Omnitetrachordality
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A scale is **omnitetrachordal** if any mode of the scale (that is, any particular octave span of the infinite scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning [[4_3|4/3]], plus a [[9_8|9/8]] that may or may not be divided into smaller steps. The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.
This definition could be difficult to understand, so take the [[5L 2s|5L+2s]] diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.
C D E F G A B C = (C D E F) + 9/8 + (G A B C) (both tetrachords are LLs)
D E F G A B C D = (D E F G) + 9/8 + (A B C D) (both tetrachords are LsL)
E F G A B C D E = (E F G A) + 9/8 + (B C D E) (both tetrachords are sLL)
F G A B C D E F = 9/8 + (G A B C) + (C D E F) (both tetrachords are LLs)
G A B C D E F G = 9/8 + (A B C D) + (D E F G) (both tetrachords are LsL) //or alternatively// (G A B C) + (C D E F) + 9/8 (both tetrachords are LLs)
A B C D E F G A = 9/8 + (B C D E) + (E F G A) (both tetrachords are sLL) //or alternatively// (A B C D) + (D E F G) + 9/8 (both tetrachords are LsL)
B C D E F G A B = (B C D E) + (E F G A) + 9/8 (both tetrachords are sLL)
Since each mode can be expressed as two tetrachords each spanning 4/3 and a leftover 9/8 (some in more than one way), the diatonic scale is omnitetrachordal.
If you understand [[MOSScales|MOS scales]] well it should be clear that any MOS of a temperament in which the period represents 2/1 and the generator represents 4/3 (including [[meantone]], [[mavila]], [[superpyth]], [[Schismatic family|schismatic]], etc.) will be omnitetrachordal. However, these are not the only possible omnitetrachordal scales. For an example of a different kind of omnitetrachordal scale, take the [[MODMOS]] of the [[2L 8s|2L+8s]] scale (in [[pajara]] for example) with the pattern LsssLsssss.
(Lsss)(Lsss)(ss)
(sssL)(ss)(sssL)
(ssLs)(ss)(ssLs)
(sLss)(ss)(sLss)
(Lsss)(ss)(Lsss)
(ss)(sssL)(sssL)
(ss)(ssLs)(ssLs)
(ss)(sLss)(sLss) OR (sssL)(sssL)(ss)
(ss)(Lsss)(Lsss) OR (ssLs)(ssLs)(ss)
(sLss)(sLss)(ss)
In this case, each 4/3 is spanned by a 5-note scale segment rather than a 4-note one, so they are more properly called "pentachords". This is why this specific MODMOS of pajara was named the "pentachordal decatonic scale" by [[Paul Erlich]] (who is believed to have originated the concept of omnitetrachordality, circa 2002). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).
See also [[Gallery of omnitetrachordal scales]].Original HTML content:
<html><head><title>Omnitetrachordality</title></head><body>A scale is <strong>omnitetrachordal</strong> if any mode of the scale (that is, any particular octave span of the infinite scale) can be expressed as two identical sequences of steps ("tetrachords") each spanning <a class="wiki_link" href="/4_3">4/3</a>, plus a <a class="wiki_link" href="/9_8">9/8</a> that may or may not be divided into smaller steps. The definition can of course be generalized to intervals of quasi-equivalence other than 4/3, but the original version is with 4/3.<br /> <br /> This definition could be difficult to understand, so take the <a class="wiki_link" href="/5L%202s">5L+2s</a> diatonic scale as an example. This scale has 7 notes and 7 different modes, so we should check each one.<br /> <br /> C D E F G A B C = (C D E F) + 9/8 + (G A B C) (both tetrachords are LLs)<br /> D E F G A B C D = (D E F G) + 9/8 + (A B C D) (both tetrachords are LsL)<br /> E F G A B C D E = (E F G A) + 9/8 + (B C D E) (both tetrachords are sLL)<br /> F G A B C D E F = 9/8 + (G A B C) + (C D E F) (both tetrachords are LLs)<br /> G A B C D E F G = 9/8 + (A B C D) + (D E F G) (both tetrachords are LsL) <em>or alternatively</em> (G A B C) + (C D E F) + 9/8 (both tetrachords are LLs)<br /> A B C D E F G A = 9/8 + (B C D E) + (E F G A) (both tetrachords are sLL) <em>or alternatively</em> (A B C D) + (D E F G) + 9/8 (both tetrachords are LsL)<br /> B C D E F G A B = (B C D E) + (E F G A) + 9/8 (both tetrachords are sLL)<br /> <br /> Since each mode can be expressed as two tetrachords each spanning 4/3 and a leftover 9/8 (some in more than one way), the diatonic scale is omnitetrachordal.<br /> <br /> If you understand <a class="wiki_link" href="/MOSScales">MOS scales</a> well it should be clear that any MOS of a temperament in which the period represents 2/1 and the generator represents 4/3 (including <a class="wiki_link" href="/meantone">meantone</a>, <a class="wiki_link" href="/mavila">mavila</a>, <a class="wiki_link" href="/superpyth">superpyth</a>, <a class="wiki_link" href="/Schismatic%20family">schismatic</a>, etc.) will be omnitetrachordal. However, these are not the only possible omnitetrachordal scales. For an example of a different kind of omnitetrachordal scale, take the <a class="wiki_link" href="/MODMOS">MODMOS</a> of the <a class="wiki_link" href="/2L%208s">2L+8s</a> scale (in <a class="wiki_link" href="/pajara">pajara</a> for example) with the pattern LsssLsssss.<br /> <br /> (Lsss)(Lsss)(ss)<br /> (sssL)(ss)(sssL)<br /> (ssLs)(ss)(ssLs)<br /> (sLss)(ss)(sLss)<br /> (Lsss)(ss)(Lsss)<br /> (ss)(sssL)(sssL)<br /> (ss)(ssLs)(ssLs)<br /> (ss)(sLss)(sLss) OR (sssL)(sssL)(ss)<br /> (ss)(Lsss)(Lsss) OR (ssLs)(ssLs)(ss)<br /> (sLss)(sLss)(ss)<br /> <br /> In this case, each 4/3 is spanned by a 5-note scale segment rather than a 4-note one, so they are more properly called "pentachords". This is why this specific MODMOS of pajara was named the "pentachordal decatonic scale" by <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> (who is believed to have originated the concept of omnitetrachordality, circa 2002). However, the property is still called "omnitetrachordality" (unless someone proposes a better name and it sticks).<br /> <br /> See also <a class="wiki_link" href="/Gallery%20of%20omnitetrachordal%20scales">Gallery of omnitetrachordal scales</a>.</body></html>