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    <title>Temperament Ontology: Temperament Ontology</title>
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    <h1 id="title">The Temperament Ontology</h1>

    <!--<h2 id="subtitle">LONG Temperament </h2>-->

    <h3 id="mymw-doctype">Working Draft &mdash; 18 Dec 2009</h3>

    <dl>

      <dt>This version:</dt>
      <dd><a href="http://purl.org/ontology/temperament/draft1-20091218/">http://purl.org/ontology/temperament/draft1-20091218/</a></dd>

      <dt>Latest version:</dt>
      <dd><a href="http://purl.org/ontology/temperament/">http://purl.org/ontology/temperament/</a></dd>

      <dt>Last Update: 18. Dec. 2009.</dt>
      <dd>Date: 18. Dec. 2009.</dd>

      <dt>Editors:</dt>
      <dd>Gyorgy Fazekas, Centre for Digital Music Queen Mary University of London</dd>

      <dt>Authors:</dt>
      <dd>Gyorgy Fazekas, Dan Tidhar, Centre for Digital Music Queen Mary University of London</dd>

      <dt>Contributors:</dt>
      <dd>See <a href="#acknowledgements">acknowledgements</a></dd>

    </dl>

    <p class="copyright">You are granted a license to use, reproduce and create derivative works of this document under <a rel="license" href="http://creativecommons.org/licenses/by/3.0/">Creative Commons</a>.</p>

    <!-- ===================================================================== -->

    <hr />

    <h2 id="abstract">Abstract</h2>
    <p>
		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 <em><a href="http://purl.org/ontology/mo/">Music Ontology</a></em>, 
		for example, when describing the tuning that was used in a particular harpsichord recording.
    </p>

    <h2 id="status">Status of this Document</h2>
    <p>
      <strong>This is a work in progress!</strong> This document is changing
      on a daily if not hourly basis. Comments are very welcome, please send
      them to <a href="mailto:gyorgy.fazekas@elec.qmul.ac.uk">gyorgy.fazekas@elec.qmul.ac.uk</a>. Thank you.
    </p>

    <h2 id="contents">Table of Contents</h2>
    <ol id="toc">
      <li><a href="#intro">Introduction</a>
        <ol>
          <li><a href="#s11">About instrument tuning</a></li>
          <li><a href="#s12">An open-ended temperament description model</a></li>
          <li>
            <a href="#s13">The many ways of describing temperaments</a>
            <ol>
              <li><a href="#s131">The Circle of Fifths</a></li>
              <li><a href="#s132">Deviation from Equal Temperament</a></li>
            </ol>
          </li>
        </ol>
      </li>
      <li><a href="#sec-glance">Temperament  ontology at a glance</a></li>
      <li>
        <a href="#overview">Temperament  ontology overview</a>
        <ol>
          <li><a href="#s31">Example</a></li>
        </ol>
      </li>
      <li><a href="#reference">Cross-reference for Temperament  classes and properties</a></li>
    </ol>

    <h3>Appendixes</h3>
    <ol id="appendix">
      <li><a href="#references">Normative References</a></li>
      <li><a href="#changes">Changes in this version</a> (Non-Normative)</li>
      <li><a href="#acknowledgements">Acknowledgements</a> (Non-Normative)</li>
    </ol>

    <hr />

    <!-- ===================================================================== -->

    <h2 id="intro">1 Introduction</h2>

    <p>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. <br><br>
	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.
	</p>

    <h3 id="s11">1.1 About instrument tuning</h3>

    <p>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. </p>


    <h3 id="s12">1.2 An open-ended temperament description model</h3>
	<p>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.</p>

	<p><center><img src="TemperamentDescription.png" alt="Using the Circle of Fifths to describe a temperament." width="850">
		<br><br>Figure 1. Overview of the temperament ontology.</center></p>


    <h3 id="s13">1.3 The many ways of describing temperaments</h3>

    <p>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.)</p>

    <h4 id="s131">1.3.1 The Circle of Fifths</h4>
	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.)

	<!--<p><center><img src="CircleOfFifths.png" alt="Using the Circle of Fifths to describe a temperament." width="350"></center></p>-->

    <p>...</p>

    <h4 id="s132">1.3.2 Deviations from Equal Temperament</h4>

    <p>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.</p>

    <!-- ===================================================================== -->

    <h2 id="sec-glance">2. Temperament  ontology at a glance</h2>

    <p>
      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.
    </p>

    <!-- The list of classes and properties goes here -->
    %s
    <!-- End of the terms list -->

    <p>...</p>

    <!-- ===================================================================== -->

    <h2 id="overview">3. Temperament  ontology overview</h2>

    <!--<p>The Temperament definitions presented here are ...</p>-->

    <h3 id="s31">3.1. Example</h3>

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

    <div class="example" style="white-space: pre-wrap;">
      <pre>
		@prefix rdf: &lt;http://www.w3.org/1999/02/22-rdf-syntax-ns#&gt;.
		@prefix rdfs: &lt;http://www.w3.org/2000/01/rdf-schema#&gt;.
		@prefix xsd: &lt;http://www.w3.org/2001/XMLSchema#&gt; .
		@prefix tm: &lt;http://purl.org/ontology/temperament/&gt; .
		@prefix pc: &lt;http://purl.org/ontology/temperament/pitchclass/&gt; .

		: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 
				] .
      </pre>
    </div>

    <p>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.</p>

	<p><center><img src="CircleOfFifths.png" alt="Using the Circle of Fifths to describe a temperament." width="350">
		<br><br>Figure 2. Describing temperaments using the circle of fifths.</center></p>


    <!-- ===================================================================== -->

    <h2 id="reference">4. Cross-reference for Temperament  classes and properties</h2>

    <p>...</p>

    <!-- The following is the script-generated list of classes and properties -->
    %s
    <!-- End of the terms list -->

    <p>...</p>

    <!-- ===================================================================== -->

    <h2 id="references">A References</h2>

    <dl>
      <dt class="label" id="ref-some">Music Ontology</dt>
      <dd><em>The <a href="http://purl.org/ontology/mo/">Music Ontology</a></em> 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.</dd>
    </dl>

    <!--
    <h3 id="references-other">A.2 Other references</h3>

    <dl>
      <dt class="label" id="ref-rfc2119">IETF RFC 2119</dt>
      <dd>
        <em><a href="http://www.ietf.org/rfc/rfc2119.txt">RFC 2119: Key words
        for use in RFCs to Indicate Requirement Levels</a></em>. Internet
        Engineering Task Force, 1997 (see http://www.ietf.org/rfc/rfc2119.txt).
      </dd>
    </dl>
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    <h2 id="changes">B Changes in this version (Non-Normative)</h2>

    <ul>
      <li><p>first published draft 18-12-2009</p></li> 
    </ul>

    <h2><a id="acknowledgements"></a>C Acknowledgements (Non-Normative)</h2>

    <p>Some modelling concepts in this ontology were borrowed form the
	<a href="http://purl.org/ontology/chord/">Chord Ontology</a></em>
	and the 
	<a href="http://purl.org/ontology/similarity/">Music Similarity Ontology</a></em>.
	</p>

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