Mailing list post
From: Keenan Pepper (2012-11-29)
Subject: An interesting cluster temperament
I wanted to write a little demo in this temperament before posting, but I don't want to just forget about it. Maybe demo will come later.
If you don't know what a "cluster temperament" is, start with http://xenharmonic.wikispaces.com/Cluster+temperament (and, needless to say, feel free to edit that).
First I'll give a sort of "construction" of slendric temperament, then do the same thing with this new temperament I'm discussing and argue that it's analogous to slendric.
A simple-as-possible JI scale that has 5 roughly equal steps is going to contain 9/8, 8/7, and 7/6 as steps. (This is because 240 cents is between 8/7 and 7/6, but you need another interval from the 2.3.7 subgroup in order to hit 2/1.) Such scales often appear with names indicating they're JI versions of slendro. What temperings of such scales make sense?
The most extreme thing to do is temper 9/8, 8/7, and 7/6 all together, obtaining 5edo. More mild forms of temperament temper 9/8 and 8/7 together (superpyth) or 8/7 and 7/6 (semaphore). But an even more accurate form of temperament is to keep 9/8, 8/7, and 7/6 all as distinct intervals, but temper them so that their sizes are in arithmetic progression. In other words the difference between 9/8 and 8/7 (64/63) is tempered together with the difference between 8/7 and 7/6 (49/48), resulting in a multi-purpose comma. This yields a cluster temperament known as slendric. All reasonably-sized slendric MOSes consist of five clusters of pitches. The clusters correspond to the steps of 5edo (as a 2.3.7 temperament), and within each cluster, the pitches are separated by the 64/63~49/48 multi-purpose comma.
The most important extensions of slendric to higher-complexity intervals are cynder, which makes [9/8, 8/7, 7/6, 6/5] equally spaced, and rodan, which makes [10/9, 9/8, 8/7, 7/6] equally spaced. So they both make the multi-purpose comma even more multi-purpose, but in different ways.
What if we start from a 7-note scale rather than a 5-note scale?
A simple-as-possible JI scale with 7 roughly equally spaced notes will contain 12/11, 11/10, 10/9, and 9/8 as steps. Examples of such scales are al-farabi_g5.scl, diaphonic_7.scl, ptolemy_hom.scl, and ptolemy_hominv.scl from the Scala archive. If we temper all of these steps together we get 7edo; if we temper only some of them together we can get a variety of things including mohaha and porcupine. But what if we keep them all distinct but temper them so that they're equally spaced from each other?
That results in this 2.3.5.11 temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=7_65&limit=2_3_5_11 . I would give this temperament a name, but it already has the systematic names "2.3.5.11 marvo" and "2.3.5.11 zarvo" and I'm hesitant to contribute to the explosion of unique names. (The name "arvo" might be nice except it might be taken to imply a connection to Arvo Part...) So I'll just refer to it as "zarvo", with the understanding that in this context that refers to 2.3.5.11 zarvo.
Zarvo is thus an analog of slendric but based around quasi-7edo scales in the 2.3.5.11 subgroup rather than quasi-5edo scales in the 2.3.7 subgroup.
All reasonably-sized zarvo MOSes (up to at least zarvo[58]!) consist of seven clusters of pitches. The pitches correspond to the steps of 7edo (as a 2.3.5.11 temperament), and within each cluster the pitches are separated by the multi-purpose comma 121/120~100/99~81/80. Here's what the 1\7 cluster looks like:
....16/15 27/25 12/11 11/10 10/9 9/8 25/22...
...114.69 131.98 149.27 166.55 183.84 201.13 218.42...
I notice that these happen to correspond quite well with some of Margo Schulter's interval categories. (http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt)
Some other features of zarvo temperament: 243/242 is tempered out, so 11/9~27/22 is a true "neutral third". Also, 4000/3993 is tempered out, so 4/3 is divided into three equal parts, each representing 11/10. Zarvo is the only 2.3.5.11 temperament with these two properties.
Zarvo has a wealth of interesting 4/3 tetrachords, including all the permutations of these:
16/15 + 11/10 + 25/22
16/15 + 10/9 + 9/8, "intense diatonic" / "5-limit diatonic"
27/25 + 12/11 + 25/22
27/25 + 11/10 + 9/8
27/25 + 10/9 + 10/9
12/11 + 12/11 + 9/8, "bayati-like" or "rast-like"
12/11 + 11/10 + 10/9, "equable diatonic"
11/10 + 11/10 + 11/10, "equal tetrachord"
Another wonderful thing about zarvo is that, since it's such an accurate temperament, many combination tones are usable, and indeed if you play with distortion you can form interesting difference tone melodies by shifting between different pitches in the same cluster.
I hope that this kind of explanation encourages people to view cluster temperaments zarvo as more than simply "unreasonably complex" temperaments with "bad" or "very uneven" MOS structure. Many people already report perceiving intervals as a generic category (usually 12edo-based) plus an intonational inflection, so temperaments that are inherently structured like that could be very natural.
If you want to experiment with zarvo temperament I would suggest simply using 65edo, but for those who love new scala files here is one:
! zarvo30.scl
!
TOP 2.3.5.11 zarvo[30]
30
!
17.28820
34.57639
149.26526
166.55346
183.84165
201.12985
315.81872
333.10692
350.39511
367.68331
384.97150
499.66037
516.94857
534.23677
551.52496
666.21383
683.50203
700.79022
718.07842
832.76729
850.05549
867.34368
884.63188
901.92007
1016.60894
1033.89714
1051.18534
1068.47353
1183.16240
1200.45060
Keenan
Full thread (4 messages)
From: Keenan Pepper (2012-11-29)
Subject: An interesting cluster temperament
I wanted to write a little demo in this temperament before posting, but I don't want to just forget about it. Maybe demo will come later.
If you don't know what a "cluster temperament" is, start with http://xenharmonic.wikispaces.com/Cluster+temperament (and, needless to say, feel free to edit that).
First I'll give a sort of "construction" of slendric temperament, then do the same thing with this new temperament I'm discussing and argue that it's analogous to slendric.
A simple-as-possible JI scale that has 5 roughly equal steps is going to contain 9/8, 8/7, and 7/6 as steps. (This is because 240 cents is between 8/7 and 7/6, but you need another interval from the 2.3.7 subgroup in order to hit 2/1.) Such scales often appear with names indicating they're JI versions of slendro. What temperings of such scales make sense?
The most extreme thing to do is temper 9/8, 8/7, and 7/6 all together, obtaining 5edo. More mild forms of temperament temper 9/8 and 8/7 together (superpyth) or 8/7 and 7/6 (semaphore). But an even more accurate form of temperament is to keep 9/8, 8/7, and 7/6 all as distinct intervals, but temper them so that their sizes are in arithmetic progression. In other words the difference between 9/8 and 8/7 (64/63) is tempered together with the difference between 8/7 and 7/6 (49/48), resulting in a multi-purpose comma. This yields a cluster temperament known as slendric. All reasonably-sized slendric MOSes consist of five clusters of pitches. The clusters correspond to the steps of 5edo (as a 2.3.7 temperament), and within each cluster, the pitches are separated by the 64/63~49/48 multi-purpose comma.
The most important extensions of slendric to higher-complexity intervals are cynder, which makes [9/8, 8/7, 7/6, 6/5] equally spaced, and rodan, which makes [10/9, 9/8, 8/7, 7/6] equally spaced. So they both make the multi-purpose comma even more multi-purpose, but in different ways.
What if we start from a 7-note scale rather than a 5-note scale?
A simple-as-possible JI scale with 7 roughly equally spaced notes will contain 12/11, 11/10, 10/9, and 9/8 as steps. Examples of such scales are al-farabi_g5.scl, diaphonic_7.scl, ptolemy_hom.scl, and ptolemy_hominv.scl from the Scala archive. If we temper all of these steps together we get 7edo; if we temper only some of them together we can get a variety of things including mohaha and porcupine. But what if we keep them all distinct but temper them so that they're equally spaced from each other?
That results in this 2.3.5.11 temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=7_65&limit=2_3_5_11 . I would give this temperament a name, but it already has the systematic names "2.3.5.11 marvo" and "2.3.5.11 zarvo" and I'm hesitant to contribute to the explosion of unique names. (The name "arvo" might be nice except it might be taken to imply a connection to Arvo Part...) So I'll just refer to it as "zarvo", with the understanding that in this context that refers to 2.3.5.11 zarvo.
Zarvo is thus an analog of slendric but based around quasi-7edo scales in the 2.3.5.11 subgroup rather than quasi-5edo scales in the 2.3.7 subgroup.
All reasonably-sized zarvo MOSes (up to at least zarvo[58]!) consist of seven clusters of pitches. The pitches correspond to the steps of 7edo (as a 2.3.5.11 temperament), and within each cluster the pitches are separated by the multi-purpose comma 121/120~100/99~81/80. Here's what the 1\7 cluster looks like:
....16/15 27/25 12/11 11/10 10/9 9/8 25/22...
...114.69 131.98 149.27 166.55 183.84 201.13 218.42...
I notice that these happen to correspond quite well with some of Margo Schulter's interval categories. (http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt)
Some other features of zarvo temperament: 243/242 is tempered out, so 11/9~27/22 is a true "neutral third". Also, 4000/3993 is tempered out, so 4/3 is divided into three equal parts, each representing 11/10. Zarvo is the only 2.3.5.11 temperament with these two properties.
Zarvo has a wealth of interesting 4/3 tetrachords, including all the permutations of these:
16/15 + 11/10 + 25/22
16/15 + 10/9 + 9/8, "intense diatonic" / "5-limit diatonic"
27/25 + 12/11 + 25/22
27/25 + 11/10 + 9/8
27/25 + 10/9 + 10/9
12/11 + 12/11 + 9/8, "bayati-like" or "rast-like"
12/11 + 11/10 + 10/9, "equable diatonic"
11/10 + 11/10 + 11/10, "equal tetrachord"
Another wonderful thing about zarvo is that, since it's such an accurate temperament, many combination tones are usable, and indeed if you play with distortion you can form interesting difference tone melodies by shifting between different pitches in the same cluster.
I hope that this kind of explanation encourages people to view cluster temperaments zarvo as more than simply "unreasonably complex" temperaments with "bad" or "very uneven" MOS structure. Many people already report perceiving intervals as a generic category (usually 12edo-based) plus an intonational inflection, so temperaments that are inherently structured like that could be very natural.
If you want to experiment with zarvo temperament I would suggest simply using 65edo, but for those who love new scala files here is one:
! zarvo30.scl
!
TOP 2.3.5.11 zarvo[30]
30
!
17.28820
34.57639
149.26526
166.55346
183.84165
201.12985
315.81872
333.10692
350.39511
367.68331
384.97150
499.66037
516.94857
534.23677
551.52496
666.21383
683.50203
700.79022
718.07842
832.76729
850.05549
867.34368
884.63188
901.92007
1016.60894
1033.89714
1051.18534
1068.47353
1183.16240
1200.45060
Keenan
From: genewardsmith (2012-11-29)
Subject: Re: An interesting cluster temperament
--- In [email protected], "Keenan Pepper" <keenanpepper@...> wrote:
> That results in this 2.3.5.11 temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=7_65&limit=2_3_5_11 . I would give this temperament a name, but it already has the systematic names "2.3.5.11 marvo" and "2.3.5.11 zarvo" and I'm hesitant to contribute to the explosion of unique names. (The name "arvo" might be nice except it might be taken to imply a connection to Arvo Part...) So I'll just refer to it as "zarvo", with the understanding that in this context that refers to 2.3.5.11 zarvo.
If you just can't do without a 7, you can take two 2.3.5.11 zarvo chains separated at half an octave, and get harry. Harry is quite temperament if you go for higher complexity temperaments. You get harry if you add tempering by 441/440 to the 2.3.5.11 zarvo mix, and it has an excellent extension to the 13-limit after adding 351/350 or 676/675. Instead of 65edo, you'd have to use 130edo, though.
From: Chris Vaisvil (2012-11-29)
Subject: Re: [tuning] An interesting cluster temperament
Hi Keenan,
put in the queue, thank you!
Chris
>
> If you want to experiment with zarvo temperament I would suggest simply
> using 65edo, but for those who love new scala files here is one:
>
> ! zarvo30.scl
> !
> TOP 2.3.5.11 zarvo[30]
> 30
> !
> 17.28820
> 34.57639
> 149.26526
> 166.55346
> 183.84165
> 201.12985
> 315.81872
> 333.10692
> 350.39511
> 367.68331
> 384.97150
> 499.66037
> 516.94857
> 534.23677
> 551.52496
> 666.21383
> 683.50203
> 700.79022
> 718.07842
> 832.76729
> 850.05549
> 867.34368
> 884.63188
> 901.92007
> 1016.60894
> 1033.89714
> 1051.18534
> 1068.47353
> 1183.16240
> 1200.45060
>
> Keenan
>
>
>
From: Keenan Pepper (2012-11-29)
Subject: Re: An interesting cluster temperament
--- In [email protected], "genewardsmith" <genewardsmith@...> wrote:
> If you just can't do without a 7, you can take two 2.3.5.11 zarvo chains separated at half an octave, and get harry. Harry is quite temperament if you go for higher complexity temperaments. You get harry if you add tempering by 441/440 to the 2.3.5.11 zarvo mix, and it has an excellent extension to the 13-limit after adding 351/350 or 676/675. Instead of 65edo, you'd have to use 130edo, though.
Indeed. This means harry is a cluster temperament with 14 clusters. That's nuts. I can't imagine actually making music using harry temperament that way, but I'd love to hear some.
Keenan