User:Rperlner - Revision history

Rperlner at 18:58, 19 December 2020

2020-12-19T18:58:53Z

← Older revision		Revision as of 18:58, 19 December 2020
Line 62:		Line 62:
	Of these, I've looked the most into Maqam music, probably because I really like the sound of neutral thirds -- they really smack one over the head with the fact that a piece is not in the tuning we in the West are used to. Unfortunately, or fascinatingly, depending on how one looks at it, there appears to be a lot of dispute about the proper tuning for Maqam music. Arabic theory uses a notation based on 24 EDO (or at the very least some sort of circle of 5ths where sharps and flats can be split exactly in half.) Turkish music uses a notation with several different accidentals analyzed in terms of 53 EDO. In both cases, multiple sources claim that actual intonation differs systematically from theory, but give only vague hints about how. It's also pretty clear that, while Turkish Makams and Arabic Maqams are related systems, with similar names for similar scales, the notational differences do reflect real differences in intonation, some of which are large enough that the Turkish and Arabic versions of a scale should often be considered completely different scales. Iranian music also has a related tradition called Dagstah, which has its own idiosyncratic notation and intonation. Even within these three traditions, there is a lot of variation from what I can tell.		Of these, I've looked the most into Maqam music, probably because I really like the sound of neutral thirds -- they really smack one over the head with the fact that a piece is not in the tuning we in the West are used to. Unfortunately, or fascinatingly, depending on how one looks at it, there appears to be a lot of dispute about the proper tuning for Maqam music. Arabic theory uses a notation based on 24 EDO (or at the very least some sort of circle of 5ths where sharps and flats can be split exactly in half.) Turkish music uses a notation with several different accidentals analyzed in terms of 53 EDO. In both cases, multiple sources claim that actual intonation differs systematically from theory, but give only vague hints about how. It's also pretty clear that, while Turkish Makams and Arabic Maqams are related systems, with similar names for similar scales, the notational differences do reflect real differences in intonation, some of which are large enough that the Turkish and Arabic versions of a scale should often be considered completely different scales. Iranian music also has a related tradition called Dagstah, which has its own idiosyncratic notation and intonation. Even within these three traditions, there is a lot of variation from what I can tell.

	As an outsider to these traditions, I suspect I have no hope of passing for an insider in using these systems. Nonetheless I aspire to take them as a source of inspiration which I can use to write music that sounds good to me personally (which is frankly what I do with traditions I know better as well.) Here's an example of a piece I wrote using 34 EDO (a system that has been occasionally proposed for approximating Turkish music) to approximate Arabic scales, and push them in a slightly more contrapuntal direction than I believe to be standard. I could probably write pages about my idiosyncratic reasoning for why I think this is a good tuning for Arabic music, but I ~~think I'll spare the reader and just give an example of what I've done with the system~~. (A little more detail is given in the file description.) All I can really say with authority is that it is a good tuning for my little song:		As an outsider to these traditions, I suspect I have no hope of passing for an insider in using these systems. Nonetheless I aspire to take them as a source of inspiration which I can use to write music that sounds good to me personally (which is frankly what I do with traditions I know better as well.) Here's an example of a piece I wrote using 34 EDO (a system that has been occasionally proposed for approximating Turkish music) to approximate Arabic scales, and push them in a slightly more contrapuntal direction than I believe to be standard. I could probably write pages about my idiosyncratic reasoning for why I think this is a good tuning for Arabic music, but I have not yet. (A little more detail is given in the file description.) All I can really say with authority is that it is a good tuning for my little song:

	[[File:A stroll through some retuned maqams.mp3\|thumb\|none\|A Stroll Through some Retuned Maqams]]		[[File:A stroll through some retuned maqams.mp3\|thumb\|none\|A Stroll Through some Retuned Maqams]]

			<H1>Xenharmony Proper</H1>
			Using alternate tunings can also open the door to new sounds and scale structures that are not part of any prior musical tradition. Here the xenharmonic community has the greatest freedom and the greatest responsibility to create its own traditions.

			One of the examples where I see the Xenharmonic community as being furthest along in creating such a new tradition is the family of Porcupine temperaments, that temper out the porcupine comma, 250/243, and usually also the 11 limit commas 55/54, 100/99, and 121/120, dividing the perfect fourth into 3 equal parts. 7 and 8 note scales are very common, and I like to think of the former as a sort of weird version of the diatonic scale, where there's one extra minor third. Many people from the xenharmonic community have worked with this system. My example of this tuning is a little lullaby in 37EDO I wrote around 2010 and subsequently transcribed into Musescore to create an mp3.

			[[File:Porcupine_Lullaby.mp3\|thumb\|none\|Porcupine Lullaby]]

			Another system which has been explored some by the xenharmonic community, but needs to be explored much much more in my opinion is 26EDO. I cannot emphasize enough how much I love this little rank 1 temperament. There are a number of ways to approach 26-edo via higher rank temperaments. It's a near optimal tuning for Flattone, Lemba, and Orgone temperaments, and it works nicely for the partially detempered Octatonic scale I discuss above. But 26-edo is much more than that. It is the smallest equal temperament that can consistently render all 13-limit harmonic intervals, and it is possible to write things in it that sound great. I've liked a lot of things I've heard in this system, but here's one of my own I like a lot. It is a scherzo and it uses both Lemba and Flattone (and is therefore designed to be played in 26 EDO and nothing else.)

			[[File:Scherzo_in_26_EDO_for_Oboe,_Horn_and_Organ.mp3\|thumb\|none\|Scherzo in 26 EDO for Oboe, Horn, and Organ]]

Rperlner at 05:15, 14 December 2020

2020-12-14T05:15:27Z

← Older revision		Revision as of 05:15, 14 December 2020
Line 55:		Line 55:

	The 12 EDO version is pleasant enough to listen to, but I find it less interesting, and at times, less harmonious than some of the nonstandard tunings. (I especially like 26 and 48 EDO here.)		The 12 EDO version is pleasant enough to listen to, but I find it less interesting, and at times, less harmonious than some of the nonstandard tunings. (I especially like 26 and 48 EDO here.)

			<H1> Non-12EDO world music </H1>

			There are a number of living musical traditions that simply do not work in 12EDO, such as Indonesian Gamelan music, and Middle Eastern Maqam/Makam/Dagstah music. Indian music is also decidedly non-12-EDO (using 5-limit JI), but is at least somewhat recognizable if forced into 12EDO, although I suspect it loses more in translation than Baroque or renaissance music written for Meantone.

			Of these, I've looked the most into Maqam music, probably because I really like the sound of neutral thirds -- they really smack one over the head with the fact that a piece is not in the tuning we in the West are used to. Unfortunately, or fascinatingly, depending on how one looks at it, there appears to be a lot of dispute about the proper tuning for Maqam music. Arabic theory uses a notation based on 24 EDO (or at the very least some sort of circle of 5ths where sharps and flats can be split exactly in half.) Turkish music uses a notation with several different accidentals analyzed in terms of 53 EDO. In both cases, multiple sources claim that actual intonation differs systematically from theory, but give only vague hints about how. It's also pretty clear that, while Turkish Makams and Arabic Maqams are related systems, with similar names for similar scales, the notational differences do reflect real differences in intonation, some of which are large enough that the Turkish and Arabic versions of a scale should often be considered completely different scales. Iranian music also has a related tradition called Dagstah, which has its own idiosyncratic notation and intonation. Even within these three traditions, there is a lot of variation from what I can tell.

			As an outsider to these traditions, I suspect I have no hope of passing for an insider in using these systems. Nonetheless I aspire to take them as a source of inspiration which I can use to write music that sounds good to me personally (which is frankly what I do with traditions I know better as well.) Here's an example of a piece I wrote using 34 EDO (a system that has been occasionally proposed for approximating Turkish music) to approximate Arabic scales, and push them in a slightly more contrapuntal direction than I believe to be standard. I could probably write pages about my idiosyncratic reasoning for why I think this is a good tuning for Arabic music, but I think I'll spare the reader and just give an example of what I've done with the system. (A little more detail is given in the file description.) All I can really say with authority is that it is a good tuning for my little song:

			[[File:A stroll through some retuned maqams.mp3\|thumb\|none\|A Stroll Through some Retuned Maqams]]

Rperlner at 16:43, 13 December 2020

2020-12-13T16:43:59Z

← Older revision		Revision as of 16:43, 13 December 2020
Line 20:		Line 20:

	In each case, I prefer the non-12EDO version.		In each case, I prefer the non-12EDO version.

			To be fair to 12 EDO, I will give one example, where I prefer the 12 EDO version of a piece. This time a short Locrian Fugue (with a 31 EDO version given for contrast):

			<table style="width:100%">
			<tr>
			<td>Locrian Fugue in 12 and 31 EDO</td>
			<td>[[File:Locrian Fugue.mp3\|thumb\|none\|Locrian Fugue in 12 EDO]]</td>
			<td>[[File:Locrian Fugue 31 edo.mp3\|thumb\|none\|Locrian Fugue in 31 EDO]]</td>
			</tr>
			</table>

	<H1>Partially De-tempered Octatonic Scale</H1>		<H1>Partially De-tempered Octatonic Scale</H1>

Rperlner: Added link

2020-12-11T02:54:27Z

Added link

← Older revision		Revision as of 02:54, 11 December 2020
Line 1:		Line 1:
	My name is Ray Perlner (the only one as far as I know.) I have a longstanding hobby of writing music for a standard acoustic piano and playing it for friends and family. I also enjoy writing and studying microtonal music, which is by definition, music which cannot be played on my chosen instrument. Why would I do this to myself? Of course, this wiki has a general page on why a person might opt for microtonalism, but I would like to give it a more personal spin using my compositions as examples.		My name is Ray Perlner (the only one as far as I know.) I have a longstanding hobby of writing music for a standard acoustic piano and playing it for friends and family. I also enjoy writing and studying microtonal music, which is by definition, music which cannot be played on my chosen instrument. Why would I do this to myself? Of course, this wiki has a general page on [[Why use alternative tunings?\|why]] a person might opt for microtonalism, but I would like to give it a more personal spin using my compositions as examples.

	First of all, sometimes music I ''can'' play on the piano may sound better in a different tuning. A well known example is that music based on the diatonic scale can be rendered in any [[meantone]] temperament, and not just 12-EDO. Baroque and Renaissance music in particular generally will sound better in tunings ranging from about sixth comma meantone (~55EDO) to quarter-comma meantone (~31edo). Indeed this was what was used when these styles were the hot new thing in Europe. I have tried my hand at this style, and I find my compositions to sound better in these tunings as well.		First of all, sometimes music I ''can'' play on the piano may sound better in a different tuning. A well known example is that music based on the diatonic scale can be rendered in any [[meantone]] temperament, and not just 12-EDO. Baroque and Renaissance music in particular generally will sound better in tunings ranging from about sixth comma meantone (~55EDO) to quarter-comma meantone (~31edo). Indeed this was what was used when these styles were the hot new thing in Europe. I have tried my hand at this style, and I find my compositions to sound better in these tunings as well.

Rperlner: added link

2020-12-11T02:52:40Z

added link

← Older revision		Revision as of 02:52, 11 December 2020
Line 1:		Line 1:
	My name is Ray Perlner (the only one as far as I know.) I have a longstanding hobby of writing music for a standard acoustic piano and playing it for friends and family. I also enjoy writing and studying microtonal music, which is by definition, music which cannot be played on my chosen instrument. Why would I do this to myself? Of course, this wiki has a general page on why a person might opt for microtonalism, but I would like to give it a more personal spin using my compositions as examples.		My name is Ray Perlner (the only one as far as I know.) I have a longstanding hobby of writing music for a standard acoustic piano and playing it for friends and family. I also enjoy writing and studying microtonal music, which is by definition, music which cannot be played on my chosen instrument. Why would I do this to myself? Of course, this wiki has a general page on why a person might opt for microtonalism, but I would like to give it a more personal spin using my compositions as examples.

	First of all, sometimes music I ''can'' play on the piano may sound better in a different tuning. A well known example is that music based on the diatonic scale can be rendered in any meantone temperament, and not just 12-EDO. Baroque and Renaissance music in particular generally will sound better in tunings ranging from about sixth comma meantone (~55EDO) to quarter-comma meantone (~31edo). Indeed this was what was used when these styles were the hot new thing in Europe. I have tried my hand at this style, and I find my compositions to sound better in these tunings as well.		First of all, sometimes music I ''can'' play on the piano may sound better in a different tuning. A well known example is that music based on the diatonic scale can be rendered in any [[meantone]] temperament, and not just 12-EDO. Baroque and Renaissance music in particular generally will sound better in tunings ranging from about sixth comma meantone (~55EDO) to quarter-comma meantone (~31edo). Indeed this was what was used when these styles were the hot new thing in Europe. I have tried my hand at this style, and I find my compositions to sound better in these tunings as well.

	<H1>Diatonic Music in Meantone</H1>		<H1>Diatonic Music in Meantone</H1>

Rperlner at 03:39, 8 December 2020

2020-12-08T03:39:05Z

← Older revision		Revision as of 03:39, 8 December 2020
Line 3:		Line 3:
	First of all, sometimes music I ''can'' play on the piano may sound better in a different tuning. A well known example is that music based on the diatonic scale can be rendered in any meantone temperament, and not just 12-EDO. Baroque and Renaissance music in particular generally will sound better in tunings ranging from about sixth comma meantone (~55EDO) to quarter-comma meantone (~31edo). Indeed this was what was used when these styles were the hot new thing in Europe. I have tried my hand at this style, and I find my compositions to sound better in these tunings as well.		First of all, sometimes music I ''can'' play on the piano may sound better in a different tuning. A well known example is that music based on the diatonic scale can be rendered in any meantone temperament, and not just 12-EDO. Baroque and Renaissance music in particular generally will sound better in tunings ranging from about sixth comma meantone (~55EDO) to quarter-comma meantone (~31edo). Indeed this was what was used when these styles were the hot new thing in Europe. I have tried my hand at this style, and I find my compositions to sound better in these tunings as well.

			<H1>Diatonic Music in Meantone</H1>
	Here are two short fugues I wrote in Phrygian and Mixolydian mode, respectively. Each is rendered in my favorite meantone tuning for that piece, as well as 12 EDO for comparison.		Here are two short fugues I wrote in Phrygian and Mixolydian mode, respectively. Each is rendered in my favorite meantone tuning for that piece, as well as 12 EDO for comparison.

Line 20:		Line 21:
	In each case, I prefer the non-12EDO version.		In each case, I prefer the non-12EDO version.

			<H1>Partially De-tempered Octatonic Scale</H1>
	A less well known example of the benefits of alternate tunings concerns music written using the Octatonic scale -- one of the better-known non-diatonic scales used in 12 EDO music. The standard treatment of this scale is as a MOS of the Rank-2 Diminished temperament, which tempers out the 7-limit commas 50/49, and 36/35. In this analysis, 12 EDO is pretty much optimal already. However, I have found that there is little downside to treating the Octatonic scale as a Rank-3 temperament that only tempers out 50/49 in the 7-limit (Jubilismic Temperament.) This allows the 6/5 and 7/6 minor thirds to be tuned differently, improving the fit to just intonation and the expressiveness of the system, Something similar is done with the Diatonic scale in Indian music, where 5-limit JI (effectively a rank 3 temperament) is used instead of Meantone temperament, rendering 9/8 and 10/9 as separate intervals. The downside of using 5-limit JI for diatonic music, in general, though, is that one is often faced with hard decisions regarding whether the D in C major, for example, should be rendered as being 9/8 or 10/9 relative to the tonic.		A less well known example of the benefits of alternate tunings concerns music written using the Octatonic scale -- one of the better-known non-diatonic scales used in 12 EDO music. The standard treatment of this scale is as a MOS of the Rank-2 Diminished temperament, which tempers out the 7-limit commas 50/49, and 36/35. In this analysis, 12 EDO is pretty much optimal already. However, I have found that there is little downside to treating the Octatonic scale as a Rank-3 temperament that only tempers out 50/49 in the 7-limit (Jubilismic Temperament.) This allows the 6/5 and 7/6 minor thirds to be tuned differently, improving the fit to just intonation and the expressiveness of the system, Something similar is done with the Diatonic scale in Indian music, where 5-limit JI (effectively a rank 3 temperament) is used instead of Meantone temperament, rendering 9/8 and 10/9 as separate intervals. The downside of using 5-limit JI for diatonic music, in general, though, is that one is often faced with hard decisions regarding whether the D in C major, for example, should be rendered as being 9/8 or 10/9 relative to the tonic.

Rperlner at 03:26, 8 December 2020

2020-12-08T03:26:35Z

← Older revision		Revision as of 03:26, 8 December 2020
Line 27:		Line 27:

	we find that while we have two different versions of a number of 9-limit consonant chords that appear in the 12-edo version, both versions are consonant in the 9-limit also in the partially de-tempered version. For example, a dominant 7th chord might either be 4:5:6:7 or 1/9:1/7:1/6:1/5. Likewise, if we treat the half octave as representing 17/12 in addition to 10/7 and 7/5, we can always render some inversion of any diminished 7th chord as 10:12:14:17. The melodic structure is also only moderately more complex than the 12-edo version, featuring a small (s) semitone, and medium (M) and large (L) wholetones in a sMsLsMsL pattern.		we find that while we have two different versions of a number of 9-limit consonant chords that appear in the 12-edo version, both versions are consonant in the 9-limit also in the partially de-tempered version. For example, a dominant 7th chord might either be 4:5:6:7 or 1/9:1/7:1/6:1/5. Likewise, if we treat the half octave as representing 17/12 in addition to 10/7 and 7/5, we can always render some inversion of any diminished 7th chord as 10:12:14:17. The melodic structure is also only moderately more complex than the 12-edo version, featuring a small (s) semitone, and medium (M) and large (L) wholetones in a sMsLsMsL pattern.

			Here is an Octatonic jazz piece based on a shorter theme I wrote for piano in three different Jubilismic tunings that distinguish between 6/5 and 7/6 minor thirds. Ranging from narrowest to widest perfect fifths, they are 26 EDO, 48 EDO, and 22 EDO:

			<table style="width:100%">
			<tr>
			<td>[[File:Octatonic Groove 26 edo.mp3\|thumb\|none\|Octatonic Groove in 26 EDO]]</td>
			<td>[[File:Octatonic Groove 48edo.mp3\|thumb\|none\|Octatonic Groove in 48 EDO]]</td>
			<td>[[File:Octatonic Groove 22 edo.mp3\|thumb\|none\|Octatonic Groove in 22 EDO]]</td>
			</tr>
			</table>

			For contrast, here is the same piece in 12 EDO:

			[[File:Octatonic Groove 12 edo.mp3\|thumb\|none\|Octatonic Groove in 12 EDO]]

			The 12 EDO version is pleasant enough to listen to, but I find it less interesting, and at times, less harmonious than some of the nonstandard tunings. (I especially like 26 and 48 EDO here.)

Rperlner at 02:06, 6 December 2020

2020-12-06T02:06:07Z

← Older revision		Revision as of 02:06, 6 December 2020
Line 20:		Line 20:
	In each case, I prefer the non-12EDO version.		In each case, I prefer the non-12EDO version.

	A less well known example concerns music written using the Octatonic scale -- one of the better-known non-diatonic scales used in 12 EDO music. The standard treatment of this scale is as a MOS of the Rank-2 Diminished temperament, which tempers out the 7-limit commas 50/49, and 36/35. In this analysis, 12 EDO is pretty much optimal already. However, I have found that there is little downside to treating the Octatonic scale as a Rank-3 temperament that only tempers out 50/49 in the 7-limit (Jubilismic Temperament.) This allows the 6/5 and 7/6 minor thirds to be tuned differently, improving the fit to just intonation and the expressiveness of the system, Something similar is done with the Diatonic scale in Indian music, where 5-limit JI (effectively a rank 3 temperament) is used instead of Meantone temperament, rendering 9/8 and 10/9 as separate intervals. The downside of using 5-limit JI for diatonic music, in general, though, is that one is often faced with hard decisions regarding whether the D in C major, for example, should be rendered as being 9/8 or 10/9 relative to the tonic.		A less well known example of the benefits of alternate tunings concerns music written using the Octatonic scale -- one of the better-known non-diatonic scales used in 12 EDO music. The standard treatment of this scale is as a MOS of the Rank-2 Diminished temperament, which tempers out the 7-limit commas 50/49, and 36/35. In this analysis, 12 EDO is pretty much optimal already. However, I have found that there is little downside to treating the Octatonic scale as a Rank-3 temperament that only tempers out 50/49 in the 7-limit (Jubilismic Temperament.) This allows the 6/5 and 7/6 minor thirds to be tuned differently, improving the fit to just intonation and the expressiveness of the system, Something similar is done with the Diatonic scale in Indian music, where 5-limit JI (effectively a rank 3 temperament) is used instead of Meantone temperament, rendering 9/8 and 10/9 as separate intervals. The downside of using 5-limit JI for diatonic music, in general, though, is that one is often faced with hard decisions regarding whether the D in C major, for example, should be rendered as being 9/8 or 10/9 relative to the tonic.

	There are no such difficult decisions in the Octatonic scale in a rank-3 Jubilismic temperament. If we render the semitone-wholetone Octatonic scale as		There are no such difficult decisions in the Octatonic scale in a rank-3 Jubilismic temperament. If we render the semitone-wholetone Octatonic scale as

Rperlner at 00:23, 6 December 2020

2020-12-06T00:23:12Z

← Older revision		Revision as of 00:23, 6 December 2020
Line 26:		Line 26:
	1 : 15/14 : 7/6 : 5/4 : 7/5=10/7 : 3/2 : 5/3 : 7/4 : 2,		1 : 15/14 : 7/6 : 5/4 : 7/5=10/7 : 3/2 : 5/3 : 7/4 : 2,

	we find that while we have two different versions of a number of 9-limit consonant chords that appear in the 12-edo version, both versions are consonant in the 9-limit also in the partially de-tempered version. For example, a dominant 7th chord might either be 4:5:6:7 or 1/9:1/7:1/6:1/5. Likewise, if we treat the half octave as representing 17/12 in addition to 10/7 and 7/5, we can always render some inversion of any diminished 7th chord as 10:12:14:17. The melodic structure is also only moderately more complex than the 12-edo version, featuring a small (s) semitone, and medium (M) and large(L) wholetones in a sMsLsMsL pattern.		we find that while we have two different versions of a number of 9-limit consonant chords that appear in the 12-edo version, both versions are consonant in the 9-limit also in the partially de-tempered version. For example, a dominant 7th chord might either be 4:5:6:7 or 1/9:1/7:1/6:1/5. Likewise, if we treat the half octave as representing 17/12 in addition to 10/7 and 7/5, we can always render some inversion of any diminished 7th chord as 10:12:14:17. The melodic structure is also only moderately more complex than the 12-edo version, featuring a small (s) semitone, and medium (M) and large (L) wholetones in a sMsLsMsL pattern.

Rperlner at 00:22, 6 December 2020

2020-12-06T00:22:12Z

← Older revision		Revision as of 00:22, 6 December 2020
Line 22:		Line 22:
	A less well known example concerns music written using the Octatonic scale -- one of the better-known non-diatonic scales used in 12 EDO music. The standard treatment of this scale is as a MOS of the Rank-2 Diminished temperament, which tempers out the 7-limit commas 50/49, and 36/35. In this analysis, 12 EDO is pretty much optimal already. However, I have found that there is little downside to treating the Octatonic scale as a Rank-3 temperament that only tempers out 50/49 in the 7-limit (Jubilismic Temperament.) This allows the 6/5 and 7/6 minor thirds to be tuned differently, improving the fit to just intonation and the expressiveness of the system, Something similar is done with the Diatonic scale in Indian music, where 5-limit JI (effectively a rank 3 temperament) is used instead of Meantone temperament, rendering 9/8 and 10/9 as separate intervals. The downside of using 5-limit JI for diatonic music, in general, though, is that one is often faced with hard decisions regarding whether the D in C major, for example, should be rendered as being 9/8 or 10/9 relative to the tonic.		A less well known example concerns music written using the Octatonic scale -- one of the better-known non-diatonic scales used in 12 EDO music. The standard treatment of this scale is as a MOS of the Rank-2 Diminished temperament, which tempers out the 7-limit commas 50/49, and 36/35. In this analysis, 12 EDO is pretty much optimal already. However, I have found that there is little downside to treating the Octatonic scale as a Rank-3 temperament that only tempers out 50/49 in the 7-limit (Jubilismic Temperament.) This allows the 6/5 and 7/6 minor thirds to be tuned differently, improving the fit to just intonation and the expressiveness of the system, Something similar is done with the Diatonic scale in Indian music, where 5-limit JI (effectively a rank 3 temperament) is used instead of Meantone temperament, rendering 9/8 and 10/9 as separate intervals. The downside of using 5-limit JI for diatonic music, in general, though, is that one is often faced with hard decisions regarding whether the D in C major, for example, should be rendered as being 9/8 or 10/9 relative to the tonic.

	There are no such difficult decisions in the Octatonic scale in a rank-3 Jubilismic temperament. If we render the semitone-wholetone Octatonic scale as 1 : 15/14 : 7/6 : 5/4 : 7/5=10/7 : 3/2 : 5/3 : 7/4 : 2, we find that while we have two different versions of a number of 9-limit consonant chords that appear in the 12-edo version, both versions are consonant in the 9-limit also in the partially de-tempered version. For example, a dominant 7th chord might either be 4:5:6:7 or 1/9:1/7:1/6:1/5. Likewise, if we treat the half octave as representing 17/12 in addition to 10/7 and 7/5, we can always render some inversion of any diminished 7th chord as 10:12:14:17. The melodic structure is also only moderately more complex than the 12-edo version, featuring a small (s) semitone, and medium (M) and large(L) wholetones in a sMsLsMsL pattern.		There are no such difficult decisions in the Octatonic scale in a rank-3 Jubilismic temperament. If we render the semitone-wholetone Octatonic scale as

			1 : 15/14 : 7/6 : 5/4 : 7/5=10/7 : 3/2 : 5/3 : 7/4 : 2,

			we find that while we have two different versions of a number of 9-limit consonant chords that appear in the 12-edo version, both versions are consonant in the 9-limit also in the partially de-tempered version. For example, a dominant 7th chord might either be 4:5:6:7 or 1/9:1/7:1/6:1/5. Likewise, if we treat the half octave as representing 17/12 in addition to 10/7 and 7/5, we can always render some inversion of any diminished 7th chord as 10:12:14:17. The melodic structure is also only moderately more complex than the 12-edo version, featuring a small (s) semitone, and medium (M) and large(L) wholetones in a sMsLsMsL pattern.