Mapping: Difference between revisions

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Now, let's consider 12edo, not as a 3-limit temperament, but as a [[5-limit]] temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is {{val| 12 19 28 }}. It's important to keep in mind that this is, technically speaking, a ''different'' regular temperament than {{val| 12 19 }}, even though they would both be referred to as "12-tone equal temperament" in common parlance.

Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D#". This choice results in two different mappings, {{val| 12 19 28 34 40 }} and {{val| 12 19 28 34 41 }}. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.

Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D#". This choice results in two different mappings, {{val| 12 19 28 34 41 }} and {{val| 12 19 28 34 42 }}. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.

(Strictly speaking, "5-limit 12edo" or even "3-limit 12edo" are also ambiguous, because {{val| 12 19 27 }}, for example, is a valid temperament even though it's much less accurate than {{val| 12 19 28 }}. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.)

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 Now, let's consider 12edo, not as a 3-limit temperament, but as a [[5-limit]] temperament. This temperament maps all the 3-limit JI intervals in the same way as above, but in addition also maps the rest of the 5-limit JI intervals. Its mapping matrix is {{val| 12 19 28 }}. It's important to keep in mind that this is, technically speaking, a ''different'' regular temperament than {{val| 12 19 }}, even though they would both be referred to as "12-tone equal temperament" in common parlance.
-Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D#". This choice results in two different mappings, {{val| 12 19 28 34 40 }} and {{val| 12 19 28 34 41 }}. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.
+Furthermore, consider 12edo as an 11-limit temperament. What is its mapping matrix? It actually depends whether you consider 11/8 a "very sharp D" or a "very flat D#". This choice results in two different mappings, {{val| 12 19 28 34 41 }} and {{val| 12 19 28 34 42 }}. The latter has a more accurate 11/8, but the former has more accurate versions of other intervals, including 12/11. In the language of regular temperament theory, these are simply two different 11-limit temperaments that both happen to have 12 steps per octave. Phrases like "11-limit 12edo" are thus ambiguous because they don't specify the mapping, and therefore don't refer to a specific temperament.
 (Strictly speaking, "5-limit 12edo" or even "3-limit 12edo" are also ambiguous, because {{val| 12 19 27 }}, for example, is a valid temperament even though it's much less accurate than {{val| 12 19 28 }}. In this temperament 5/4 would be represented as 3 steps of 12edo, or 300 cents. For practical purposes, of course, the ambiguity doesn't appear until higher limits.)