User talk:Xenwolf: Difference between revisions

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311 EDO is a one-of-a-kind extremely consistent system to the point it can be said to be the first true melodic model of the harmonic series itself. Therefore to understand the structure of the harmonic series (at least in the 41-limit add-73) without making it impractical, it would be useful to use 311 EDO, and 311 EDO is a great alternative to cents for this reason (which requires smaller numbers and only round numbers) because of being distinctly consistent in the [[23-odd-limit]] and consistent in the 41-prime-limited 77-odd-limit with prime harmonic 73 included as the only prime over 41 and under 81 which is more in tune than out of tune (which is to say its step error is less than a quarter), but furthermore, because of it fitting the way 311 EDO maps the harmonic series by filling a gap and because of it being an odd prime with less error than all previous primes. 311 EDO is therefore a natural system to inspect when looking at ways of modelling higher-limit systems and has important connections with many simpler systems. Just one example would be [[Canousmic_temperaments#Superlimmal|Superlimmal]], the 29-limit 80&311 temperament. Another reason it is a one-of-a-kind system is that in terms of having rooted harmonics with less than 25% error each, I did a search before in Python 3 and there is nothing that beats it until over 20 thousand, meaning it is simultaneously the first and last EDO to truly approximate the harmonic series, and the fact it does it up to a very large subset of the 77-odd-limit makes it worth inspecting the structure of for anyone wishing to learn medium-to-high-complexity tempered JI/odd-limits. Sorry if I've overstressed my point. Also the other table has been there for as long as I remember, and I felt it was both incomplete and messy so I wanted a table from a more systematic approach. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 23:17, 27 December 2021 (UTC)

Oh one other thing, feel free to suggest ways of making the table prettier. For example, I was initially only putting the intervals of the form a/b for a>b and for a and b from 39 to 78 (not including primes >41 other than 73 and corresponding thusly to the 41-prime-limited add-73 77-odd-limit) but then I noticed the end of the table was sparse and some intervals were missing, so instead I put the octave-complement of each such interval too, but this made the organisation messier. It does mean however that the interval table could potentially have its size halved without any loss of information and with the only caveat that you'd have to calculate octave-complements of intervals yourself (both in terms of number of steps and in terms of JI intervals). I think my former method which lead to a more orderly organisation of odd-limit intervals per interval of 311 EDO was maybe better especially considering it lead to simpler code too. Once I or others decide which is the better method I may start hand-prettifying the table by adding consistently-mapped odd-limit intervals or commas for empty intervals of 311 EDO in the table. One example could be that 9801/9800, 441/440 and 385/384 are all mapped to 1\311, although my concern for giving interpretations like that is there is a gargantuan number of intervals that are mapped to one or two steps of 311 EDO, so maybe leaving them blank is more elegant. --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 23:26, 27 December 2021 (UTC)

Revision as of 23:17, 27 December 2021 view source Godtone (talk \| contribs) autopatrolled, emailconfirmed 2,289 edits m →311 EDO huge table ← Older edit		Revision as of 23:26, 27 December 2021 view source Godtone (talk \| contribs) autopatrolled, emailconfirmed 2,289 edits m →311 EDO huge table: added question about organisation Newer edit →
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	311 EDO is a one-of-a-kind extremely consistent system to the point it can be said to be the first true melodic model of the harmonic series itself. Therefore to understand the structure of the harmonic series (at least in the 41-limit add-73) without making it impractical, it would be useful to use 311 EDO, and 311 EDO is a great alternative to cents for this reason (which requires smaller numbers and only round numbers) because of being distinctly consistent in the [[23-odd-limit]] and consistent in the 41-prime-limited 77-odd-limit with prime harmonic 73 included as the only prime over 41 and under 81 which is more in tune than out of tune (which is to say its step error is less than a quarter), but furthermore, because of it fitting the way 311 EDO maps the harmonic series by filling a gap and because of it being an odd prime with less error than all previous primes. 311 EDO is therefore a natural system to inspect when looking at ways of modelling higher-limit systems and has important connections with many simpler systems. Just one example would be [[Canousmic_temperaments#Superlimmal\|Superlimmal]], the 29-limit 80&311 temperament. Another reason it is a one-of-a-kind system is that in terms of having rooted harmonics with less than 25% error each, I did a search before in Python 3 and there is nothing that beats it until over 20 thousand, meaning it is simultaneously the first and last EDO to truly approximate the harmonic series, and the fact it does it up to a very large subset of the 77-odd-limit makes it worth inspecting the structure of for anyone wishing to learn medium-to-high-complexity tempered JI/odd-limits. Sorry if I've overstressed my point. Also the other table has been there for as long as I remember, and I felt it was both incomplete and messy so I wanted a table from a more systematic approach. --[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 23:17, 27 December 2021 (UTC)		311 EDO is a one-of-a-kind extremely consistent system to the point it can be said to be the first true melodic model of the harmonic series itself. Therefore to understand the structure of the harmonic series (at least in the 41-limit add-73) without making it impractical, it would be useful to use 311 EDO, and 311 EDO is a great alternative to cents for this reason (which requires smaller numbers and only round numbers) because of being distinctly consistent in the [[23-odd-limit]] and consistent in the 41-prime-limited 77-odd-limit with prime harmonic 73 included as the only prime over 41 and under 81 which is more in tune than out of tune (which is to say its step error is less than a quarter), but furthermore, because of it fitting the way 311 EDO maps the harmonic series by filling a gap and because of it being an odd prime with less error than all previous primes. 311 EDO is therefore a natural system to inspect when looking at ways of modelling higher-limit systems and has important connections with many simpler systems. Just one example would be [[Canousmic_temperaments#Superlimmal\|Superlimmal]], the 29-limit 80&311 temperament. Another reason it is a one-of-a-kind system is that in terms of having rooted harmonics with less than 25% error each, I did a search before in Python 3 and there is nothing that beats it until over 20 thousand, meaning it is simultaneously the first and last EDO to truly approximate the harmonic series, and the fact it does it up to a very large subset of the 77-odd-limit makes it worth inspecting the structure of for anyone wishing to learn medium-to-high-complexity tempered JI/odd-limits. Sorry if I've overstressed my point. Also the other table has been there for as long as I remember, and I felt it was both incomplete and messy so I wanted a table from a more systematic approach. --[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 23:17, 27 December 2021 (UTC)

			Oh one other thing, feel free to suggest ways of making the table prettier. For example, I was initially only putting the intervals of the form a/b for a>b and for a and b from 39 to 78 (not including primes >41 other than 73 and corresponding thusly to the 41-prime-limited add-73 77-odd-limit) but then I noticed the end of the table was sparse and some intervals were missing, so instead I put the octave-complement of each such interval too, but this made the organisation messier. It does mean however that the interval table could potentially have its size halved without any loss of information and with the only caveat that you'd have to calculate octave-complements of intervals yourself (both in terms of number of steps and in terms of JI intervals). I think my former method which lead to a more orderly organisation of odd-limit intervals per interval of 311 EDO was maybe better especially considering it lead to simpler code too. Once I or others decide which is the better method I may start hand-prettifying the table by adding consistently-mapped odd-limit intervals or commas for empty intervals of 311 EDO in the table. One example could be that 9801/9800, 441/440 and 385/384 are all mapped to 1\311, although my concern for giving interpretations like that is there is a gargantuan number of intervals that are mapped to one or two steps of 311 EDO, so maybe leaving them blank is more elegant. --[[User:Godtone\|Godtone]] ([[User talk:Godtone\|talk]]) 23:26, 27 December 2021 (UTC)