pitchclass/Bs

Bs - pitch class Bs

RDF Type:
tm:PitchClass

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Instance: C

URI: http://purl.org/ontology/temperament/pitchclass/C

C - pitch class C

RDF Type:
tm:PitchClass

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Instance: Cb

URI: http://purl.org/ontology/temperament/pitchclass/Cb

Cb - pitch class Cb

RDF Type:
tm:PitchClass

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Instance: Cs

URI: http://purl.org/ontology/temperament/pitchclass/Cs

Cs - pitch class Cs

RDF Type:
tm:PitchClass

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Instance: D

URI: http://purl.org/ontology/temperament/pitchclass/D

D - pitch class D

RDF Type:
tm:PitchClass

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Instance: Db

URI: http://purl.org/ontology/temperament/pitchclass/Db

Db - pitch class Db

RDF Type:
tm:PitchClass

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Instance: Diaschisma

URI: http://purl.org/ontology/temperament/Diaschisma

Diaschisma - One Diaschisma is 19.55 cents.

RDF Type:
tm:Comma

[back to top]

Instance: DidymusComma

URI: http://purl.org/ontology/temperament/DidymusComma

Didymus Comma - The Didymus is equivalent to the Syntonic comma (21.51 cents).

RDF Type:
tm:Comma

[back to top]

Instance: DitonicComma

URI: http://purl.org/ontology/temperament/DitonicComma

Ditonic Comma - The Ditonic comma is equivalent to the Pythagorean comma.

RDF Type:
tm:Comma

[back to top]

Instance: Ds

URI: http://purl.org/ontology/temperament/pitchclass/Ds

Ds - pitch class Ds

RDF Type:
tm:PitchClass

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Instance: E

URI: http://purl.org/ontology/temperament/pitchclass/E

E - pitch class E

RDF Type:
tm:PitchClass

[back to top]

Instance: Eb

URI: http://purl.org/ontology/temperament/pitchclass/Eb

Eb - pitch class Eb

RDF Type:
tm:PitchClass

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Instance: Es

URI: http://purl.org/ontology/temperament/pitchclass/Es

Es - pitch class Es

RDF Type:
tm:PitchClass

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Instance: F

URI: http://purl.org/ontology/temperament/pitchclass/F

F - pitch class F

RDF Type:
tm:PitchClass

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Instance: Fb

URI: http://purl.org/ontology/temperament/pitchclass/Fb

Fb - pitch class Fb

RDF Type:
tm:PitchClass

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Instance: FifthCommaMeantone

URI: http://purl.org/ontology/temperament/FifthCommaMeantone

FifthCommaMeantone -

RDF Type:
tm:MeantoneTemperament
tm:RegularTemperament

[back to top]

Instance: Fs

URI: http://purl.org/ontology/temperament/pitchclass/Fs

Fs - pitch class Fs

RDF Type:
tm:PitchClass

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Instance: G

URI: http://purl.org/ontology/temperament/pitchclass/G

G - pitch class G

RDF Type:
tm:PitchClass

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Instance: Gb

URI: http://purl.org/ontology/temperament/pitchclass/Gb

Gb - pitch class Gb

RDF Type:
tm:PitchClass

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Instance: Gs

URI: http://purl.org/ontology/temperament/pitchclass/Gs

Gs - pitch class Gs

RDF Type:
tm:PitchClass

[back to top]

Instance: KirnbergerIII

URI: http://purl.org/ontology/temperament/KirnbergerIII

KirnbergerIII -

RDF Type:
tm:WellTemperament

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Instance: Neidhardt

URI: http://purl.org/ontology/temperament/Neidhardt

Neidhardt -

RDF Type:
tm:WellTemperament

[back to top]

Instance: PythagoreanComma

URI: http://purl.org/ontology/temperament/PythagoreanComma

Pythagorean Comma - One Pythagorean Comma, the difference between 7 octaves and 12 pure fifths, is 23.5 cents. It is equivalent to the Ditonic comma.

RDF Type:
tm:Comma

[back to top]

Instance: QuarterCommaMeantone

URI: http://purl.org/ontology/temperament/QuarterCommaMeantone

QuarterCommaMeantone -

RDF Type:
tm:MeantoneTemperament
tm:RegularTemperament

[back to top]

Instance: Schisma

URI: http://purl.org/ontology/temperament/Schisma

Schisma - One Schisma is 1.95 cents.

RDF Type:
tm:Comma

[back to top]

Instance: SixthCommaMeantone

URI: http://purl.org/ontology/temperament/SixthCommaMeantone

SixthCommaMeantone -

RDF Type:
tm:MeantoneTemperament
tm:RegularTemperament

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Instance: Stanhope

URI: http://purl.org/ontology/temperament/Stanhope

Stanhope -

RDF Type:
tm:WellTemperament

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Instance: SyntonicComma

URI: http://purl.org/ontology/temperament/SyntonicComma

Syntonic Comma - One Syntonic Comma is 21.51 cents.

RDF Type:
tm:Comma

[back to top]

Instance: Valotti

URI: http://purl.org/ontology/temperament/Valotti

Valotti -

RDF Type:
tm:WellTemperament

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Instance: WerckmeisterIII

URI: http://purl.org/ontology/temperament/WerckmeisterIII

WerckmeisterIII -

RDF Type:
tm:WellTemperament

[back to top]

Instance: YoungII

URI: http://purl.org/ontology/temperament/YoungII

YoungII -

RDF Type:
tm:WellTemperament

[back to top]

...

A References

Music Ontology
The Music Ontology is a thorough and mature ontology for describing music-related data, including musicological information. This ontology is intended to be used in conjunction with the Music Ontology.

B Changes in this version (Non-Normative)

C Acknowledgements (Non-Normative)

Some modelling concepts in this ontology were borrowed form the Chord Ontology and the Music Similarity Ontology.