Temperament addition: Difference between revisions

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We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243=20000/19683 and 80/81 ÷ 250/243=25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET.

We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243=20000/19683 and 80/81 ÷ 250/243=25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET. (Note that these examples are all given in canonical form, which is why we're seeing the meantone comma as 80/81 instead of the more common 81/80; for the reason why, see [[Temperament arithmetic#Negation]].)

Temperament arithmetic is simplest for temperaments which can be represented by single vectors such as demonstrated in these examples. In other words, it is simplest for temperaments that are either rank-1 ([[equal temperament]]s, or ETs for short) or nullity-1 (having only a single comma). Because [[grade]] <math>g</math> is the generic term for rank <math>r</math> and nullity <math>n</math>, we could define the minimum grade <math>g_{\text{min}}</math> of a temperament as the minimum of its rank and nullity <math>\min(r,n)</math>, and so for convenience in this article we will refer to <math>r=1</math> (read "rank-1") or <math>n=1</math> (read "nullity-1") temperaments as <math>g_{\text{min}}=1</math> (read "min-grade-1") temperaments. We'll also use <math>g_{\text{max}}</math> (read "max-grade"), which naturally is equal to <math>\max(r,n)</math>.

Revision as of 18:07, 4 January 2022 view source Cmloegcmluin (talk \| contribs) autopatrolled, emailconfirmed 5,160 edits m →Introductory examples: I feel like I'm going to be fixing these equal signs problems forever hehe ← Older edit		Revision as of 18:16, 4 January 2022 view source Cmloegcmluin (talk \| contribs) autopatrolled, emailconfirmed 5,160 edits →Introductory examples: acknowledge potentially surprising effects of canonical form earlier Newer edit →
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	We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243=20000/19683 and 80/81 ÷ 250/243=25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET.		We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243=20000/19683 and 80/81 ÷ 250/243=25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET. (Note that these examples are all given in canonical form, which is why we're seeing the meantone comma as 80/81 instead of the more common 81/80; for the reason why, see [[Temperament arithmetic#Negation]].)

	Temperament arithmetic is simplest for temperaments which can be represented by single vectors such as demonstrated in these examples. In other words, it is simplest for temperaments that are either rank-1 ([[equal temperament]]s, or ETs for short) or nullity-1 (having only a single comma). Because [[grade]] <math>g</math> is the generic term for rank <math>r</math> and nullity <math>n</math>, we could define the minimum grade <math>g_{\text{min}}</math> of a temperament as the minimum of its rank and nullity <math>\min(r,n)</math>, and so for convenience in this article we will refer to <math>r=1</math> (read "rank-1") or <math>n=1</math> (read "nullity-1") temperaments as <math>g_{\text{min}}=1</math> (read "min-grade-1") temperaments. We'll also use <math>g_{\text{max}}</math> (read "max-grade"), which naturally is equal to <math>\max(r,n)</math>.		Temperament arithmetic is simplest for temperaments which can be represented by single vectors such as demonstrated in these examples. In other words, it is simplest for temperaments that are either rank-1 ([[equal temperament]]s, or ETs for short) or nullity-1 (having only a single comma). Because [[grade]] <math>g</math> is the generic term for rank <math>r</math> and nullity <math>n</math>, we could define the minimum grade <math>g_{\text{min}}</math> of a temperament as the minimum of its rank and nullity <math>\min(r,n)</math>, and so for convenience in this article we will refer to <math>r=1</math> (read "rank-1") or <math>n=1</math> (read "nullity-1") temperaments as <math>g_{\text{min}}=1</math> (read "min-grade-1") temperaments. We'll also use <math>g_{\text{max}}</math> (read "max-grade"), which naturally is equal to <math>\max(r,n)</math>.