19edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~hearneg~~|~~hearneg~~]] and made on <tt>2016-09-~~08 11~~:45:20 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-09-21 22:15:43 UTC</tt>.

: The original revision id was <tt>~~591425460~~</tt>.

: The original revision id was <tt>593016010</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for [[xenharmonic/Meantone family|meantone]] temperament. It is also a suitable for [[xenharmonic/Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its //twelfths.// For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[xenharmonic/31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[xenharmonic/41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.

However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[xenharmonic/Harmonic Limit|5-limit]] music in a tolerable manner, and is the fifth (after 12) [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. It is less successful with [[xenharmonic/7-limit|7-limit]] (but still better than 12-et), as it eliminates the distinction between a septimal minor third ([[xenharmonic/7_6|7/6]]), and a septimal whole tone ([[xenharmonic/8_7|8/7]]). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The [[xenharmonic/Graham complexity|Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone~~/flattone~~, 11 for triton, 12 for magic/muggles and 13 for sensi.

However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[xenharmonic/Harmonic Limit|5-limit]] music in a tolerable manner, and is the fifth (after 12) [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. It is less successful with [[xenharmonic/7-limit|7-limit]] (but still better than 12-et), as it eliminates the distinction between a septimal minor third ([[xenharmonic/7_6|7/6]]), and a septimal whole tone ([[xenharmonic/8_7|8/7]]). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The [[xenharmonic/Graham complexity|Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the 13-limit is represented relatively well, and practically [[19edo|19-edo]] can be used //adaptively// on instruments which are allowing you to bend notes up: by different amounts, the 3rd, 5th, 7th and 13th harmonics are all tuned flat. The same cannot be said of 12edo, in which the 5th, 7th and 11th are - not only farther than they are in 19, but fairly sharp already. Another option would be to use a stretched octave; the [[The Riemann Zeta Function and Tuning|zeta function]]-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them.

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The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone%20family">meantone</a> temperament. It is also a suitable for <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Regular%20Temperaments#magic">magic/muggles</a> temperament, because five of its major thirds are equivalent to one of its twelfths. For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is <a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo">31 equal temperament</a>. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; <a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo">41 equal temperament</a> more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.

However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Harmonic%20Limit">5-limit</a> music in a tolerable manner, and is the fifth (after 12) <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. It is less successful with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit">7-limit</a> (but still better than 12-et), as it eliminates the distinction between a septimal minor third (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/7_6">7/6</a>), and a septimal whole tone (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/8_7">8/7</a>). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Graham%20complexity">Graham complexity</a> of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone~~/flattone~~, 11 for triton, 12 for magic/muggles and 13 for sensi.

However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Harmonic%20Limit">5-limit</a> music in a tolerable manner, and is the fifth (after 12) <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. It is less successful with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit">7-limit</a> (but still better than 12-et), as it eliminates the distinction between a septimal minor third (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/7_6">7/6</a>), and a septimal whole tone (<a class="wiki_link" href="http://xenharmonic.wikispaces.com/8_7">8/7</a>). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Graham%20complexity">Graham complexity</a> of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the 13-limit is represented relatively well, and practically <a class="wiki_link" href="/19edo">19-edo</a> can be used adaptively on instruments which are allowing you to bend notes up: by different amounts, the 3rd, 5th, 7th and 13th harmonics are all tuned flat. The same cannot be said of 12edo, in which the 5th, 7th and 11th are - not only farther than they are in 19, but fairly sharp already. Another option would be to use a stretched octave; the <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning">zeta function</a>-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2016-09-08 11:45:20 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2016-09-21 22:15:43 UTC</tt>.<br>
-: The original revision id was <tt>591425460</tt>.<br>
+: The original revision id was <tt>593016010</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 16: / Line 16: @@
 The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for [[xenharmonic/Meantone family|meantone]] temperament. It is also a suitable for [[xenharmonic/Regular Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its //twelfths.// For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[xenharmonic/31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[xenharmonic/41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.
-However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[xenharmonic/Harmonic Limit|5-limit]] music in a tolerable manner, and is the fifth (after 12) [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. It is less successful with [[xenharmonic/7-limit|7-limit]] (but still better than 12-et), as it eliminates the distinction between a septimal minor third ([[xenharmonic/7_6|7/6]]), and a septimal whole tone ([[xenharmonic/8_7|8/7]]). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The [[xenharmonic/Graham complexity|Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone/flattone, 11 for triton, 12 for magic/muggles and 13 for sensi.
+However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[xenharmonic/Harmonic Limit|5-limit]] music in a tolerable manner, and is the fifth (after 12) [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. It is less successful with [[xenharmonic/7-limit|7-limit]] (but still better than 12-et), as it eliminates the distinction between a septimal minor third ([[xenharmonic/7_6|7/6]]), and a septimal whole tone ([[xenharmonic/8_7|8/7]]). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The [[xenharmonic/Graham complexity|Graham complexity]] of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.
 Being a zeta integral tuning, the 13-limit is represented relatively well, and practically [[19edo|19-edo]] can be used //adaptively// on instruments which are allowing you to bend notes up: by different amounts, the 3rd, 5th, 7th and 13th harmonics are all tuned flat. The same cannot be said of 12edo, in which the 5th, 7th and 11th are - not only farther than they are in 19, but fairly sharp already. Another option would be to use a stretched octave; the [[The Riemann Zeta Function and Tuning|zeta function]]-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them.
@@ Line 192: / Line 192: @@
 The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Meantone%20family"&gt;meantone&lt;/a&gt; temperament. It is also a suitable for &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Regular%20Temperaments#magic"&gt;magic/muggles&lt;/a&gt; temperament, because five of its major thirds are equivalent to one of its &lt;em&gt;twelfths.&lt;/em&gt; For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/31edo"&gt;31 equal temperament&lt;/a&gt;. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/41edo"&gt;41 equal temperament&lt;/a&gt; more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.&lt;br /&gt;
 &lt;br /&gt;
-However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Harmonic%20Limit"&gt;5-limit&lt;/a&gt; music in a tolerable manner, and is the fifth (after 12) &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;. It is less successful with &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit"&gt;7-limit&lt;/a&gt; (but still better than 12-et), as it eliminates the distinction between a septimal minor third (&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7_6"&gt;7/6&lt;/a&gt;), and a septimal whole tone (&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/8_7"&gt;8/7&lt;/a&gt;). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Graham%20complexity"&gt;Graham complexity&lt;/a&gt; of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone/flattone, 11 for triton, 12 for magic/muggles and 13 for sensi.&lt;br /&gt;
+However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Harmonic%20Limit"&gt;5-limit&lt;/a&gt; music in a tolerable manner, and is the fifth (after 12) &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;. It is less successful with &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7-limit"&gt;7-limit&lt;/a&gt; (but still better than 12-et), as it eliminates the distinction between a septimal minor third (&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7_6"&gt;7/6&lt;/a&gt;), and a septimal whole tone (&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/8_7"&gt;8/7&lt;/a&gt;). 19-EDO also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles and triton/liese, and fairly decent for sensi. Keemun and Negri are of particular note for being very simple 7-limit temperaments, with their MOS scales in 19-EDO offering a great abundance of septimal tetrads. The &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Graham%20complexity"&gt;Graham complexity&lt;/a&gt; of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.&lt;br /&gt;
 &lt;br /&gt;
 Being a zeta integral tuning, the 13-limit is represented relatively well, and practically &lt;a class="wiki_link" href="/19edo"&gt;19-edo&lt;/a&gt; can be used &lt;em&gt;adaptively&lt;/em&gt; on instruments which are allowing you to bend notes up: by different amounts, the 3rd, 5th, 7th and 13th harmonics are all tuned flat. The same cannot be said of 12edo, in which the 5th, 7th and 11th are - not only farther than they are in 19, but fairly sharp already. Another option would be to use a stretched octave; the &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning"&gt;zeta function&lt;/a&gt;-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them.&lt;br /&gt;