Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625349103 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c. | We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c. | ||
In certain edos, the up symbol's cents can be directly related to the sharp's cents. The same can be done for rank-2 pergens if and only if the enharmonic is an A1. | |||
15-edo: # ``=`` 240¢, ^ ``=`` 80¢ | This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples: | ||
15-edo: # ``=`` 240¢, ^ ``=`` 80¢ (^ = 1/3 #) | |||
16-edo: # ``=`` -75¢ | 16-edo: # ``=`` -75¢ | ||
17-edo: # ``=`` 141¢, ^ ``=`` 71¢ | 17-edo: # ``=`` 141¢, ^ ``=`` 71¢ (^ = 1/2 #) | ||
18b-edo: # ``=`` -133¢, ^ = 67¢ | 18b-edo: # ``=`` -133¢, ^ = 67¢ (^ = 1/2 #) | ||
19-edo: # = 63¢ | 19-edo: # = 63¢ | ||
21-edo: ^ = 57¢ | 21-edo: ^ = 57¢ (if used, # = 0¢) | ||
22-edo: # ``=`` 164¢, ^ = 55¢ | 22-edo: # ``=`` 164¢, ^ = 55¢ (^ = 1/3 #) | ||
quarter-comma meantone: # = 76¢ | quarter-comma meantone: # = 76¢ | ||
fifth-comma meantone: # = 84¢ | fifth-comma meantone: # = 84¢ | ||
third-comma archy: # = 177¢ | third-comma archy: # = 177¢ | ||
eighth-comma porcupine: # ``=`` 157¢, ^ = 52¢ | eighth-comma porcupine: # ``=`` 157¢, ^ = 52¢ (^ = 1/3 #) | ||
sixth-comma srutal: # ``=`` 139¢, ^ = 33¢ | sixth-comma srutal: # ``=`` 139¢, ^ = 33¢ (no fixed relationship between ^ and #) | ||
third-comma injera: # ``=`` 63¢, ^ = 31¢ (third-comma = 1/3 of 81/80) | third-comma injera: # ``=`` 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma = 1/3 of 81/80) | ||
eighth-comma hedgehog: # ``=`` 157¢, ^ = 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243) | eighth-comma hedgehog: # ``=`` 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma = 1/8 of 250/243) | ||
Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different. | Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different. | ||
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There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts: | There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts: | ||
* For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < |y| <= n/2 | * For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 < |x| <= m/2 and 0 < |y| <= n/2 | ||
* For false doubles using single-pair notation, E = E', but x and y are usually different | * x is the count for E with P8, and xE is P8's **multi-enharmonic**, or **multi-E** for short | ||
* For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics | |||
* The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE", and P8 = mP + xE" | * The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE", and P8 = mP + xE" | ||
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To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2. | To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2. | ||
A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3). | A false-double pergen can optionally use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3). | ||
Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span>``//``ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain: | Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span>``//``ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain: | ||
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</span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4 | </span><span style="display: block; text-align: center;">P1 -- /m2 -- ``//``d3=\\A2 -- \M3 -- P4 | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span> | </span><span style="display: block; text-align: center;">C -- Db/ -- Ebb``//``=D#\\ -- E\ -- F</span> | ||
Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2 | To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period). | ||
For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-1,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3dd2 = P5. Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, so ^ = 67¢ + 8.67·c, about a third-tone. | |||
Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd. | |||
Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic. | ||
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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics. | ||
==Chord names and scale names== | ==Chord names and scale names== | ||
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||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* || | ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 || | ||
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[[image:alt-pergenLister.png width="704" height="460"]] | [[image:alt-pergenLister.png width="704" height="460"]] | ||
Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up: | ||
If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3) | If z > 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3) | ||
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== | ==Various proofs (unfinished)== | ||
Given: | Given: | ||
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==Misc notes== | |||
**__Tipping points and sweet spots__** | |||
As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to "tip over". That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly. | |||
Does the temperament's "sweet spot", where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which contains a tipping point, but it remains an open question whether one exists. | |||
The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a __very__ unlikely 5th, and tipping is impossible. As noted, at least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on. | |||
The boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by "far worse" is meant an error twice as large, meantone's sweet spot is from | |||
The tipping point can be found directly from the vanishing comma and the mapping comma. The vanishing comma is simply mapped to the 3-limit to create a 3-limit comma, which implies an edo, which implies a tipping point. For a 5-limit comma (a,b,c), using 81/80 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of this comma. At exactly the tipping point, the error of the 5th from just is C / (b+4c). | |||
__**Combining pergens**__ | |||
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). | |||
General rules for combining pergens: | |||
* (P8/m, M/n) + (P8, P5) = (P8/m, M/n) | |||
* (P8/m, P5) + (P8, M/n) = (P8/m, M/n) | |||
* (P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m') | |||
* (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | |||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. | |||
__**Alternate keyspans and stepspans**__ | |||
One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important. | |||
__** | __**Expanding gedras to 5-limit**__ | ||
Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]: | Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]: | ||
k = 12a + 19b + 28c + 34d | k = 12a + 19b + 28c + 34d | ||
s = 7a + 11b + 14c + 20d | s = 7a + 11b + 14c + 20d | ||
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d = -r | d = -r | ||
The LCM of the pergen's two splitting fractions | __**Height of a pergen**__ | ||
The LCM of the pergen's two splitting fractions could be called the **height** of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The __minimum__ number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed. | |||
__**Credits**__ | |||
Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:52 --> </h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen</title></head><body><!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:52 --> </h1> | ||
<!-- ws:start:WikiTextTocRule: | <!-- ws:start:WikiTextTocRule:88:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 1em;"><a href="#toc0"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 1em;"><a href="#Derivation">Derivation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 1em;"><a href="#Applications">Applications</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 1em;"><a href="#Further Discussion">Further Discussion</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 2em;"><a href="#Further Discussion-Extremely large multigens">Extremely large multigens</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 2em;"><a href="#Further Discussion-Singles and doubles">Singles and doubles</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding an example temperament">Finding an example temperament</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 2em;"><a href="#Further Discussion-Ratio and cents of the accidentals">Ratio and cents of the accidentals</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 2em;"><a href="#Further Discussion-Finding a notation for a pergen">Finding a notation for a pergen</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Further Discussion-Alternate enharmonics">Alternate enharmonics</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#toc12"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Further Discussion-Various proofs (unfinished)">Various proofs (unfinished)</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --></div> | ||
<!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextHeadingRule:54:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:54 --><u><strong>Definition</strong></u></h1> | |||
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We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br /> | We can assign cents to each accidental symbol. First let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. Since the enharmonic = 0¢, we can derive the cents of the accidental. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic. #1 always equals 100¢ + c.<br /> | ||
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In certain edos, the up symbol's cents can be directly related to the sharp's cents. The same can be done for rank-2 pergens if and only if the enharmonic is an A1.<br /> | |||
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15-edo: # <!-- ws:start:WikiTextRawRule:036:``=`` -->=<!-- ws:end:WikiTextRawRule:036 --> 240¢, ^ <!-- ws:start:WikiTextRawRule:037:``=`` -->=<!-- ws:end:WikiTextRawRule:037 --> 80¢<br /> | This suggests a simple format for describing the tuning at the top of the score: the name of the tuning, and the cents of any accidental pairs used, rounded off to the nearest integer. This gives the musician, who may not be well-versed in microtonal theory, the necessary information to play the score. Examples:<br /> | ||
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15-edo: # <!-- ws:start:WikiTextRawRule:036:``=`` -->=<!-- ws:end:WikiTextRawRule:036 --> 240¢, ^ <!-- ws:start:WikiTextRawRule:037:``=`` -->=<!-- ws:end:WikiTextRawRule:037 --> 80¢ (^ = 1/3 #)<br /> | |||
16-edo: # <!-- ws:start:WikiTextRawRule:038:``=`` -->=<!-- ws:end:WikiTextRawRule:038 --> -75¢<br /> | 16-edo: # <!-- ws:start:WikiTextRawRule:038:``=`` -->=<!-- ws:end:WikiTextRawRule:038 --> -75¢<br /> | ||
17-edo: # <!-- ws:start:WikiTextRawRule:039:``=`` -->=<!-- ws:end:WikiTextRawRule:039 --> 141¢, ^ <!-- ws:start:WikiTextRawRule:040:``=`` -->=<!-- ws:end:WikiTextRawRule:040 --> 71¢<br /> | 17-edo: # <!-- ws:start:WikiTextRawRule:039:``=`` -->=<!-- ws:end:WikiTextRawRule:039 --> 141¢, ^ <!-- ws:start:WikiTextRawRule:040:``=`` -->=<!-- ws:end:WikiTextRawRule:040 --> 71¢ (^ = 1/2 #)<br /> | ||
18b-edo: # <!-- ws:start:WikiTextRawRule:041:``=`` -->=<!-- ws:end:WikiTextRawRule:041 --> -133¢, ^ = 67¢<br /> | 18b-edo: # <!-- ws:start:WikiTextRawRule:041:``=`` -->=<!-- ws:end:WikiTextRawRule:041 --> -133¢, ^ = 67¢ (^ = 1/2 #)<br /> | ||
19-edo: # = 63¢<br /> | 19-edo: # = 63¢<br /> | ||
21-edo: ^ = 57¢<br /> | 21-edo: ^ = 57¢ (if used, # = 0¢)<br /> | ||
22-edo: # <!-- ws:start:WikiTextRawRule:042:``=`` -->=<!-- ws:end:WikiTextRawRule:042 --> 164¢, ^ = 55¢<br /> | 22-edo: # <!-- ws:start:WikiTextRawRule:042:``=`` -->=<!-- ws:end:WikiTextRawRule:042 --> 164¢, ^ = 55¢ (^ = 1/3 #)<br /> | ||
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quarter-comma meantone: # = 76¢<br /> | quarter-comma meantone: # = 76¢<br /> | ||
fifth-comma meantone: # = 84¢<br /> | fifth-comma meantone: # = 84¢<br /> | ||
third-comma archy: # = 177¢<br /> | third-comma archy: # = 177¢<br /> | ||
eighth-comma porcupine: # <!-- ws:start:WikiTextRawRule:043:``=`` -->=<!-- ws:end:WikiTextRawRule:043 --> 157¢, ^ = 52¢<br /> | eighth-comma porcupine: # <!-- ws:start:WikiTextRawRule:043:``=`` -->=<!-- ws:end:WikiTextRawRule:043 --> 157¢, ^ = 52¢ (^ = 1/3 #)<br /> | ||
sixth-comma srutal: # <!-- ws:start:WikiTextRawRule:044:``=`` -->=<!-- ws:end:WikiTextRawRule:044 --> 139¢, ^ = 33¢<br /> | sixth-comma srutal: # <!-- ws:start:WikiTextRawRule:044:``=`` -->=<!-- ws:end:WikiTextRawRule:044 --> 139¢, ^ = 33¢ (no fixed relationship between ^ and #)<br /> | ||
third-comma injera: # <!-- ws:start:WikiTextRawRule:045:``=`` -->=<!-- ws:end:WikiTextRawRule:045 --> 63¢, ^ = 31¢ (third-comma = 1/3 of 81/80)<br /> | third-comma injera: # <!-- ws:start:WikiTextRawRule:045:``=`` -->=<!-- ws:end:WikiTextRawRule:045 --> 63¢, ^ = 31¢ (no fixed relationship between ^ and #, third-comma = 1/3 of 81/80)<br /> | ||
eighth-comma hedgehog: # <!-- ws:start:WikiTextRawRule:046:``=`` -->=<!-- ws:end:WikiTextRawRule:046 --> 157¢, ^ = 49¢, / = 52¢ (eighth-comma = 1/8 of 250/243)<br /> | eighth-comma hedgehog: # <!-- ws:start:WikiTextRawRule:046:``=`` -->=<!-- ws:end:WikiTextRawRule:046 --> 157¢, ^ = 49¢, / = 52¢ (/ = 1/3 #, eighth-comma = 1/8 of 250/243)<br /> | ||
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Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.<br /> | Hedgehog is half-8ve third-4th. While the best tuning of a specific temperament is found by minimizing the mistuning of certain ratios, the best tuning of a general pergen is less obvious. Both srutal and injera are half-8ve, but their optimal tunings are very different.<br /> | ||
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There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br /> | There are multiple notations for a given pergen, depending on the enharmonic interval(s). Preferably, the enharmonic's degree will be a unison or a 2nd, because equating two notes a 3rd or 4th apart is very disconcerting. If it's a unison, it will always be an A1. (P1 would be pointless, d1 would be inverted to A1, and AA1 would be split into two A1's.) If it's a 2nd, preferably it will be a m2 or a d2 or a dd2, and not a M2 or an A2 or a ddd2. There is an easy method for finding such a pergen, if one exists. First, some terminology and basic concepts:<br /> | ||
<ul><li>For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>For false doubles using single-pair notation, E = E', but x and y are usually different</li><li>The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE&quot;, and P8 = mP + xE&quot;</li></ul><br /> | <ul><li>For (P8/m, M/n), P8 = mP + xE and M = nG + yE', with 0 &lt; |x| &lt;= m/2 and 0 &lt; |y| &lt;= n/2</li><li>x is the count for E with P8, and xE is P8's <strong>multi-enharmonic</strong>, or <strong>multi-E</strong> for short</li><li>For false doubles using single-pair notation, E = E', but x and y are usually different, making different multi-enharmonics</li><li>The unreduced pergen is (P8/m, M'/n'), with M' = n'G' + zE&quot;, and P8 = mP + xE&quot;</li></ul><br /> | ||
The <strong>keyspan</strong> of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The <strong>stepspan</strong> of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.<br /> | The <strong>keyspan</strong> of an interval is the number of keys or frets or semitones that the interval spans in 12-edo. Most musicians know that a minor 2nd is one key or fret and a major 2nd is two keys or frets. The keyspans of larger intervals aren't as well known. The concept can easily be expanded to other edos, but we'll assume 12-edo for now. The <strong>stepspan</strong> of an interval is simply the degree minus one. M2, m2, A2 and d2 all have a stepspan of 1. P5, d5 and A5 all have stepspan 4. The stepspan can be thought of as the 7-edo keyspan. This concept can be expanded to include pentatonicism, octotonicism, etc., but we'll assume heptatonicism for now.<br /> | ||
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To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.<br /> | To find the double-pair notation for a true double pergen, find each pair from each half of the pergen. Each pair has its own enharmonic. For (P8/2, P4/2), the split octave implies P = vA4 and E = ^^d2, and the split 4th implies G = /M2 and E' = \\m2.<br /> | ||
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A false-double pergen can use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br /> | A false-double pergen can optionally use double-pair notation, which is found as if it were a true double. Double-pair notation is often preferable when the single-pair enharmonic is not a unison or a 2nd, as with (P8/2, P4/3).<br /> | ||
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Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span><!-- ws:start:WikiTextRawRule:047:``//`` -->//<!-- ws:end:WikiTextRawRule:047 -->ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:<br /> | Even single-split pergens may benefit from double-pair notation. For example, (P8, P11/4) has an enharmonic that's a 3rd: P11/4 = [17,10]/4 = [4,2] = M3, and P11 - 4⋅M3 = [1,2] = dd3. So E = v<span style="vertical-align: super;">4</span>dd3, and G = ^M3. But by using double-pair notation, we can avoid that. We find P11/2, which equals two generators: P11/2 = 2⋅G = [17,10]/2 = [8,5] = m6. The bare enharmonic is P11 - 2⋅m6 = [1,0] = A1. For this second enharmonic, we use the second pair of accidentals: E' = \\A1 and 2⋅G = /m6 or \M6. The sum or difference of two enharmonic intervals is also an enharmonic: E + E' = v<span style="vertical-align: super;">4</span>\\d3 = 2·vv\m2, and E - E' = v<span style="vertical-align: super;">4</span><!-- ws:start:WikiTextRawRule:047:``//`` -->//<!-- ws:end:WikiTextRawRule:047 -->ddd3 = 2·vv/d2. Thus vv\m2 and vv/d2 are equivalent enharmonics, and v\4 and v/d4 are equivalent generators. Here is the genchain:<br /> | ||
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</span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:050:``//`` -->//<!-- ws:end:WikiTextRawRule:050 -->d3=\\A2 -- \M3 -- P4<br /> | </span><span style="display: block; text-align: center;">P1 -- /m2 -- <!-- ws:start:WikiTextRawRule:050:``//`` -->//<!-- ws:end:WikiTextRawRule:050 -->d3=\\A2 -- \M3 -- P4<br /> | ||
</span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:051:``//`` -->//<!-- ws:end:WikiTextRawRule:051 -->=D#\\ -- E\ -- F</span><br /> | </span><span style="display: block; text-align: center;">C -- Db/ -- Ebb<!-- ws:start:WikiTextRawRule:051:``//`` -->//<!-- ws:end:WikiTextRawRule:051 -->=D#\\ -- E\ -- F</span><br /> | ||
Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2 | To search for alternate enharmonics, convert E to a gedra, then multiply it by the count to get the multi-E (multi-enharmonic). The count is always the number of ups or downs in the generator (or period). The count is positive only if G (or P) is upped and E is downed, or vice versa. Add or subtract the splitting fraction n (or m) to/from either half of the gedra as desired to get a new multi-E. If the stepspan is negative, or if it's zero and the keyspan is negative, invert it. If the two halves of the new gedra have a common factor, simplify the gedra by this factor, which becomes the new count. Convert the simplified gedra to a 3-limit interval. Add n (or m) ups or downs, this is the new E. Choose between ups and downs according to whether the 5th falls in the enharmonic's upping or downing range, see above. Add n·count ups or downs to the new multi-E. Add or subtract the new multi-E from the multigen (or the octave) to get an interval which splits cleanly into m or n parts. Each part is the new generator (or period).<br /> | ||
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For example, (P8, P5/3) has n = 3, G = ^M2, and E = v<span style="vertical-align: super;">3</span>m2 = [1,1]. G is upped only once, so the count is 1, and the multi-E is also [1,1]. Subtract n from the gedra's keyspan to make a new multi-E [-2,1]. This can't be simplified, so the new E is also [-2,1] = d<span style="vertical-align: super;">3</span>2. Assuming a reasonably just 5th, E needs to be upped, so E = ^<span style="vertical-align: super;">3</span>d<span style="vertical-align: super;">3</span>2. Add the multi-E ^<span style="vertical-align: super;">3</span>[-1,1] to the multigen P5 = [7,4] to get ^<span style="vertical-align: super;">3</span>[5,3]. This isn't divisible by n, so we must subtract instead: [7,4] - ^<span style="vertical-align: super;">3</span>[-2,1] = v<span style="vertical-align: super;">3</span>[9,3] = 3·v[3,1], and G = vA2. Use the equation M = n·G + y·E to check: 3·vA2 + 1·^3dd2 = P5. Here are the genchains: P1 -- vA2=^^dd3 -- ^d4 -- P5 and C -- D#v -- Fb^ -- G. d<span style="vertical-align: super;">3</span>2 = -200¢ - 26·c, so ^ = 67¢ + 8.67·c, about a third-tone.<br /> | |||
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Approaching pergens in a higher-primes-agnostic way, independently of specific commas or temperaments, one enharmonic (and one notation) will usually be obviously superior. But sometimes the temperament being notated implies a certain enharmonic. Specifically, the comma tempered out should map to the enharmonic, or some multiple of it. For example, consider semaphore (2.3.7 and 49/48), which is half-4th. Assuming 7/4 is a m7, the comma is a m2, and a vvm2 enharmonic makes sense. G = ^M2 and the genchain is C -- D^=Ebv -- F. But consider double-large deep yellow, which tempers out (-22, 11, 2). This temperament is also half-4th. The comma is a descending dd2, thus E = ^^dd2, G = vA2, and the genchain is C -- D#v=Ebb^ -- F. This is the best notation because the (-10, 5, 1) generator is an augmented 2nd.<br /> | |||
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Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br /> | Of course, the 2nd comma is much more obscure than 49/48, and much harder to pump. Most half-4th commas are in fact minor 2nds, and the vvm2 enharmonic is indeed a superior notation. But there is another situation in which alternate harmonics arise. There is no consensus on how to map primes 11 and 13 to the 3-limit. 11/8 can be either P4 or A4, and 13/8 can be either m6 or M6. The choice can affect the choice of enharmonic.<br /> | ||
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This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br /> | This is all for single-comma temperaments. Each comma of a multiple-comma temperament also implies an enharmonic, and they may conflict. True double pergens, which are always multi-comma, have multiple notations. For example, the half-everything pergen has 3 possible notations, all equally valid. Even single-split pergens can have multiple commas that imply different enharmonics.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:74:&lt;h2&gt; --><h2 id="toc11"><a name="Further Discussion-Chord names and scale names"></a><!-- ws:end:WikiTextHeadingRule:74 -->Chord names and scale names</h2> | ||
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Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | Using pergens, all rank-2 chords can be named using ups and downs, and if needed highs and lows as well. See the <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a> page for chord naming conventions. The genchain and/or the perchain creates a lattice in which each note and each interval has its own name. The many enharmonic equivalents allow proper chord spelling.<br /> | ||
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Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.<br /> | Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Pergens and MOS scales"></a><!-- ws:end:WikiTextHeadingRule:78 -->Pergens and MOS scales</h2> | ||
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Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br /> | Every rank-2 pergen generates certain MOS scales. This of course depends on the exact size of the generator. In this table, the 5th is assumed to be between 4\7 and 3\5. Sometimes the genchain is too short to generate the multigen. For example, (P8/3, P4/2) [6] has 3 genchains, each with only 2 notes, and thus only 1 step. But it takes 2 steps to make a 4th, so the scale doesn't actually contain any 4ths. Such scales are marked with an asterisk.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:80:&lt;h2&gt; --><h2 id="toc14"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:80 -->Pergens and EDOs</h2> | ||
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Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | ||
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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | ||
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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | ||
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If z &gt; 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)<br /> | If z &gt; 0, then y is at least ceiling (x·(3/2 - z/2)) and at most floor (x·5/3)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Various proofs (unfinished)"></a><!-- ws:end:WikiTextHeadingRule:84 -->Various proofs (unfinished)</h2> | ||
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Given:<br /> | Given:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:86 -->Misc notes</h2> | |||
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<strong><u>Tipping points and sweet spots</u></strong><br /> | |||
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As noted above, in the chord names section, the 5th of pajara (half-8ve) tends to be sharp, thus it has E = ^^d2. But injera, also half-8ve, has a flat 5th, and thus E = vvd2. The tipping point for half-octave with a d2 enharmonic is 700¢, 12-edo's 5th. It is fine for two temperaments with the same pergen to be on opposite sides of the tipping point. But it isn't acceptable for a single temperament to &quot;tip over&quot;. That would lead to the up symbol sometimes meaning down in pitch. Or even worse, the direction of ups and downs for a piece would reverse if the tuning is adjusted slightly.<br /> | |||
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Does the temperament's &quot;sweet spot&quot;, where the damage to those JI ratios likely to occur in chords is minimized, ever contain the tipping point? No single-comma temperament has yet been found which contains a tipping point, but it remains an open question whether one exists. <br /> | |||
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The tipping point is affected by the choice of enharmonic. Half-8ve can be notated with E = vvM2. The tipping point becomes 600¢, a <u>very</u> unlikely 5th, and tipping is impossible. As noted, at least for single-comma temperaments, the choice of E is usually dictated by the 3-limit mapping of the comma. Thus for 5-limit and 7-imit temperaments, the choice of E is a given. However, the mapping of primes 11 and 13 is not agreed on.<br /> | |||
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The boundaries of the sweet spot can be defined loosely such that going beyond them makes the error of some such JI ratios better and others far worse. If by &quot;far worse&quot; is meant an error twice as large, meantone's sweet spot is from<br /> | |||
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The tipping point can be found directly from the vanishing comma and the mapping comma. The vanishing comma is simply mapped to the 3-limit to create a 3-limit comma, which implies an edo, which implies a tipping point. For a 5-limit comma (a,b,c), using 81/80 = (-4,4,-1) for mapping, the 3-limit comma is (a,b,c) + c·(-4,4,-1) = (a-4c, b+4c). This is often a descending interval, and the comma will need to be inverted. Let C = the cents of this comma. At exactly the tipping point, the error of the 5th from just is C / (b+4c).<br /> | |||
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<u><strong>Combining pergens</strong></u><br /> | |||
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Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br /> | |||
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General rules for combining pergens:<br /> | |||
<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li></ul><br /> | |||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<br /> | |||
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<u><strong>Alternate keyspans and stepspans</strong></u><br /> | |||
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One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.<br /> | |||
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<u><strong> | <u><strong>Expanding gedras to 5-limit</strong></u><br /> | ||
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Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:<br /> | Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:<br /> | ||
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k = 12a + 19b + 28c + 34d<br /> | k = 12a + 19b + 28c + 34d<br /> | ||
s = 7a + 11b + 14c + 20d<br /> | s = 7a + 11b + 14c + 20d<br /> | ||
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d = -r<br /> | d = -r<br /> | ||
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The LCM of the pergen's two splitting fractions | <u><strong>Height of a pergen</strong></u><br /> | ||
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The LCM of the pergen's two splitting fractions could be called the <strong>height</strong> of the pergen. For example, (P8, P5) has height 1, and (P8/2, M2/4) has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The <u>minimum</u> number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.<br /> | |||
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<u><strong>Credits</strong></u><br /> | |||
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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.</body></html></pre></div> | |||