User:CritDeathX/Sam's 19-note Well Temperament - Revision history

CritDeathX: /* Choosing Our Intervals */

2020-09-05T15:34:12Z

Choosing Our Intervals

← Older revision		Revision as of 15:34, 5 September 2020
Line 145:		Line 145:
	\|A3		\|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=1&horiz=3&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 ==		== The Scale ==

CritDeathX: /* The Math Behind It */

2020-07-22T16:04:03Z

The Math Behind It

← Older revision		Revision as of 16:04, 22 July 2020
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	== The Math Behind It ==		== 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:		Starting with 5/4, we can choose the intervals we want and end up with this equation:

CritDeathX: /* The Math Behind It */

2020-07-22T15:59:04Z

The Math Behind It

← Older revision		Revision as of 15:59, 22 July 2020
Line 25:		Line 25:
	Multiplying our answer by 1200 gives 443.129594209c. This is around the same as 19EDO.		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"		{\| class="wikitable"
	!1/1		!1/1

CritDeathX: /* The Math Behind It */

2020-07-22T15:58:10Z

The Math Behind It

← Older revision		Revision as of 15:58, 22 July 2020
Line 78:		Line 78:
	Just by this simple interval chain alone, we can notice a couple things:		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.		# 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.		# 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.		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.

CritDeathX: /* The Scale */

2020-07-22T15:56:05Z

The Scale

← Older revision		Revision as of 15:56, 22 July 2020
Line 342:		Line 342:
	\|		\|
	\|1140.55		\|1140.55
	\| -3.51		\| +3.51
	\|		\|
	\|27/14		\|27/14

CritDeathX: /* Choosing Our Intervals */

2020-07-22T15:52:48Z

Choosing Our Intervals

← Older revision		Revision as of 15:52, 22 July 2020
Line 145:		Line 145:
	\|A3		\|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=9&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=1&horiz=3&colors=&waveform=triangle&ampenv=organ our 19-note well temperament tuning].

	== The Scale ==		== The Scale ==

CritDeathX: /* Choosing Our Intervals */

2020-07-22T15:48:26Z

Choosing Our Intervals

← Older revision		Revision as of 15:48, 22 July 2020
Line 145:		Line 145:
	\|A3		\|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=9&horiz=1&colors=&waveform=triangle&ampenv=organ our 19-note well temperament tuning].

	== The Scale ==		== The Scale ==

CritDeathX: /* Explaining Why I Wanna Temper 19EDO To The Feds */

2020-07-22T07:48:52Z

Explaining Why I Wanna Temper 19EDO To The Feds

← Older revision		Revision as of 07:48, 22 July 2020
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.		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.		Well, we use two.

CritDeathX: Created page with "== Explaining Why I Wanna Temper 19EDO To The Feds == ''"Sam, 19EDO is already close enough to JI! Why would you need to make a Well Temperamen..."

2020-07-22T07:45:52Z

Created page with "== Explaining Why I Wanna Temper 19EDO To The Feds == ''"Sam, 19EDO is already close enough to JI! Why would you need to make a Well Temperamen..."

New page

== Explaining Why I Wanna Temper 19EDO To The Feds ==
''"Sam, [[19edo|19EDO]] is already close enough to [[Just intonation|JI]]! Why would you need to make a [[Well Temperament|well temperament]] for it?!?!?!?!!!"'' Good question, I'll answer that.

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.

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?

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.

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

''(1log2(5/4) + 2log2(14/9) + 4log2(12/5) + 5log2(3/1)) / (1^2 + 2^2 + 4^2 + 5^2)''

Multiplying our answer by 1200, we get 380.18447121c, which admittedly is still a bit flat, but its better than what 19EDO gives.

For 9/7, because I'm lazy, lets assume we just want to approximate [[9/8]] & 7/4. We get this equation:

''(1log2(9/7) + 5log2(32/9) + 13log2(28/1)) / (1^2 + 5^2 + 13^2)''

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(''note; these are just the closest approximations i could find. they're not representative of the actual limit of this tuning''):
{| class="wikitable"
!1/1
!5/4
![[14/9]]
![[27/14]]
![[6/5]]
![[3/2]]
![[15/8]]
!7/6
![[13/9]]
![[9/5]]
!9/8
|-
|0.0c
|380.18c
|760.37c
|1140.55c
|320.74c
|700.92c
|1081.11c
|261.29c
|641.48c
|1021.66c
|201.85c
|}
{| class="wikitable"
!1/1
!9/7
![[5/3]]
![[14/13]]
![[25/18]]
!9/5
![[15/13]]
!3/2
!27/14
!5/4
![[13/8]]
|-
|0.0c
|443.13c
|886.26c
|129.39c
|572.52c
|1015.65c
|258.78c
|701.91c
|1145.04c
|388.17c
|831.3c
|}
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)
# 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.

== Choosing Our Intervals ==
To demonstrate what intervals we're gonna use, I'll represent our 5/4 chain using A and our 9/7 chain using B. How ever many generators up and down will be represented by an integer beside it.
{| class="wikitable"
!Degree
!Generator Stack
|-
|0
|
|-
|1
|B11
|-
|2
|B3
|-
|3
|B14
|-
|4
|A7
|-
|5
|A4
|-
|6
|B9
|-
|7
|B1
|-
|8
|B-7
|-
|9
|B4
|-
|10
|A8
|-
|11
|B7
|-
|12
|A2
|-
|13
|B10
|-
|14
|B2
|-
|15
|A12
|-
|16
|B5
|-
|17
|A6
|-
|18
|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].

== The Scale ==
{| class="wikitable"
!Degree
!
!Cents
!Difference
!
!JI Approximation
!
!19EDO
!Difference
|-
|0
|
|0.0
|+0
|
|1/1
|
|0.0
|+0
|-
|1
|
|74.43
| +3.76
|
|[[25/24]]
|
|63.16
| -7.48
|-
|2
|
|129.39
| +1.1
|
|14/13
|
|126.32
| -1.97
|-
|3
|
|203.81
| -0.1
|
|9/8
|
|189.47
| -14.44
|-
|4
|
|261.29
| -5.58
|
|7/6
|
|252.63
| -14.24
|-
|5
|
|320.74
| +5.1
|
|6/5
|
|315.79
| +0.15
|-
|6
|
|388.17
| +1.86
|
|5/4
|
|378.95
| -7.36
|-
|7
|
|443.13
| +8.13
|
|9/7
|
|442.11
| +7.11
|-
|8
|
|498.09
| +0.04
|
|4/3
|
|505.26
| +7.24
|-
|9
|
|572.52
| +3.8
|
|25/18
|
|568.42
| -0.03
|-
|10
|
|641.48
| +10.2
|
|36/25
|
|631.58
| +0.3
|-
|11
|
|701.91
| -0.04
|
|3/2
|
|694.74
| -7.21
|-
|12
|
|760.37
| -4.55
|
|14/9
|
|757.89
| -7.03
|-
|13
|
|831.29
| -9.24
|
|13/8
|
|821.05
| -19.48
|-
|14
|
|886.26
| +1.9
|
|5/3
|
|884.21
| -0.16
|-
|15
|
|962.21
| -6.62
|
|7/4
|
|947.37
| -21.46
|-
|16
|
|1015.65
| -1.94
|
|9/5
|
|1010.53
| -7.06
|-
|17
|
|1081.11
| -7.16
|
|15/8
|
|1073.68
| -14.59
|-
|18
|
|1140.55
| -3.51
|
|27/14
|
|1136.84
| -0.2
|-
|19
|
|1200
|+0
|
|2/1
|
|1200
|+0
|}

== Little Observations ==
I think I'm confident enough to say that this belongs in the 2.3.5.7.13 [[Just intonation subgroups|subgroup]], the same as 19EDO.

The average difference for 19EDO is 7.649444...c, while our temperament's average is 4.1444...c.