Just intonation: Difference between revisions

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In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit]] tuning. ''Extended just intonation'', a term coined by [[Ben Johnston]], usually refers to higher prime limits,<ref>[https://marsbat.space/pdfs/EJItext.pdf Sabat, Marc. ''On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation'']</ref> such as the [[7-limit]], the [[11-limit]] and the [[13-limit]]. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].

The structure of just intonation has several implications on music composition. [[Wolf interval]]s and [[comma]]s, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]] and [[temperament]].

The structure of just intonation has several implications on music composition. [[Wolf interval]]s and [[comma]]s, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include pitch shifts, [[adaptive just intonation]] and [[temperament]].

== Just intonation explained ==

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Of course we are describing an ideal tone - in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our "fuzzy" and "flawed" reality, spectra adhere to this pattern with impressive consistency.

In ~~a tuning "according to~~ the ~~natural scale", we have for example a "~~perfect fifth~~" as~~ simply the ratio between the third ~~partial and the second partial~~: "3~~:2"~~. In our example tone, that would be the ratio of ~~300~~ Hz to ~~200~~ Hz. ~~Were we to want a~~ just intonation perfect fifth above our original tone~~, its~~ fundamental frequency ~~would be found at~~ 3/2 times the fundamental frequency of our original tone. ~~So,~~ 3/2 times 100 gives us ~~150. Our example perfect fifth has a fundamental frequency at~~ 150 Hz.

In just intonation, the perfect fifth is simply the ratio between the second and third harmonics: 2:3. In our example tone, that would be the ratio of 200 Hz to 300 Hz. A just intonation perfect fifth above our original tone would have a fundamental frequency 3/2 times the fundamental frequency of our original tone. 3/2 times 100 Hz gives us 150 Hz.

Now, let us play our two example tones together, and we shall see why the German term is ''Reine'', "pure", and why you'll hear "pure" used in English and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "~~Sol~~".

Now, let us play our two example tones together, and we shall see why the German term is ''Reine'', "pure", and why you'll hear "pure" used in English and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "So".

<pre>Tone Frequencies of partials (Hz)

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So 150 300 450 600 750 900 1050 1200 1350 ...</pre>

You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of ''Tonverschmelzung'' in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).

You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of ''Tonverschmelzung'' in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, but would beat. That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).

One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.

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== Instruments ==

~~{{todo|expand|comment=Expand~~ the instruments ~~section with more~~ examples}}

Any tunable instrument can be tuned to just intonation, with the exception of fretted instruments like guitars, in which each string's notes have a "baked in" tuning. Even those instruments can be refretted to just intonation. A few noteworthy examples:

* The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a ~~versatile~~ just intonation instrument, covering a 5 octave range.

* All fretless string instruments.

* The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a just intonation instrument, covering a 5 octave range.

* Most of [[Harry Partch]]'s instruments were designed to be for just intonation.

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 In the context of Western music theory prior to the 20th century, the term ''just intonation'' used alone usually refers to [[5-limit]] tuning. ''Extended just intonation'', a term coined by [[Ben Johnston]], usually refers to higher prime limits,<ref>[https://marsbat.space/pdfs/EJItext.pdf Sabat, Marc. ''On Ben Johnston’s Notation and the Performance Practice of Extended Just Intonation'']</ref> such as the [[7-limit]], the [[11-limit]] and the [[13-limit]]. The practice of just intonation without any particular constraint is sometimes referred to as '''rational intonation''' ('''RI''') or as [[free style JI]].
-The structure of just intonation has several implications on music composition. [[Wolf interval]]s and [[comma]]s, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]] and [[temperament]].
+The structure of just intonation has several implications on music composition. [[Wolf interval]]s and [[comma]]s, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include pitch shifts, [[adaptive just intonation]] and [[temperament]].
 == Just intonation explained ==
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 Of course we are describing an ideal tone - in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our "fuzzy" and "flawed" reality, spectra adhere to this pattern with impressive consistency.
-In a tuning "according to the natural scale", we have for example a "perfect fifth" as simply the ratio between the third partial and the second partial: "3:2". In our example tone, that would be the ratio of 300 Hz to 200 Hz. Were we to want a just intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.
+In just intonation, the perfect fifth is simply the ratio between the second and third harmonics: 2:3. In our example tone, that would be the ratio of 200 Hz to 300 Hz. A just intonation perfect fifth above our original tone would have a fundamental frequency 3/2 times the fundamental frequency of our original tone. 3/2 times 100 Hz gives us 150 Hz.
-Now, let us play our two example tones together, and we shall see why the German term is ''Reine'', "pure", and why you'll hear "pure" used in English and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "Sol".
+Now, let us play our two example tones together, and we shall see why the German term is ''Reine'', "pure", and why you'll hear "pure" used in English and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "So".
 <pre>Tone   Frequencies of partials (Hz)
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 So     150  300  450  600  750  900  1050  1200  1350   ...</pre>
-You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of ''Tonverschmelzung'' in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).
+You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of ''Tonverschmelzung'' in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, but would beat. That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).
 One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.
@@ Line 76: / Line 76: @@
 == Instruments ==
-{{todo|expand|comment=Expand the instruments section with more examples}}
+Any tunable instrument can be tuned to just intonation, with the exception of fretted instruments like guitars, in which each string's notes have a "baked in" tuning. Even those instruments can be refretted to just intonation. A few noteworthy examples:
-* The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a versatile just intonation instrument, covering a 5 octave range.
+* All fretless string instruments.
+* The [[Kalimba#Array mbira|array mbira]] was designed by [[Bill Wesley]] as a just intonation instrument, covering a 5 octave range.
 * Most of [[Harry Partch]]'s instruments were designed to be for just intonation.