Crossbone tuning
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<span style="background-color: #ffffff; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Tuning is a coordinated tuning (intended to be used simultaneously) system utilizing pairs of twin primes (primes separated by the integer 2) separated by the integer 12. Crossbone is capable of being applied to any set of prime twins separated by this constant, though higher orders often yield less musical results.</span> //**__<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Example:</span>__**// <span style="background-color: #ffffff; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Twin primes (5,7) and (17,19) represent a set of twin primes capable of being Crossbone'd, as both 17-5=12 and 19-17 = 12. Once a pair of permissible twin primes is composed, a temperament is devised by taking each member of the lesser set and raising it to (1/x) where x the corresponding member of the greater set.</span> <span style="background-color: #ffffff; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">For (5,7) and (17,19), the Tuning would utilize both 5^1/17 and 7^1/19, or 17EDP(Equally-divided pentave) and 19EDS(Equally-divided septave).</span> //**__<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Background:</span>__**// <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px; vertical-align: baseline;">Using a prime number wheel -a face with 24 repeating integers equally spaced over 2pi- the relationship between twin primes separated by 12 is easily visualized. We are familiar with the special relationships prime numbers</span> [[image:xenharmonic/crossbones.png width="424" align="right"]]<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px; line-height: 1.5;"> have to music, and we understand the relationship certain primes share with each other, so it is not unfeasible that prime relationships in communication with one another also share musical significance.</span> //__**<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Temperament (1st Order):</span>**__// <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">It is such that the 19th root of 7 and the 17th root of 5 share a unique correspondence, being both reasonable in harmonic range and distance, the pentave and septave ending between the 2nd and 3rd octave in 17 or 19 steps respectively. Because the temperaments are coordinated in Crossbone, each combination of septave and pentave I refer to as a 'sepent'.</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Because the septave and pentave are not based around the octave, each individual septave and pentave will be completely unique and be found to approximate different intervals. Notice below that many of the intervals approximated happen to be harmonics! Those which are not harmonics happen to be readily usable just intervals no greater than 7-limit in the first sepent and 11-limit in the second sepent.</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">One beautiful alignment of this temperament is that it uses a combination of 17 (pentave) and 19 (septave) tones per sepent. We notice that the standard 12-tone keyboard has 12 keys broken into a grouping of 5 notes and a grouping of 7 notes. This means that Crossbones tuning can be easily realized on any 'standard' keyboard, the first 5 7 5 grouping of keys representing the pentave, the second 7 5 7 grouping of keys representing the septave.. a sepent spanning a total key range of 3 12-tone octaves, coinciding beautifully with the fact that the natural sepent occurs between the 2nd and 3rd standard octave naturally. Note that below, the intervals represented are octave-reduced, though the true harmonic range of the septave and pentave are true and preserved on the keyboard, representing the absolute pitch approximation I find to be much more intuitive in comparison to the octave-equivalent versions (ex. a twelfth being represented as 3/2 though sounding as a twelfth).</span> [[image:crossbonepiano.png width="993" height="199"]] __//**<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Lattice:</span>**//__ <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">The Crossbone Lattice is a 4-dimensional lattice generated using the 5-limit, 7-limit, 17-limit, and 19-limit. By not including the 3-limit (but still implying the octave) and skipping over the 11 and 13 limits, we can arrive at very unique and interesting destinations.</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Because the lattice is octave-based and generated using the Crossbone primes, represented as harmonics, and doesn't enforce a relationship between them -as in Crossbone Temperament-, the relationship between the lattice and the temperament should be considered tangential, and the usage of the lattice </span> [[image:tessa.png width="435" align="right"]] <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">considered </span><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px; line-height: 1.5;">novel. I personally use it simply to add 17 and 19 limit 'spice' to certain intervals generated using my 3rds and 7ths. Arriving at destinations by combination of 3rd and 7ths without piggybacking on the 5th I find to be very satisfying.</span> __//**<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Scale (1st sepent):</span>**//__ <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">The Crossbone Scale is an octave-repeating simplified version of the Crossbone Temperament, forcing the intervals found within the sepent into a 12-tone octave-repeating scale. Within the first sepent, when one simply neglects including the harmonic approximations (and the 128/125 anamoly), he is left with 12 just 7-limit ratios between 1/1 and 2/1:</span> [[image:crossbonepiano12scale.png width="216" height="202" align="right"]] **<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•1/1</span>**<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;"> - tonic</span> **<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•16/15</span>**<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;"> - major 5-limit half-step</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**7/6** - septimal minor third</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**5/4** - 5-limit major third</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**9/7** - septimal major third</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**10/7** - septimal tritone</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**3/2** - perfect fifth</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**14/9** - septimal minor sixth</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**5/3** - 5-limit major sixth</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**12/7** - septimal major sixth</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**7/4** - septimal minor seventh</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•**15/8** - 5-limit major seventh</span> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Because the scale is octave-repeating (though derived from the sepent), all sorts of 7-limit inversional fun can be had, and because it is 12-tone, it can be played on a standard keyboard.</span> __//**<span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Final Thoughts:</span>**//__ <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">In the near future it would be wise to create a partner scale to the Crossbone Scale based on the 11-limit 2nd sepent as well as extend the current Crossbone Temperament to a full four sepents (the near limit of human hearing). Currently working on a 'Crossbone Extended' using another subset of the 12 relationship, this form in 31 root 19 and 29 root 17. Language on current Crossbone implementation to be clarified. Also uploading spreadsheet with errors and approximations as well as scala files for Crossbone Temperament, Crossbone Extended, Crossbone Scale.</span>
Original HTML content:
<html><head><title>Crossbone Tuning</title></head><body><span style="background-color: #ffffff; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Tuning is a coordinated tuning (intended to be used simultaneously) system utilizing pairs of twin primes (primes separated by the integer 2) separated by the integer 12. Crossbone is capable of being applied to any set of prime twins separated by this constant, though higher orders often yield less musical results.</span><br /> <br /> <br /> <em><strong><u><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Example:</span></u></strong></em><br /> <br /> <span style="background-color: #ffffff; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Twin primes (5,7) and (17,19) represent a set of twin primes capable of being Crossbone'd, as both 17-5=12 and 19-17 = 12. Once a pair of permissible twin primes is composed, a temperament is devised by taking each member of the lesser set and raising it to (1/x) where x the corresponding member of the greater set.</span><br /> <br /> <span style="background-color: #ffffff; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">For (5,7) and (17,19), the Tuning would utilize both 5^1/17 and 7^1/19, or 17EDP(Equally-divided pentave) and 19EDS(Equally-divided septave).</span><br /> <br /> <br /> <em><strong><u><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Background:</span></u></strong></em><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px; vertical-align: baseline;">Using a prime number wheel -a face with 24 repeating integers equally spaced over 2pi- the relationship between twin primes separated by 12 is easily visualized. We are familiar with the special relationships prime numbers</span><br /> <!-- ws:start:WikiTextLocalImageRule:0:<img src="http://xenharmonic.wikispaces.com/file/view/crossbones.png/516326650/crossbones.png" alt="" title="" style="width: 424px;" align="right" /> --><img src="http://xenharmonic.wikispaces.com/file/view/crossbones.png/516326650/crossbones.png" alt="crossbones.png" title="crossbones.png" style="width: 424px;" align="right" /><!-- ws:end:WikiTextLocalImageRule:0 --><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px; line-height: 1.5;"> have to music, and we understand the relationship certain primes share with each other, so it is not unfeasible that prime relationships in communication with one another also share musical significance.</span><br /> <br /> <em><u><strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Temperament (1st Order):</span></strong></u></em><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">It is such that the 19th root of 7 and the 17th root of 5 share a unique correspondence, being both reasonable in harmonic range and distance, the pentave and septave ending between the 2nd and 3rd octave in 17 or 19 steps respectively. Because the temperaments are coordinated in Crossbone, each combination of septave and pentave I refer to as a 'sepent'.</span><br /> <br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Because the septave and pentave are not based around the octave, each individual septave and pentave will be completely unique and be found to approximate different intervals. Notice below that many of the intervals approximated happen to be harmonics! Those which are not harmonics happen to be readily usable just intervals no greater than 7-limit in the first sepent and 11-limit in the second sepent.</span><br /> <br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">One beautiful alignment of this temperament is that it uses a combination of 17 (pentave) and 19 (septave) tones per sepent. We notice that the standard 12-tone keyboard has 12 keys broken into a grouping of 5 notes and a grouping of 7 notes. This means that Crossbones tuning can be easily realized on any 'standard' keyboard, the first 5 7 5 grouping of keys representing the pentave, the second 7 5 7 grouping of keys representing the septave.. a sepent spanning a total key range of 3 12-tone octaves, coinciding beautifully with the fact that the natural sepent occurs between the 2nd and 3rd standard octave naturally. Note that below, the intervals represented are octave-reduced, though the true harmonic range of the septave and pentave are true and preserved on the keyboard, representing the absolute pitch approximation I find to be much more intuitive in comparison to the octave-equivalent versions (ex. a twelfth being represented as 3/2 though sounding as a twelfth).</span><br /> <br /> <!-- ws:start:WikiTextLocalImageRule:1:<img src="/file/view/crossbonepiano.png/516324958/993x199/crossbonepiano.png" alt="" title="" style="height: 199px; width: 993px;" /> --><img src="/file/view/crossbonepiano.png/516324958/993x199/crossbonepiano.png" alt="crossbonepiano.png" title="crossbonepiano.png" style="height: 199px; width: 993px;" /><!-- ws:end:WikiTextLocalImageRule:1 --><br /> <br /> <br /> <u><em><strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Lattice:</span></strong></em></u><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">The Crossbone Lattice is a 4-dimensional lattice generated using the 5-limit, 7-limit, 17-limit, and 19-limit. By not including the 3-limit (but still implying the octave) and skipping over the 11 and 13 limits, we can arrive at very unique and interesting destinations.</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Because the lattice is octave-based and generated using the Crossbone primes, represented as harmonics, and doesn't enforce a relationship between them -as in Crossbone Temperament-, the relationship between the lattice and the temperament should be considered tangential, and the usage of the lattice </span><br /> <!-- ws:start:WikiTextLocalImageRule:2:<img src="/file/view/tessa.png/516325086/tessa.png" alt="" title="" style="width: 435px;" align="right" /> --><img src="/file/view/tessa.png/516325086/tessa.png" alt="tessa.png" title="tessa.png" style="width: 435px;" align="right" /><!-- ws:end:WikiTextLocalImageRule:2 --><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">considered </span><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px; line-height: 1.5;">novel. I personally use it simply to add 17 and 19 limit 'spice' to certain intervals generated using my 3rds and 7ths. Arriving at destinations by combination of 3rd and 7ths without piggybacking on the 5th I find to be very satisfying.</span><br /> <br /> <u><em><strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Crossbone Scale (1st sepent):</span></strong></em></u><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">The Crossbone Scale is an octave-repeating simplified version of the Crossbone Temperament, forcing the intervals found within the sepent into a 12-tone octave-repeating scale. Within the first sepent, when one simply neglects including the harmonic approximations (and the 128/125 anamoly), he is left with 12 just 7-limit ratios between 1/1 and 2/1:</span><br /> <!-- ws:start:WikiTextLocalImageRule:3:<img src="/file/view/crossbonepiano12scale.png/516325150/216x202/crossbonepiano12scale.png" alt="" title="" style="height: 202px; width: 216px;" align="right" /> --><img src="/file/view/crossbonepiano12scale.png/516325150/216x202/crossbonepiano12scale.png" alt="crossbonepiano12scale.png" title="crossbonepiano12scale.png" style="height: 202px; width: 216px;" align="right" /><!-- ws:end:WikiTextLocalImageRule:3 --><br /> <strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•1/1</span></strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;"> - tonic</span><br /> <strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•16/15</span></strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;"> - major 5-limit half-step</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>7/6</strong> - septimal minor third</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>5/4</strong> - 5-limit major third</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>9/7</strong> - septimal major third</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>10/7</strong> - septimal tritone</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>3/2</strong> - perfect fifth</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>14/9</strong> - septimal minor sixth</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>5/3</strong> - 5-limit major sixth</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>12/7</strong> - septimal major sixth</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>7/4</strong> - septimal minor seventh</span><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">•<strong>15/8</strong> - 5-limit major seventh</span><br /> <br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Because the scale is octave-repeating (though derived from the sepent), all sorts of 7-limit inversional fun can be had, and because it is 12-tone, it can be played on a standard keyboard.</span><br /> <br /> <u><em><strong><span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">Final Thoughts:</span></strong></em></u><br /> <span style="background-color: #ffffff; color: #444444; font-family: 'Open Sans','Helvetica Neue',Arial,Helvetica,sans-serif; font-size: 20px;">In the near future it would be wise to create a partner scale to the Crossbone Scale based on the 11-limit 2nd sepent as well as extend the current Crossbone Temperament to a full four sepents (the near limit of human hearing). Currently working on a 'Crossbone Extended' using another subset of the 12 relationship, this form in 31 root 19 and 29 root 17. Language on current Crossbone implementation to be clarified. Also uploading spreadsheet with errors and approximations as well as scala files for Crossbone Temperament, Crossbone Extended, Crossbone Scale.</span></body></html>