Xenharmonic Wiki

Tonality diamond: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 249965572 - Original comment: **
Nick Vuci (talk | contribs)
autopatrolled, emailconfirmed
388 edits
Updated applet to v2
 
(43 intermediate revisions by 12 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = Tonalitätsdiamant
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-09-01 01:30:55 UTC</tt>.<br>
| en = Tonality diamond
: The original revision id was <tt>249965572</tt>.<br>
}}
: The revision comment was: <tt></tt><br>
{{Wikipedia|Tonality diamond}}
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">The q-odd-limit tonality diamond is the [[Diamonds|diamond]] function applied to the odd numbers from 1 to q: diamond({1, 3, 5, ..., q}). Another way of defining it is in terms of the most common number theoretic height function on rational numbers: H(N/M) = max(|M|, |N|); as all rational numbers which are the quotient of two positive odd integers N/M with H(N/M) &lt;= q, reduced to the octave.


[[http://en.wikipedia.org/wiki/Tonality_diamond|Wikipedia article on the tonality diamond]] </pre></div>
A '''tonality diamond''' is a symmetric organization of [[otonality and utonality|otonal and utonal]] [[chord]]s based around a central note and bounded by an [[odd limit]]. First formalized in the [[7-odd-limit]] by {{w|Max Friedrich Meyer|Max F. Meyer}} in 1929,<ref name="meyer1929">Meyer, Max F. (1929) [https://archive.org/details/max-f-meyer-the-musicians-arithmetic/page/22/mode/2up ''The Musician’s Arithmetic: Drill Problems for an Introduction to the Scientific Study of Musical Composition'']. The University of Missouri Studies. Vol. 4, no. 1. University of Missouri. January 1, 1929. p. 22.</ref> the idea became central to the music and theories of [[Harry Partch]],<ref>Harry Partch (1949), ''Genesis of a Music'', University of Wisconsin Press</ref> who built his tonal system around the [[11-odd-limit]] tonality diamond. Tonality diamonds have been used both conceptually (such as for [[target tuning|targets]] of [[temperaments]]) and practically (such as for instrument layouts) in xenharmonics ever since.
<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;Tonality diamond&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The q-odd-limit tonality diamond is the &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; function applied to the odd numbers from 1 to q: diamond({1, 3, 5, ..., q}). Another way of defining it is in terms of the most common number theoretic height function on rational numbers: H(N/M) = max(|M|, |N|); as all rational numbers which are the quotient of two positive odd integers N/M with H(N/M) &amp;lt;= q, reduced to the octave.&lt;br /&gt;
[https://tonalitydiamondapplet.nickvuci.com/ Play some tonality diamonds on your browser here.]
&lt;br /&gt;
 
&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Tonality_diamond" rel="nofollow"&gt;Wikipedia article on the tonality diamond&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:How to tonality diamond 1.png|'''Step 1: Take the numbers of an odd limit and arrange them along two axes.'''
File:How to tonality diamond 2.png|'''Step 2: Using one axis as the numerator and the other as the denominator, fill in the cells with the ratios they form.'''
File:How to tonality diamond 3.png|'''Step 3: Octave-reduce the ratios (i.e. make sure the decimal form of each ratio is between 1 and 2; if it is not, double one of the numbers until it is).'''
File:How to tonality diamond 4.png|'''Optional step: to make the rows play rooted chords, one half of the diamond (not including the middle unison row) must be lowered by an octave (represented by grey cells in image).'''
</gallery>
 
Note: the numbers of the odd limit are generally arranged in one of three ways:  
* Numerically: (1, 3, 5, 7, 9, 11) as in Meyer's 7-odd-limit diamond.
* Tonally: (1, 9, 5, 11, 3, 7) as in Partch's 11-odd-limit diamond.
* Chordally: (1, 5, 3, 7, 9, 11) as in the layout for the Diamond Marimba. This creates a 4:5:6:7:9:11 extended 11th chord on the diagonal, arranged in thirds.
 
Here is a short video illustrating the interlocking nature of the otonal and utonal chords and constant presence of the 1/1 interval in the 5-limit tonality diamond:
 
[[File:5-Limit Tonality Diamond original format.mp4|1000x400px]]
 
== History ==
The tonality diamond was first formally explained by Max F. Meyer in his 1929 publication ''The Musician's Arithmetic'' using the 7-odd-limit.<ref name="meyer1929"/>
 
Harry Partch is the person most associated with the tonality diamond, and claimed to have invented it. However, it is likely that he plagiarized the idea from Meyer.<ref>Forster, Cris (2015). [https://web.archive.org/web/20221207160002/https://www.chrysalis-foundation.org/the-partch-hoax-doctrines/ ''The Partch Hoax Doctrines'']. Self-published.</ref> Regardless, his extending of the concept to the 11-odd-limit (as well as his other extensions and uses of it) was an extremely important and foundational moment in the history of xenharmonic music.
 
[[Erv Wilson]] in particular was inspired by Partch's use of the tonality diamond and its extended form. He developed a number of "diamonds" himself,<ref>Wilson, Erv. (1965–1970) [https://anaphoria.com/diamond.pdf ''Letters on Diamond Lattices''] (PDF) Self-published.</ref> as well as other concepts inspired by Partch's use of the extended tonality diamond such as [[constant structure]].<ref>Wilson, Erv. (1964-2002) [https://www.anaphoria.com/Partchpapers.pdf ''The Partch Papers''] (collection of documents on Harry Partch's 11-odd-limit diamond and its extensions, PDF). Self-published.</ref> A related idea of Wilson's is the [[cross-set scale|cross-set]], of which the tonality diamond is a special case.   
 
The first novel xenharmonic temperament—[[George Secor]]'s later-named [[miracle]] temperament—was made to approximate Partch's 11-odd-limit diamond.<ref>Secor, George (1975). [https://www.anaphoria.com/SecorMiracle.pdf ''A New Look at the Partch Monophonic Fabric.''] Xenharmonicon. Vol. 3</ref><ref>Secor, George. (2006) [https://www.anaphoria.com/SecorMiracle.pdf ''The Miracle Temperament and Decimal Keyboard'']. Xenharmonikon. Vol. 18. 2006. pp. 5–15</ref>
 
=== Instrument layout ===
The most famous example of the tonality diamond as a practical layout for an instrument is Harry Partch's "Diamond Marimba," which uses the 11-odd-limit tonality diamond exactly. This idea was explored further with Partch's "Quadrangularis Reversum," and by [[Cris Forster]] with his [[13-odd-limit]] Diamond Marimba.<ref>[https://web.archive.org/web/20220901100217/https://www.chrysalis-foundation.org/instruments-and-music/diamond-marimba-i/ Diamond Marimba I – The Chrysalis Foundation]</ref>[[File:Diamond_marimba_layout.png|thumb|Layout of the Diamond Marimba. Ratios are shown unreduced to highlight the structure. [https://sintel.website/posts/diamond_marimba.html Click here to play the Diamond Marimba on your browser.]|488x488px|none]]
 
== Music ==
; [[Banjo Boogie]]
* [https://www.youtube.com/watch?v=1K227vkfBdc Demonstration and improvisation on a banjo drumset tuned to the 7-limit tonality diamond]
 
; [[Cris Forster]]
* [https://www.youtube.com/watch?v=jDFRGgWQp4I ''Dream Time'']
* [https://www.youtube.com/watch?v=6zyCZklRrnI ''The Harbor'']
* [https://www.youtube.com/watch?v=foqWB37nW7w ''Wild Flower'']
 
; [[Harry Partch]]
* [https://www.youtube.com/watch?v=kCuYcS_Lcro ''Castor & Pollux'']
* [https://www.youtube.com/watch?v=J_trV1AWU0Y "Diamond Marimba"] from ''The World of Harry Partch''
* [https://www.youtube.com/watch?v=qZybJAEPu18 ''Sonata Dementia''] (1950)
* [https://www.youtube.com/watch?v=gZiTiveqbDw ''Three Dances''] (1952)
 
; [[David Paulick]]
* [https://www.youtube.com/watch?v=-c3hYWunKps ''Improvisation using a Web MIDI Tonality Diamond''] (2022)
 
; [[T.J Troy]]
* [https://www.youtube.com/watch?v=4Q-sq9UwSgY ''Five-Corner Square'']
 
== Related scales ==
* [[Diamond5|5-odd-limit diamond]]
* [[Diamond7|7-odd-limit diamond]]
* [[Diamond9|9-odd-limit diamond]]
* [[Diamond11|11-odd-limit diamond]]
* [[Diamond13|13-odd-limit diamond]]
* [[Diamond15|15-odd-limit diamond]]
 
== See also ==
* [[Odd limit]]
* [[Cross-set scale]]
* [[Diamond function]]
* [[Lattice]]
 
== External links ==
* [http://www.tonalsoft.com/enc/t/tonality-diamond.aspx Tonality diamond – arrangement of musical frequency ratios showing the dual identity of each ratio] on [[Tonalsoft Encyclopedia]]
* [https://www.youtube.com/watch?v=jsBsnNGkdcc Harry Partch's Diamond Marimba, as demonstrated by John Schneider]
* [https://www.youtube.com/watch?v=N57Wt0mpSu4 <nowiki>"What is the Tonality Diamond? (Harry Partch's Theories, Explained) [Harry Partch, Pt. 2/2]"</nowiki>] on [https://www.youtube.com/@ClassicalNerd Classical Nerd YouTube Channel]
* [https://web.archive.org/web/20221226001701/https://tonalitydiamond.com/ David Paulick's webapp of various tonality diamond inspired layouts for the Novation Launchpad] (now only available on archive.org)
 
== References ==
<references/>
 
[[Category:Diamond]]
[[Category:Pitch space]]
Retrieved from "https://en.xen.wiki/w/Tonality_diamond"