16edo
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=16 tone equal temperament= ==Theory== 16-tone equal temperament is the division of the octave into sixteen narrow chromatic semitones. It can be thought of as a Diminished Temperament for it's 1/4 octave period. Also as a rough Slendro temperament with a supermajor second generator (250cents [ideally 233cents]), or as a Pelog or Mavila temperament generated by (fifths greater than 600 and less than 686 cents). The temperament could be popular for it's easy manageability of 150 cent intervals 3/4, 9/4 and 21/4-tones. The 25 cent difference in the steps can have a similar effect the scales of Olympos have with buried enharmonic genera. It can be It can be treated as 4 interwoven diminished seventh arpeggios, or as 2 interwoven 8-edo scales (narrow 11-limit whole tones which when stacked produce traditional 300 cent minor third intervals). 16-tone has the same stacked minor thirds diminished seventh scale/chord available in 12, and It is often cited that the most consonant chords involve the tritone. There are two minor seventh intervals, a harmonic seventh at step 13, a 7/4 ratio approximation, off by 3.5879 cents, followed by an undecimal 11/6 ratio or neutral seventh. The septimal can be the 9/4th tone or septimal semi diminished fourth (35/27 ratio) , semi-augmented fifth (54/35), harmonic seventh (7/4), and septimal whole tone 8/7. The Undecimal intervals are the 3/4 tone or undecimal neutral second (12/11), and the 21/4th tone or undecimal neutral seventh (11/6). One neat xenharmonic aspect of 16-tone is how the 11-limit whole tone scale using the neutral second, interlocks with the diminished scale, similar to the augmented scale and whole tone relationship in 12-tone (the whole tone divides the major third in 12, in 16-it's the minor third). In 16-tone, because of the 25 cent difference in the steps from 100 in 12-tone, a western "twelve tone ear" hears dissonance with more complexity and less familiarity than even 24-tone, yet within a more manageable number of tones and a strange familiarity - the diminished. Hence, why 16-tone is a truly Xenharmonic system. 16-edo Diatonic scales are dissonant because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. The septimal semi diminished fourth can be more desirable. Perhaps using Moment of Symmetry Scales an alternative temperament families like the "Anti-Diatonic" Mavila (which reverses step sizes of diatonic), Diminished, Happy, Rice, Grumpy, Mosh, Magic, Lemba, Cynder, and Decatonic can be more interesting and suitable. Cycle of 7/4 (Armodue): 1 8 6# 5 3 1# 8# 7 5# 3# 2 9 7# 6 4 2# 1 <span class="text_exposed_show"> Diminished family of scales (1 3 1 3 1 3 1 3, 1 1 2 1 1 2 1 1 2 1 1 2) Magic family of scales (1 4 1 4 1 4 1, 1 3 1 1 3 1 1 1 3 1, 1 1 2 1 1 1 2 1 1 1 2 1 1) Cynder family (3 3 4 3 3, 3 3 1 3 3 3, 1 2 1 2 1 2 1 2 1 2 1) Lemba family (3 2 3 3 2 3, 2 1 2 1 2 2 1 2 1 2)</span> Like the conventional 12-tet diatonic and pentatonic (meantone) scales, these arise from tempering out a unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:<span class="text_exposed_show">128, instead of 81:80. </span> 0. 1/1 C 1. 75.00 cents C# Dbb 2. 150.00 cents Cx Db 3. 225.00 cents D 4. 300.00 cents D# Ebb 5. 375.00 cents Dx Eb 6. 450.00 cents E Fb 7. 525.00 cents F 8. 600.00 cents F# Gbb 9. 675.00 cents Fx Gb 10. 750.00 cents G Abb 11. 825.00 cents G# Ab 12. 900.00 cents A 13. 975.00 cents A# Bbb 14. 1050.00 cents Ax Bb 15. 1125.00 cents B Cb 16. 2/1 C 1 octave into 8 equal parts = 2 2 2 2 2 2 2 2 = 3/4 tone Neutral Second Progression 2 octaves into 8 equal parts = 4 4 4 4 4 4 4 4 = Classic Minor Third Progression 3 octaves into 8 equal parts = 6 6 6 6 6 6 6 6 = 9/4tone or Septimal semi-dim Fourth Progression 4 octaves into 8 equal parts = 8 8 8 8 8 8 8 8 = Tritone Progression 5 octaves into 8 equal parts = 10 10 10 10 10 10 10 10 = Septimal semi-aug Fifth Progression 6 octaves into 8 equal parts = 12 12 12 12 12 12 12 12 = Classic Sixth Progression 7 octaves into 8 equal parts = 14 14 14 14 14 14 14 14 = 21/4 tone or Neutral Seventh Progression 8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression 9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression [[http://www.armodue.com/ricerche.htm|Armodue]]: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions - translation, anyone? [[image:http://ronsword.com/images/ESG_sm.jpg width="120" height="161"]] Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning). Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning) ==Compositions== [[http://www.io.com/%7Ehmiller/midi/16tet.mid|Etude in 16-tone equal tuning]] by Herman Miller [[http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3|Armodue78]] by [[@http://www.jeanpierrepoulin.com/|Jean-Pierre Poulin]] [[@http://ronsword.com/sounds/16chordscale_improv.mp3|Chord-scale Improvisation in 16-tet]] by Ron Sword [[@http://www.ronsword.com/sounds/ron_sword_16_improv.mp3|Chromatic 16-tet Improvisation]] by Ron Sword [[@http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3|16-tet Acoustic Improvisation]] by Ron Sword [[@http://www.ronsword.com/sounds/ronsword_miracle528_part3.mp3|16-tet Magic Drone]] by Ron Sword
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<html><head><title>16edo</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x16 tone equal temperament"></a><!-- ws:end:WikiTextHeadingRule:0 -->16 tone equal temperament</h1> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x16 tone equal temperament-Theory"></a><!-- ws:end:WikiTextHeadingRule:2 -->Theory</h2> 16-tone equal temperament is the division of the octave into sixteen narrow chromatic semitones. It can be thought of as a Diminished Temperament for it's 1/4 octave period. Also as a rough Slendro temperament with a supermajor second generator (250cents [ideally 233cents]), or as a Pelog or Mavila temperament generated by (fifths greater than 600 and less than 686 cents). The temperament could be popular for it's easy manageability of 150 cent intervals 3/4, 9/4 and 21/4-tones.<br /> The 25 cent difference in the steps can have a similar effect the scales of Olympos have with buried enharmonic genera.<br /> <br /> It can be It can be treated as 4 interwoven diminished seventh arpeggios, or as 2 interwoven 8-edo scales (narrow 11-limit whole tones which when stacked produce traditional 300 cent minor third intervals). 16-tone has the same stacked minor thirds diminished seventh scale/chord available in 12, and It is often cited that the most consonant chords involve the tritone. There are two minor seventh intervals, a harmonic seventh at step 13, a 7/4 ratio approximation, off by 3.5879 cents, followed by an undecimal 11/6 ratio or neutral seventh. The septimal can be the 9/4th tone or septimal semi diminished fourth (35/27 ratio) , semi-augmented fifth (54/35), harmonic seventh (7/4), and septimal whole tone 8/7. The Undecimal intervals are the 3/4 tone or undecimal neutral second (12/11), and the 21/4th tone or undecimal neutral seventh (11/6).<br /> <br /> One neat xenharmonic aspect of 16-tone is how the 11-limit whole tone scale using the neutral second, interlocks with the diminished scale, similar to the augmented scale and whole tone relationship in 12-tone (the whole tone divides the major third in 12, in 16-it's the minor third).<br /> <br /> In 16-tone, because of the 25 cent difference in the steps from 100 in 12-tone, a western "twelve tone ear" hears dissonance with more complexity and less familiarity than even 24-tone, yet within a more manageable number of tones and a strange familiarity - the diminished. Hence, why 16-tone is a truly Xenharmonic system.<br /> 16-edo Diatonic scales are dissonant because of the 25 cent raised superfourth in conjunction with the 25 cent subtracted fifth / poor 3/2 approximation. The septimal semi diminished fourth can be more desirable. Perhaps using Moment of Symmetry Scales an alternative temperament families like the "Anti-Diatonic" Mavila (which reverses step sizes of diatonic), Diminished, Happy, Rice, Grumpy, Mosh, Magic, Lemba, Cynder, and Decatonic can be more interesting and suitable.<br /> <br /> <br /> Cycle of 7/4 (Armodue):<br /> 1 8 6# 5 3 1# 8# 7 5# 3# 2 9 7# 6 4 2# 1<br /> <br /> <span class="text_exposed_show"> Diminished family of scales (1 3 1 3 1 3 1 3, 1 1 2 1 1 2 1 1 2 1 1 2)<br /> Magic family of scales (1 4 1 4 1 4 1, 1 3 1 1 3 1 1 1 3 1, 1 1 2 1 1 1 2 1 1 1 2 1 1)<br /> Cynder family (3 3 4 3 3, 3 3 1 3 3 3, 1 2 1 2 1 2 1 2 1 2 1)<br /> Lemba family (3 2 3 3 2 3, 2 1 2 1 2 2 1 2 1 2)</span><br /> <br /> Like the conventional 12-tet diatonic and pentatonic (meantone) scales, these arise from tempering out a unison vector from Fokker periodicity blocks. Only in 16-EDO, that unison vector is 135:<span class="text_exposed_show">128, instead of 81:80. </span><br /> <br /> 0. 1/1 C<br /> 1. 75.00 cents C# Dbb<br /> 2. 150.00 cents Cx Db<br /> 3. 225.00 cents D<br /> 4. 300.00 cents D# Ebb<br /> 5. 375.00 cents Dx Eb<br /> 6. 450.00 cents E Fb<br /> 7. 525.00 cents F<br /> 8. 600.00 cents F# Gbb<br /> 9. 675.00 cents Fx Gb<br /> 10. 750.00 cents G Abb<br /> 11. 825.00 cents G# Ab<br /> 12. 900.00 cents A<br /> 13. 975.00 cents A# Bbb<br /> 14. 1050.00 cents Ax Bb<br /> 15. 1125.00 cents B Cb<br /> 16. 2/1 C<br /> <br /> <br /> 1 octave into 8 equal parts = 2 2 2 2 2 2 2 2 = 3/4 tone Neutral Second Progression<br /> 2 octaves into 8 equal parts = 4 4 4 4 4 4 4 4 = Classic Minor Third Progression<br /> 3 octaves into 8 equal parts = 6 6 6 6 6 6 6 6 = 9/4tone or Septimal semi-dim Fourth Progression<br /> 4 octaves into 8 equal parts = 8 8 8 8 8 8 8 8 = Tritone Progression<br /> 5 octaves into 8 equal parts = 10 10 10 10 10 10 10 10 = Septimal semi-aug Fifth Progression<br /> 6 octaves into 8 equal parts = 12 12 12 12 12 12 12 12 = Classic Sixth Progression<br /> 7 octaves into 8 equal parts = 14 14 14 14 14 14 14 14 = 21/4 tone or Neutral Seventh Progression<br /> 8 octaves into 8 equal parts = 16 16 16 16 16 16 16 16 = Octave Progression<br /> 9 octaves into 8 equal parts = 18 18 18 18 18 18 18 18 = Ninth Progression<br /> <br /> <br /> <br /> <br /> <br /> <a class="wiki_link_ext" href="http://www.armodue.com/ricerche.htm" rel="nofollow">Armodue</a>: Italian pages of theory for 16-tone (esadekaphonic) system, including compositions - translation, anyone?<br /> <br /> <!-- ws:start:WikiTextRemoteImageRule:6:<img src="http://ronsword.com/images/ESG_sm.jpg" alt="" title="" style="height: 161px; width: 120px;" /> --><img src="http://ronsword.com/images/ESG_sm.jpg" alt="external image ESG_sm.jpg" title="external image ESG_sm.jpg" style="height: 161px; width: 120px;" /><!-- ws:end:WikiTextRemoteImageRule:6 --><br /> Sword, Ronald. "Hexadecaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: Feb, 2010. (superfourth tuning).<br /> Sword, Ronald. "Esadekaphonic Scales for Guitar." IAAA Press, UK-USA. First Ed: April, 2009. (semi-diminished fourth tuning)<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x16 tone equal temperament-Compositions"></a><!-- ws:end:WikiTextHeadingRule:4 -->Compositions</h2> <br /> <a class="wiki_link_ext" href="http://www.io.com/%7Ehmiller/midi/16tet.mid" rel="nofollow">Etude in 16-tone equal tuning</a> by Herman Miller<br /> <a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3" rel="nofollow">Armodue78</a> by <a class="wiki_link_ext" href="http://www.jeanpierrepoulin.com/" rel="nofollow" target="_blank">Jean-Pierre Poulin</a><br /> <br /> <a class="wiki_link_ext" href="http://ronsword.com/sounds/16chordscale_improv.mp3" rel="nofollow" target="_blank">Chord-scale Improvisation in 16-tet</a> by Ron Sword<br /> <a class="wiki_link_ext" href="http://www.ronsword.com/sounds/ron_sword_16_improv.mp3" rel="nofollow" target="_blank">Chromatic 16-tet Improvisation</a> by Ron Sword<br /> <a class="wiki_link_ext" href="http://www.ronsword.com/sounds/Ron%20Sword%20-%2016-tone%20acoustic%20improvisation.mp3" rel="nofollow" target="_blank">16-tet Acoustic Improvisation</a> by Ron Sword<br /> <a class="wiki_link_ext" href="http://www.ronsword.com/sounds/ronsword_miracle528_part3.mp3" rel="nofollow" target="_blank">16-tet Magic Drone</a> by Ron Sword</body></html>