Keenan's comma pump page: Difference between revisions

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:keenanpepper|keenanpepper]] and made on <tt>2011-08-26 01:33:24 UTC</tt>.<br>

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-26 21:02:25 UTC</tt>.<br>

: The original revision id was <tt>~~248581957~~</tt>.<br>

: The original revision id was <tt>248734771</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>

<h4>Original Wikitext content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Candidate 5-limit commas:

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Why do we temper intervals? The only truly essential reasons are [[Why Microtonality?#Why Microtonality?-5. For "puns"|puns]] and [[comma pump]]s.

Comma pumps are essential to many kinds of harmony. If you don't believe me, try to retune "I've Got Rhythm" to just intonation.

If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.

Good commas to use for comma pumps should be fairly small (here I'm using some arbitrary comma size cutoff), and also simple, i.e. have numerator and denominator that are not too large. [[81_80|81/80]] is an ideal comma pump comma in many ways.

==Candidate 5-limit commas==

81/80 (meantone)

128/125 (augmented)

Line 22:

Line 30:

32805/32768 (helmholtz)

Certain comma pumps, for example diminished, occur often enough in 12edo common practice music. Others, such as ripple or helmholtz, do not occur significantly in common practice music, but since they are compatible with 12edo they do not violate an internal 12edo-based perception. While such comma pumps could definitely be useful (especially srutal when used in conjunction with decatonic scales), I will limit my first investigations to commas that are incompatible with 12-equal and force a perceptual shift.

5-limit commas NOT tempered out by 12-equal:

135/128 (mavila)

Line 33:

Line 42:

This seems like a really good list.

Candidate 7-limit commas:

==Mavila==

135/128 = 2/1 * 5/4 / (3/4)^3

The simplest mavila comma pumps involve 3 root motions by 4/3 and one by 5/4. Bringing 6/5 into the picture can only make the progressions more complicated.

In this way mavila comma pumps have exactly the same structure as meantone comma pumps, except with the "wrong" kinds of thirds (6/5 and 5/4 are swapped). This is an example of a well-known mapping between mavila and meantone (see [[http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments]]).

==Porcupine==

250/243 = (4/3)^2 / (6/5)^3

The simplest porcupine comma pumps involve 2 root motions by 4/3 and three by 6/5. Bringing 5/4 into the picture only makes things more complicated.

==Blackwood==

256/243 = (4/3)^5 / (2/1)^2

Blackwood comma pumps involve 5 motions by 4/3 around a "circle of fifths" that's about as small as it can reasonably get.

==Candidate 7-limit commas==

49/48 (2.3.7 semaphore)

50/49 (jubilisma)

Line 123:

Line 146:

33614/32805</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Keenan's comma pump page</title></head><body>Candidate 5-limit commas:<br />

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Keenan's comma pump page</title></head><body>Why do we temper intervals? The only truly essential reasons are <a class="wiki_link" href="/Why%20Microtonality%3F#Why Microtonality?-5. For &quot;puns&quot;">puns</a> and <a class="wiki_link" href="/comma%20pump">comma pump</a>s.<br />

81/80 (meantone)<br />

<br />

Comma pumps are essential to many kinds of harmony. If you don't believe me, try to retune &quot;I've Got Rhythm&quot; to just intonation.<br />

<br />

If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.<br />

<br />

Good commas to use for comma pumps should be fairly small (here I'm using some arbitrary comma size cutoff), and also simple, i.e. have numerator and denominator that are not too large. <a class="wiki_link" href="/81_80">81/80</a> is an ideal comma pump comma in many ways.<br />

<br />

<h2 id="toc0"><a name="x-Candidate 5-limit commas"></a>Candidate 5-limit commas</h2>

81/80 (meantone)<br />

128/125 (augmented)<br />

135/128 (mavila)<br />

Line 139:

Line 170:

32805/32768 (helmholtz)<br />

<br />

Certain comma pumps, for example diminished, occur often enough in 12edo common practice music. Others, such as ripple or helmholtz, do not occur significantly in common practice music, but since they are compatible with 12edo they do not violate an internal 12edo-based perception. While such comma pumps could definitely be useful (especially srutal when used in conjunction with decatonic scales), I will limit my first investigations to commas that are incompatible with 12-equal and force a perceptual shift.<br />

5-limit commas NOT tempered out by 12-equal:<br />

135/128 (mavila)<br />

Line 150:

Line 182:

This seems like a really good list.<br />

<br />

Candidate 7-limit commas:<br />

<h2 id="toc1"><a name="x-Mavila"></a>Mavila</h2>

49/48 (2.3.7 semaphore)<br />

135/128 = 2/1 * 5/4 / (3/4)^3<br />

The simplest mavila comma pumps involve 3 root motions by 4/3 and one by 5/4. Bringing 6/5 into the picture can only make the progressions more complicated.<br />

<br />

In this way mavila comma pumps have exactly the same structure as meantone comma pumps, except with the &quot;wrong&quot; kinds of thirds (6/5 and 5/4 are swapped). This is an example of a well-known mapping between mavila and meantone (see <a class="wiki_link_ext" href="http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments" rel="nofollow">http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments</a>).<br />

<br />

<h2 id="toc2"><a name="x-Porcupine"></a>Porcupine</h2>

250/243 = (4/3)^2 / (6/5)^3<br />

The simplest porcupine comma pumps involve 2 root motions by 4/3 and three by 6/5. Bringing 5/4 into the picture only makes things more complicated.<br />

<br />

<h2 id="toc3"><a name="x-Blackwood"></a>Blackwood</h2>

256/243 = (4/3)^5 / (2/1)^2<br />

Blackwood comma pumps involve 5 motions by 4/3 around a &quot;circle of fifths&quot; that's about as small as it can reasonably get.<br />

<br />

<h2 id="toc4"><a name="x-Candidate 7-limit commas"></a>Candidate 7-limit commas</h2>

49/48 (2.3.7 semaphore)<br />

50/49 (jubilisma)<br />

64/63 (archytas)<br />

@@ 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:keenanpepper|keenanpepper]] and made on <tt>2011-08-26 01:33:24 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-08-26 21:02:25 UTC</tt>.<br>
-: The original revision id was <tt>248581957</tt>.<br>
+: The original revision id was <tt>248734771</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>
 <h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Candidate 5-limit commas:
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Why do we temper intervals? The only truly essential reasons are [[Why Microtonality?#Why Microtonality?-5. For "puns"|puns]] and [[comma pump]]s.
+Comma pumps are essential to many kinds of harmony. If you don't believe me, try to retune "I've Got Rhythm" to just intonation.
+If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.
+Good commas to use for comma pumps should be fairly small (here I'm using some arbitrary comma size cutoff), and also simple, i.e. have numerator and denominator that are not too large. [[81_80|81/80]] is an ideal comma pump comma in many ways.
+==Candidate 5-limit commas==
 /80 (meantone)
 /125 (augmented)
@@ Line 22: / Line 30: @@
 /32768 (helmholtz)
+Certain comma pumps, for example diminished, occur often enough in 12edo common practice music. Others, such as ripple or helmholtz, do not occur significantly in common practice music, but since they are compatible with 12edo they do not violate an internal 12edo-based perception. While such comma pumps could definitely be useful (especially srutal when used in conjunction with decatonic scales), I will limit my first investigations to commas that are incompatible with 12-equal and force a perceptual shift.
 -limit commas NOT tempered out by 12-equal:
 /128 (mavila)
@@ Line 33: / Line 42: @@
 This seems like a really good list.
-Candidate 7-limit commas:
+==Mavila==
+/128 = 2/1 * 5/4 / (3/4)^3
+The simplest mavila comma pumps involve 3 root motions by 4/3 and one by 5/4. Bringing 6/5 into the picture can only make the progressions more complicated.
+In this way mavila comma pumps have exactly the same structure as meantone comma pumps, except with the "wrong" kinds of thirds (6/5 and 5/4 are swapped). This is an example of a well-known mapping between mavila and meantone (see [[http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments]]).
+==Porcupine==
+/243 = (4/3)^2 / (6/5)^3
+The simplest porcupine comma pumps involve 2 root motions by 4/3 and three by 6/5. Bringing 5/4 into the picture only makes things more complicated.
+==Blackwood==
+/243 = (4/3)^5 / (2/1)^2
+Blackwood comma pumps involve 5 motions by 4/3 around a "circle of fifths" that's about as small as it can reasonably get.
+==Candidate 7-limit commas==
 /48 (2.3.7 semaphore)
 /49 (jubilisma)
@@ Line 123: / Line 146: @@
 /32805</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Keenan's comma pump page&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Candidate 5-limit commas:&lt;br /&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Keenan's comma pump page&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Why do we temper intervals? The only truly essential reasons are &lt;a class="wiki_link" href="/Why%20Microtonality%3F#Why Microtonality?-5. For &amp;quot;puns&amp;quot;"&gt;puns&lt;/a&gt; and &lt;a class="wiki_link" href="/comma%20pump"&gt;comma pump&lt;/a&gt;s.&lt;br /&gt;
-/80 (meantone)&lt;br /&gt;
+&lt;br /&gt;
+Comma pumps are essential to many kinds of harmony. If you don't believe me, try to retune &amp;quot;I've Got Rhythm&amp;quot; to just intonation.&lt;br /&gt;
+&lt;br /&gt;
+If we want to truly understand different temperament systems and make beautiful music with them, we must understand comma pumps.&lt;br /&gt;
+&lt;br /&gt;
+Good commas to use for comma pumps should be fairly small (here I'm using some arbitrary comma size cutoff), and also simple, i.e. have numerator and denominator that are not too large. &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt; is an ideal comma pump comma in many ways.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Candidate 5-limit commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Candidate 5-limit commas&lt;/h2&gt;
+/80 (meantone)&lt;br /&gt;
 /125 (augmented)&lt;br /&gt;
 /128 (mavila)&lt;br /&gt;
@@ Line 139: / Line 170: @@
 /32768 (helmholtz)&lt;br /&gt;
 &lt;br /&gt;
+Certain comma pumps, for example diminished, occur often enough in 12edo common practice music. Others, such as ripple or helmholtz, do not occur significantly in common practice music, but since they are compatible with 12edo they do not violate an internal 12edo-based perception. While such comma pumps could definitely be useful (especially srutal when used in conjunction with decatonic scales), I will limit my first investigations to commas that are incompatible with 12-equal and force a perceptual shift.&lt;br /&gt;
 -limit commas NOT tempered out by 12-equal:&lt;br /&gt;
 /128 (mavila)&lt;br /&gt;
@@ Line 150: / Line 182: @@
 This seems like a really good list.&lt;br /&gt;
 &lt;br /&gt;
-Candidate 7-limit commas:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Mavila"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Mavila&lt;/h2&gt;
-/48 (2.3.7 semaphore)&lt;br /&gt;
+/128 = 2/1 * 5/4 / (3/4)^3&lt;br /&gt;
+The simplest mavila comma pumps involve 3 root motions by 4/3 and one by 5/4. Bringing 6/5 into the picture can only make the progressions more complicated.&lt;br /&gt;
+&lt;br /&gt;
+In this way mavila comma pumps have exactly the same structure as meantone comma pumps, except with the &amp;quot;wrong&amp;quot; kinds of thirds (6/5 and 5/4 are swapped). This is an example of a well-known mapping between mavila and meantone (see &lt;a class="wiki_link_ext" href="http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments" rel="nofollow"&gt;http://soundcloud.com/mikebattagliamusic/sets/the-mavila-experiments&lt;/a&gt;).&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Porcupine"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Porcupine&lt;/h2&gt;
+/243 = (4/3)^2 / (6/5)^3&lt;br /&gt;
+The simplest porcupine comma pumps involve 2 root motions by 4/3 and three by 6/5. Bringing 5/4 into the picture only makes things more complicated.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Blackwood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Blackwood&lt;/h2&gt;
+/243 = (4/3)^5 / (2/1)^2&lt;br /&gt;
+Blackwood comma pumps involve 5 motions by 4/3 around a &amp;quot;circle of fifths&amp;quot; that's about as small as it can reasonably get.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x-Candidate 7-limit commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Candidate 7-limit commas&lt;/h2&gt;
+/48 (2.3.7 semaphore)&lt;br /&gt;
 /49 (jubilisma)&lt;br /&gt;
 /63 (archytas)&lt;br /&gt;