Minimum-ambiguity DE scales
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These [[distributionally even]] scales have maximum distinguishability, where "distinguishability" is defined as the minimum separation between different notes or intervals. If "ambiguity" (confusability) is defined as the inverse of distinguishability, then they have minimum ambiguity. The scale that has minimum ambiguity between notes is always an [[equal]] scale. The DE scale of N notes that has strictly minimum ambiguity between intervals is always a subscale of (2N-P)-EDO, where P is the number of periods per octave. (Octave-repeating scales are assumed here, but in fact the minimum ambiguity property is insensitive to any stretching or compressing, so the same scale patterns apply as equal divisions of any interval whatsoever.) Scales are also listed that have some totally ambiguous (coinciding) intervals, but have minimum ambiguity over all other intervals. ==3 notes== Minimum ambiguity between notes: [[3edo]] Minimum ambiguity between intervals: 3-note DE scales in [[5edo]] * 1 2 2 * 1 1 3 One ambiguous interval allowed: 1 1 2 in [[4edo]] ==4 notes== Minimum ambiguity between notes: [[4edo]] (always N-edo for N notes; hereafter omitted) Minimum ambiguity between intervals: * 2 periods per octave in [[6edo]] ** 1 2 1 2 * 1 period per octave in [[7edo]] ** 1 2 2 2 (a [[neutral third scale]]) ** 1 1 1 4 One ambiguous interval allowed: * 1 period per octave in [[6edo]] ** 1 1 1 3 ==5 notes== Minimum ambiguity between intervals: [[9edo]] * 1 2 2 2 2 ([[superpelog]]-like) * 1 1 3 1 3 ([[mavila]]- or [[pelog]]-like) * 1 1 1 1 5 One ambiguous interval allowed: [[8edo]] * 1 2 1 2 2 ([[father]]-like) * 1 1 1 1 4 Two ambiguous intervals allowed: [[7edo]] * 1 1 2 1 2 (2s ambiguous with L, 2L+s ambiguous with L+3s) * 1 1 1 1 3 ==6 notes== Minimum ambiguity between intervals: * 3 periods per octave in [[9edo]] ** 1 2 1 2 1 2 * 2 periods per octave in [[10edo]] ** 1 2 2 1 2 2 ([[decimal]]-like) ** 1 1 3 1 1 3 * 1 period per octave in [[11edo]] ** 1 2 2 2 2 2 ([[machine]]-like) ** 1 1 1 1 1 6 ==7 notes== Minimum ambiguity between intervals: [[13edo]] * 1 2 2 2 2 2 2 ([[13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches|archeotonic]]) * 1 1 3 1 3 1 3 * 1 1 1 4 1 1 4 * 1 1 1 1 1 1 7 One ambiguous interval allowed: [[12edo]] * 1 2 2 1 2 2 2 (familiar [[meantone]]-like diatonic scale) * 1 1 1 1 1 1 6 Two ambiguous intervals allowed: [[11edo]]) * 1 2 1 2 1 2 2 ([[orgone]]-like) * 1 1 1 3 1 1 3 * 1 1 1 1 1 1 5
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<html><head><title>Minimum-ambiguity DE scales</title></head><body>These <a class="wiki_link" href="/distributionally%20even">distributionally even</a> scales have maximum distinguishability, where "distinguishability" is defined as the minimum separation between different notes or intervals. If "ambiguity" (confusability) is defined as the inverse of distinguishability, then they have minimum ambiguity.<br /> <br /> The scale that has minimum ambiguity between notes is always an <a class="wiki_link" href="/equal">equal</a> scale.<br /> <br /> The DE scale of N notes that has strictly minimum ambiguity between intervals is always a subscale of (2N-P)-EDO, where P is the number of periods per octave. (Octave-repeating scales are assumed here, but in fact the minimum ambiguity property is insensitive to any stretching or compressing, so the same scale patterns apply as equal divisions of any interval whatsoever.)<br /> <br /> Scales are also listed that have some totally ambiguous (coinciding) intervals, but have minimum ambiguity over all other intervals.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-3 notes"></a><!-- ws:end:WikiTextHeadingRule:0 -->3 notes</h2> Minimum ambiguity between notes: <a class="wiki_link" href="/3edo">3edo</a><br /> Minimum ambiguity between intervals: 3-note DE scales in <a class="wiki_link" href="/5edo">5edo</a><br /> <ul><li>1 2 2</li><li>1 1 3</li></ul>One ambiguous interval allowed: 1 1 2 in <a class="wiki_link" href="/4edo">4edo</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-4 notes"></a><!-- ws:end:WikiTextHeadingRule:2 -->4 notes</h2> Minimum ambiguity between notes: <a class="wiki_link" href="/4edo">4edo</a> (always N-edo for N notes; hereafter omitted)<br /> Minimum ambiguity between intervals:<br /> <ul><li>2 periods per octave in <a class="wiki_link" href="/6edo">6edo</a><ul><li>1 2 1 2</li></ul></li><li>1 period per octave in <a class="wiki_link" href="/7edo">7edo</a><ul><li>1 2 2 2 (a <a class="wiki_link" href="/neutral%20third%20scale">neutral third scale</a>)</li><li>1 1 1 4</li></ul></li></ul>One ambiguous interval allowed:<br /> <ul><li>1 period per octave in <a class="wiki_link" href="/6edo">6edo</a><ul><li>1 1 1 3</li></ul></li></ul><br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-5 notes"></a><!-- ws:end:WikiTextHeadingRule:4 -->5 notes</h2> Minimum ambiguity between intervals: <a class="wiki_link" href="/9edo">9edo</a><br /> <ul><li>1 2 2 2 2 (<a class="wiki_link" href="/superpelog">superpelog</a>-like)</li><li>1 1 3 1 3 (<a class="wiki_link" href="/mavila">mavila</a>- or <a class="wiki_link" href="/pelog">pelog</a>-like)</li><li>1 1 1 1 5</li></ul>One ambiguous interval allowed: <a class="wiki_link" href="/8edo">8edo</a><br /> <ul><li>1 2 1 2 2 (<a class="wiki_link" href="/father">father</a>-like)</li><li>1 1 1 1 4</li></ul>Two ambiguous intervals allowed: <a class="wiki_link" href="/7edo">7edo</a><br /> <ul><li>1 1 2 1 2 (2s ambiguous with L, 2L+s ambiguous with L+3s)</li><li>1 1 1 1 3</li></ul><br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-6 notes"></a><!-- ws:end:WikiTextHeadingRule:6 -->6 notes</h2> Minimum ambiguity between intervals:<br /> <ul><li>3 periods per octave in <a class="wiki_link" href="/9edo">9edo</a><ul><li>1 2 1 2 1 2</li></ul></li><li>2 periods per octave in <a class="wiki_link" href="/10edo">10edo</a><ul><li>1 2 2 1 2 2 (<a class="wiki_link" href="/decimal">decimal</a>-like)</li><li>1 1 3 1 1 3</li></ul></li><li>1 period per octave in <a class="wiki_link" href="/11edo">11edo</a><ul><li>1 2 2 2 2 2 (<a class="wiki_link" href="/machine">machine</a>-like)</li><li>1 1 1 1 1 6</li></ul></li></ul><br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-7 notes"></a><!-- ws:end:WikiTextHeadingRule:8 -->7 notes</h2> Minimum ambiguity between intervals: <a class="wiki_link" href="/13edo">13edo</a><br /> <ul><li>1 2 2 2 2 2 2 (<a class="wiki_link" href="/13edo#Igliashon%27s%2013-EDO%20diatonic%20approaches">archeotonic</a>)</li><li>1 1 3 1 3 1 3</li><li>1 1 1 4 1 1 4</li><li>1 1 1 1 1 1 7</li></ul>One ambiguous interval allowed: <a class="wiki_link" href="/12edo">12edo</a><br /> <ul><li>1 2 2 1 2 2 2 (familiar <a class="wiki_link" href="/meantone">meantone</a>-like diatonic scale)</li><li>1 1 1 1 1 1 6</li></ul>Two ambiguous intervals allowed: <a class="wiki_link" href="/11edo">11edo</a>)<br /> <ul><li>1 2 1 2 1 2 2 (<a class="wiki_link" href="/orgone">orgone</a>-like)</li><li>1 1 1 3 1 1 3</li><li>1 1 1 1 1 1 5</li></ul></body></html>