User:CritDeathX/Sam's 19-note Well Temperament: Difference between revisions

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First off, I don't even know if this is an actual well temperament. It probably is, but the way I get to it certainly doesn't scream it to me. So if it isn't, then I'm sorry.

Secondly, the short answer is simple: I wanna make it more accurate. Yeah, 19EDO is close to 1/3-comma [[meantone]], but the [[5/4]] is -7c. For a tuning that everyone says is so pure, I'd rather have a -3c 5/4 than a -7c 5/4. A couple of other JI intervals are also wonky enough to question whether they should be used to approximate the intervals listed, like 4\19 for [[7/6]] (-14c) and 15\19 for [[7/4]] (-21c). Like I said for my [[User:CritDeathX/Sam's 17-note Well Temperament|17-note well temperament]], I don't want to use JI nor use a large EDO. The answer to this dilemma is to make your own temperament.

Secondly, the short answer is simple: I wanna make it more accurate. Yeah, 19EDO is close to 1/3-comma [[meantone]], but the [[5/4]] is -7c. For a tuning that everyone says is so pure, I'd rather have a -3c 5/4 than a -7c 5/4. A couple of other JI intervals are also wonky enough to question whether they should be used to approximate the intervals listed, like 4\19 for [[7/6]] (-14c) and 15\19 for [[7/4]] (-21c). Like I said for my [[User:CritDeathX/Sam's 17-note Well Temperament|17-note well temperament]], I don't want to use JI nor use a large [[EDO]]. The answer to this dilemma is to make your own temperament.

~~So we~~ know what we need to do. So we choose what intervals we want and then we make our tuning, right? Well, I find that there's a bunch of temperaments that don't give me the intervals I want to use. A bit of [https://en.xen.wiki/w/19edo#Linear_temperaments the temperaments listed for 19EDO] have something making it to where around 250c represents 7/6, 950c represents 7/4, etc etc. This doesn't feel right with me, which is kind of a shame. So what do we do if we can't use a straight generator?

We know what we need to do. So we choose what intervals we want and then we make our tuning, right? Well, I find that there's a bunch of temperaments that don't give me the intervals I want to use. A bit of [https://en.xen.wiki/w/19edo#Linear_temperaments the temperaments listed for 19EDO] have something making it to where around 250c represents 7/6, 950c represents 7/4, etc etc. This doesn't feel right with me, which is kind of a shame. So what do we do if we can't use a straight generator?

Well, we use two.

== The Math Behind It ==

Now, since we want to use two generators, we must need to figure out what generators to use by this point. The intervals I have in mind as generators are [[9/7]] and 5/4, just so that we can keep everything in check. Lets use 6\19 for 5/4 and 7\19 for [[9/7]], per tradition.

Now, since we want to use two generators, we must need to figure out what generators to use by this point. The intervals I have in mind as generators are [[9/7]] and 5/4, just so that we can keep everything in check. Lets use 6\19 for 5/4 and 7\19 for 9/7, per tradition.

Starting with 5/4, we can choose the intervals we want and end up with this equation:

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Multiplying our answer by 1200 gives 443.129594209c. This is around the same as 19EDO.

The next question comes to what stacks do we use from each generator? Well, lets see what our generators give~~<sub>(''note; these are just the closest approximations i could find. they're not representative of the actual limit of this tuning'')</sub>~~:

The next question comes to what stacks do we use from each generator? Well, lets see what our generators give:

{| class="wikitable"

!1/1

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Just by this simple interval chain alone, we can notice a couple things:

# Our 9/7 generator has a more accurate 5/4 than our 5/4 generator, strangely enough.

# We have not one, not two, but ''four'' different types of major 2nds; in our 9/7 chain, we can either go 5 generators down (184.35c) or go 14 generators up (203.82c). In our 5/4 chain, we can go 10 generators up (201.85c) or 9 generators up (178.34c)

# We have not one, not two, but ''four'' different types of major 2nds; in our 9/7 chain, we can either go 5 generators down (184.35c) or go 14 generators up (203.82c). In our 5/4 chain, we can go 10 generators up (201.85c) or 9 generators down (178.34c)

# Some intervals here differ in ''very'' small amounts, with our P5s in both chains only differing by a cent.

By these observations, we realize that we've run into a problem: there's too many intervals. How do we decide what stacks go to which interval? I'll do the work for y'all.

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|A3

|}

And we have our wonderful, unequal scale; [https://sevish.com/scaleworkshop/?name=19-note%20Well%20Temperament&data=74.425536299%0A129.388782627%0A203.814318926%0A261.29129847%0A320.73788484%0A388.166347881%0A443.129594209%0A498.092840537%0A572.518376836%0A641.47576968%0A701.907159463%0A760.36894242%0A831.29594209%0A886.259188418%0A962.21365452%0A1015.64797104%0A1081.10682726%0A1140.55341363%0A1200.0&freq=261.625565&midi=60&vert=5&horiz=1&colors=&waveform=triangle&ampenv=organ our 19-note well temperament tuning].

And we have our wonderful, unequal scale; [https://sevish.com/scaleworkshop/?name=19-note%20Well%20Temperament&data=74.425536299%0A129.388782627%0A203.814318926%0A261.29129847%0A320.73788484%0A388.166347881%0A443.129594209%0A498.092840537%0A572.518376836%0A641.47576968%0A701.907159463%0A760.36894242%0A831.29594209%0A886.259188418%0A962.21365452%0A1015.64797104%0A1081.10682726%0A1140.55341363%0A1200.0&freq=261.625565&midi=60&vert=2&horiz=3&colors=&waveform=triangle&ampenv=organ our 19-note well temperament tuning].

== The Scale ==

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|

|1140.55

| -3.51

| +3.51

|

|27/14

@@ Line 4: / Line 4: @@
 First off, I don't even know if this is an actual well temperament. It probably is, but the way I get to it certainly doesn't scream it to me. So if it isn't, then I'm sorry.
-Secondly, the short answer is simple: I wanna make it more accurate. Yeah, 19EDO is close to 1/3-comma [[meantone]], but the [[5/4]] is -7c. For a tuning that everyone says is so pure, I'd rather have a -3c 5/4 than a -7c 5/4. A couple of other JI intervals are also wonky enough to question whether they should be used to approximate the intervals listed, like 4\19 for [[7/6]] (-14c) and 15\19 for [[7/4]] (-21c). Like I said for my [[User:CritDeathX/Sam's 17-note Well Temperament|17-note well temperament]], I don't want to use JI nor use a large EDO. The answer to this dilemma is to make your own temperament.
+Secondly, the short answer is simple: I wanna make it more accurate. Yeah, 19EDO is close to 1/3-comma [[meantone]], but the [[5/4]] is -7c. For a tuning that everyone says is so pure, I'd rather have a -3c 5/4 than a -7c 5/4. A couple of other JI intervals are also wonky enough to question whether they should be used to approximate the intervals listed, like 4\19 for [[7/6]] (-14c) and 15\19 for [[7/4]] (-21c). Like I said for my [[User:CritDeathX/Sam's 17-note Well Temperament|17-note well temperament]], I don't want to use JI nor use a large [[EDO]]. The answer to this dilemma is to make your own temperament.
-So we know what we need to do. So we choose what intervals we want and then we make our tuning, right? Well, I find that there's a bunch of temperaments that don't give me the intervals I want to use. A bit of [https://en.xen.wiki/w/19edo#Linear_temperaments the temperaments listed for 19EDO] have something making it to where around 250c represents 7/6, 950c represents 7/4, etc etc. This doesn't feel right with me, which is kind of a shame. So what do we do if we can't use a straight generator?
+We know what we need to do. So we choose what intervals we want and then we make our tuning, right? Well, I find that there's a bunch of temperaments that don't give me the intervals I want to use. A bit of [https://en.xen.wiki/w/19edo#Linear_temperaments the temperaments listed for 19EDO] have something making it to where around 250c represents 7/6, 950c represents 7/4, etc etc. This doesn't feel right with me, which is kind of a shame. So what do we do if we can't use a straight generator?
 Well, we use two.
 == The Math Behind It ==
-Now, since we want to use two generators, we must need to figure out what generators to use by this point. The intervals I have in mind as generators are [[9/7]] and 5/4, just so that we can keep everything in check. Lets use 6\19 for 5/4 and 7\19 for [[9/7]], per tradition.
+Now, since we want to use two generators, we must need to figure out what generators to use by this point. The intervals I have in mind as generators are [[9/7]] and 5/4, just so that we can keep everything in check. Lets use 6\19 for 5/4 and 7\19 for 9/7, per tradition.
 Starting with 5/4, we can choose the intervals we want and end up with this equation:
@@ Line 25: / Line 25: @@
 Multiplying our answer by 1200 gives 443.129594209c. This is around the same as 19EDO.
-The next question comes to what stacks do we use from each generator? Well, lets see what our generators give<sub>(''note; these are just the closest approximations i could find. they're not representative of the actual limit of this tuning'')</sub>:
+The next question comes to what stacks do we use from each generator? Well, lets see what our generators give:
 {| class="wikitable"
 !1/1
@@ Line 78: / Line 78: @@
 Just by this simple interval chain alone, we can notice a couple things:
 # Our 9/7 generator has a more accurate 5/4 than our 5/4 generator, strangely enough.
-# We have not one, not two, but ''four'' different types of major 2nds; in our 9/7 chain, we can either go 5 generators down (184.35c) or go 14 generators up (203.82c). In our 5/4 chain, we can go 10 generators up (201.85c) or 9 generators up (178.34c)
+# We have not one, not two, but ''four'' different types of major 2nds; in our 9/7 chain, we can either go 5 generators down (184.35c) or go 14 generators up (203.82c). In our 5/4 chain, we can go 10 generators up (201.85c) or 9 generators down (178.34c)
 # Some intervals here differ in ''very'' small amounts, with our P5s in both chains only differing by a cent.
 By these observations, we realize that we've run into a problem: there's too many intervals. How do we decide what stacks go to which interval? I'll do the work for y'all.
@@ Line 145: / Line 145: @@
 |A3
 |}
-And we have our wonderful, unequal scale; [https://sevish.com/scaleworkshop/?name=19-note%20Well%20Temperament&data=74.425536299%0A129.388782627%0A203.814318926%0A261.29129847%0A320.73788484%0A388.166347881%0A443.129594209%0A498.092840537%0A572.518376836%0A641.47576968%0A701.907159463%0A760.36894242%0A831.29594209%0A886.259188418%0A962.21365452%0A1015.64797104%0A1081.10682726%0A1140.55341363%0A1200.0&freq=261.625565&midi=60&vert=5&horiz=1&colors=&waveform=triangle&ampenv=organ our 19-note well temperament tuning].
+And we have our wonderful, unequal scale; [https://sevish.com/scaleworkshop/?name=19-note%20Well%20Temperament&data=74.425536299%0A129.388782627%0A203.814318926%0A261.29129847%0A320.73788484%0A388.166347881%0A443.129594209%0A498.092840537%0A572.518376836%0A641.47576968%0A701.907159463%0A760.36894242%0A831.29594209%0A886.259188418%0A962.21365452%0A1015.64797104%0A1081.10682726%0A1140.55341363%0A1200.0&freq=261.625565&midi=60&vert=2&horiz=3&colors=&waveform=triangle&ampenv=organ our 19-note well temperament tuning].
 == The Scale ==
@@ Line 342: / Line 342: @@
 |
 |1140.55
-| -3.51
+| +3.51
 |
 |27/14