Just intonation: Difference between revisions

Line 6:

}}

== Just Intonation explained ==

Just Intonation (JI) describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of [[Wikipedia: Rational number|rational numbers]]) between the frequencies of pitches<ref>. Just Intonation is sometimes distinguished from ''rational intonation'', by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]], [[Benedetti height]], etc.) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit [[microtempering]] system because of very accurate approximations to the octave and to seven limit intervals: (6/5)2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)9 is less than a cent short of an octave, while (27/25)2 is almost precisely 7/6, the septimal minor third.</ref>

Just Intonation (JI) describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of [[Wikipedia: Rational number|rational numbers]]) between the frequencies of pitches<ref>Just Intonation is sometimes distinguished from ''rational intonation'', by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]], [[Benedetti height]], etc.) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit [[microtempering]] system because of very accurate approximations to the octave and to seven limit intervals: (6/5)2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)9 is less than a cent short of an octave, while (27/25)2 is almost precisely 7/6, the septimal minor third.</ref>.

If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), you may need to study the relation between frequency and [[Wikipedia: Pitch (music)|pitch]]. Kyle Gann's ''[http://www.kylegann.com/tuning.html Just Intonation Explained]'' is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].

Line 22:

If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...

The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.<ref>All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.</ref>

The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum<ref>All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.</ref>.

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.

@@ Line 6: / Line 6: @@
 }}
 == Just Intonation explained ==
-Just Intonation (JI) describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of [[Wikipedia: Rational number|rational numbers]]) between the frequencies of pitches<ref>. Just Intonation is sometimes distinguished from ''rational intonation'', by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]], [[Benedetti height]], etc.) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit [[microtempering]] system because of very accurate approximations to the octave and to seven limit intervals: (6/5)<sup>2</sup>/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)<sup>9</sup> is less than a cent short of an octave, while (27/25)<sup>2</sup> is almost precisely 7/6, the septimal minor third.</ref>
+Just Intonation (JI) describes [[Gallery of Just Intervals|intervals]] between pitches by specifying ratios (of [[Wikipedia: Rational number|rational numbers]]) between the frequencies of pitches<ref>Just Intonation is sometimes distinguished from ''rational intonation'', by requiring that the ratios be lower than some arbitrary complexity (as for example measured by [[Tenney height]], [[Benedetti height]], etc.) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit [[microtempering]] system because of very accurate approximations to the octave and to seven limit intervals: (6/5)<sup>2</sup>/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)<sup>9</sup> is less than a cent short of an octave, while (27/25)<sup>2</sup> is almost precisely 7/6, the septimal minor third.</ref>.
 If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), you may need to study the relation between frequency and [[Wikipedia: Pitch (music)|pitch]]. Kyle Gann's ''[http://www.kylegann.com/tuning.html Just Intonation Explained]'' is one good reference. A transparent illustration and one of just intonation's acoustic bases is the [[OverToneSeries|harmonic series]].
@@ Line 22: / Line 22: @@
 If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...
-The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum.<ref>All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.</ref>
+The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum<ref>All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.</ref>.
 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.