7 to 10 note proper scales with lots of harmony

This is a range of scale sizes small enough to be heard as actual scales. When scales of this size are both proper and well-stocked with harmony (the diatonic scale being the most famous example) they are both harmonically and melodically interesting, or at least I think so. I've put up a large collection here:

http://xenharmonic.wikispaces.com/Scalesmith

These scales all employ approximation but JI proper scales in this range are an alternative which someone might want to catalog.

Thanks for sharing this! I think it's a great idea to document it, and I'm sure quite a massive pile of possibilities are just waiting to be tried out in this innovative method of scale creation... i will do my best to see what I can borrow from it.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> This is a range of scale sizes small enough to be heard as actual scales. When scales of this size are both proper and well-stocked with harmony (the diatonic scale being the most famous example) they are both harmonically and melodically interesting, or at least I think so. I've put up a large collection here:
>
> http://xenharmonic.wikispaces.com/Scalesmith
>
> These scales all employ approximation but JI proper scales in this range are an alternative which someone might want to catalog.
>

As an old JI regular I particularly appreciate the "transversal" traduction you gave lately with some of your tunings. Why not also for those ?
- - - - - - - -
Jacques

Gene wrote :
> This is a range of scale sizes small enough to be heard as actual > scales. When scales of this size are both proper and well-stocked > with harmony (the diatonic scale being the most famous example) > they are both harmonically and melodically interesting, or at least > I think so. I've put up a large collection here:
> http://xenharmonic.wikispaces.com/Scalesmith
> These scales all employ approximation but JI proper scales in this > range are an alternative which someone might want to catalog.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> As an old JI regular I particularly appreciate the "transversal"
> traduction you gave lately with some of your tunings. Why not also
> for those ?

I'm afraid these don't really have any, as they are not regular temperament constructions.

--- In tuning@yahoogroups.com, "dasdastri" <dasdastri@...> wrote:
>
>
> Thanks for sharing this! I think it's a great idea to document it, and I'm sure quite a massive pile of possibilities are just waiting to be tried out in this innovative method of scale creation... i will do my best to see what I can borrow from it.

The possibilties are indeed numbingly massive, but insisting on a lot of dyads near to just intervals cuts down on it a lot. This also helps understanding the structure.

The diatonic scale is the locus classicus for this kind of scale. It's a seven note scale, and since it is so well-supplied with harmony, it should not surprise us that the cycle of the 2/7 pitch class, the cycle of thirds, has all consonances. If M is the majr third and m is the minor third, this cycle goes MmMmmMm. This leads to a corresponding cycle of triads, where the mm triad is the diminished triad, an approximate 6/5-6/5-10/7 chord implying the 126/125 comma.

We can try the same kind of analysis on the seven-note scales in my collection, and a similar analysis for the other scales. Consider prop7a. This has a 2/7 cycle which goes MmmmmMf, where M is a slightly sharp major third, m a slightly flat minor third, and f a flat major third. The corresponding cycle of triads goes MdddMaa, where M is a major triad, d is a diminished triad, and a is an augmented triad.

Prop7b: The 2/7 cycle is MfMSMMS, where S is an approximate 8/7. The corresponding triads are augmented, augmented, 5/4-8/7-10/7, 8/7-5/4-10/7, 5/4-5/4-11/7, 5/4-8/7-10/7, 8/7-5/4-10/7.

Prop7c: The 2/7 cycle is MmMsMMs, where s is a subminor third, an approximate 7/6. The corresponding triads are major, minor, 5/4-7/6-16/11, 7/6-5/4-16/11, augmented, 5/4-7/6-16/11, 7/6-5/4-16/11.

Prop7d: The 2/7 cycle is MmzmMmM, where z is approximately 19/16 but functions more as 25/21. Triads are major, 6/5-z-10/7, z-6/5-10/7, minor, major, minor, augmented.

Prop7e: the 2/7 cycle is MmmMMSM, where S is approximately 8/7. The corresponding triads are major, diminished, minor, augmented, 5/4-8/7-10/7, 8/7-5/4-10/7, augmented.

Prop7f. This has a 2/7 cycle of MMSMMtM, where t is an approximate 15/13, M is a major third and S a supermajor second. The triads are augmented, 5/4-8/7-10/7, 8/7-5/4-10/7, augmented, 5/4-15/13-13/9, 15/13-5/4-13/9, augmented.

Prop7g. This has a 2/7 cycle consisting of MMSMMSM, where M is a major third and S a supermajor second: an approximate 8/7. The corresponding triads are simply what you get by assuming the intervals are just, and the whole cycle completes based on a 3136/3125 comma.

Prop7h: The 2/7 cycle is Mmnnnnm, where n is a neutral third. The triads are major, 6/5-11/9-19/13, neutral, neutral, neutral, 11/9-6/5-19/13, minor.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Prop7c: The 2/7 cycle is MmMsMMs, where s is a subminor third, an approximate 7/6. The corresponding triads are major, minor, 5/4-7/6-16/11, 7/6-5/4-16/11, augmented, 5/4-7/6-16/11, 7/6-5/4-16/11.

Here "augmented" really means 5/4-5/4-11/7. From this we get a 176/175 comma, from 5/4-7/6-16/11 we get 385/384, and from the whole cycle and the octave, 6144/6125. We can regularize by assuming all three commas are always in effect, giving zeus temperament:

http://xenharmonic.wikispaces.com/Porwell+family

> Prop7e: the 2/7 cycle is MmmMMSM, where S is approximately 8/7. The corresponding triads are major, diminished, minor, augmented, 5/4-8/7-10/7, 8/7-5/4-10/7, augmented.

Here "augmented" means the 5/4-5/4-14/9 chord implying 225/224. Diminished implies 126/125, and putting them together gives septimal meantone.

> Prop7f. This has a 2/7 cycle of MMSMMtM, where t is an approximate 15/13, M is a major third and S a supermajor second. The triads are augmented, 5/4-8/7-10/7, 8/7-5/4-10/7, augmented, 5/4-15/13-13/9, 15/13-5/4-13/9, augmented.

The two augmented chords are 5/4-5/4-14/9, implying 225/224. 5/4-15/13-13/9 implies 676/675, and the whole cycle implies 46875/46592. Putting it all together gives a temperament in the kleismic rank three family

http://xenharmonic.wikispaces.com/Kleismic+rank+three+family

which has normal comma list [15625/15552, 4375/4374, 325/324] and which probably deserves a writeup as catakleismic+ or something, since it's popping up giving proper seven note scales.

> Prop7g. This has a 2/7 cycle consisting of MMSMMSM, where M is a major third and S a supermajor second: an approximate 8/7. The corresponding triads are simply what you get by assuming the intervals are just, and the whole cycle completes based on a 3136/3125 comma.

This time "augmented" means an honest 5/4-5/4-25/16 chord, and we are looking at hemimean (3136/3125) temperament.