Yahoo Tuning Groups Ultimate Backup tuning-math the first six criteria

Scales that pass all of the first six criteria
----------------------------------------------

07- Diatonic scale in meantone
[0 5 10 15 18 23 28 31]
3. efficiency 0.76
4. strictly proper
5. yes; 5th is 3:2 in 6 of 7 modes
6. yes; 3rd is 6:5 or 5:4 in 7 of 7 modes

10- Paul Erlich's pentachordal decatonic in 22-tet
[0 2 4 7 9 11 13 16 18 20 22]
3. efficiency 0.62
4. strictly proper
5. yes; 7th is 3:2 in 8 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes

07- Dave Keenan's heptatonic MOS in porcupine
[0 3 6 9 12 15 18 22]
3. efficiency 0.76
4. strictly proper
5. yes; 5th is 3:2 in 4 of 7 modes
6. yes; 3rd is 6:5 or 5:4 in 7 of 7 modes

08- Gene Smith's Euclidean-reduced scale in 46-tet
[0 3 12 15 22 27 34 37 46]
3. efficiency 0.55
4. strictly proper
5. yes; 7th is 5:3 in 6 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 7 of 8 modes

10- Carl Lumma's x2 scale in meantone
[0 4 6 10 12 16 18 22 25 28 31]
3. efficiency 0.53
4. strictly proper
5. yes; 9th is 7:4 in 8 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 7 of 10 modes

10- Dave Keenan's decatonic MOS in quadrafourths
[0 3 6 9 11 14 17 20 23 26 29]
3. efficiency 0.73
4. strictly proper
5. yes; 7th is 3:2 in 6 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes

09- Paul Hahn's 4-3-3 trichordal scale in 31-tet
[0 4 7 10 15 18 21 25 28 31]
3. efficiency 0.49
4. strictly proper
5. yes; 4th is 5:4 in 6 of 9 modes
6. yes; 8th is 5:3, 12:7, 7:4 in 9 of 9 modes

Scales that pass most of the first six criteria
-----------------------------------------------

10- Paul Erlich's symmetrical decatonic in 22-tet
[0 2 4 7 9 11 13 15 18 20 22]
3. efficiency 0.51 (ambiguous key)
4. strictly proper
5. yes; 7th is 3:2 in 8 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes

10- Easley Blackwood's decatonic in 15-tet
[0 2 3 5 6 8 9 11 12 14 15]
3. efficiency 0.27 (ambiguous key)
4. strictly proper
5. yes; 7th is 3:2 in 10 of 10 modes
6. yes; 4th is 6:5 or 5:4 in 10 of 10 modes

08- octatonic scale in 12-tet
[0 1 3 4 6 7 9 10 12]
3. efficiency 0.33 (ambiguous key)
4. strictly proper
5. yes; 3rd is 6:5 in 8 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 8 of 8 modes

06- hexatonic scale in 12-tet
[0 1 4 5 8 9 12]
3. efficiency 0.42 (ambiguous key)
4. strictly proper
5. yes; 3rd is 5:4 in 6 of 6 modes
6. yes; 4th is 4:3 or 3:2 in 6 of 6 modes

09- David Rothenberg's generalized diatonic in 31-tet
[0 5 8 11 14 17 22 25 28 31]
3. efficiency 0.74
4. strictly proper
5. no, but 9th is 15:8 in 7 of 9 modes
6. yes; 8th is 5:3 or 7:4 in 9 of 9 modes

08- Carl Lumma's octatonic subset of kleismic
[0 1 5 6 10 11 15 16 19]
3. efficiency 0.75
4. proper, but R. stability only 0.36
5. yes; 3rd is 6:5 in 6 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 5 of 8 modes

09- Balzano's generalized diatonic in 20-tet
[0 2 5 7 9 11 14 16 18 20]
3. efficiency 0.74
4. strictly proper
5. no.
6. yes; 8th is 5:3 or 7:4 in 9 of 9 modes
9th is 9:5 or 15:8 in 9 of 9 modes

08- Ken Wauchope's minor
[1/1 21/20 7/6 5/4 7/5 3/2 5/3 7/4 2/1]
3. efficiency 0.42
4. strictly proper
5. no, but 6th is 3:2 in 4 of 8 modes
3rd is ~ 6:5 in 6 of 8 modes
6. yes; 4th is 5:4 or 4:3 in 7 of 8 modes

Scales that fail the first six criteria
---------------------------------------

06- Hexatonic scale in MAGIC
[0 11 13 24 26 39 41]
3. efficiency 0.78
4. proper, but R. stability only 0.40
5. yes; 3rd is 5:4 in 4 of 6 modes
6. no.

07- Neutral thirds scale in 31-tet
[0 5 9 13 18 22 27 31]
3. efficiency 0.76
4. strictly proper
5. yes; 5th is 3:2 in 5 of 7 modes
6. no.

08- Ken Wauchope's major
[1/1 21/20 5/4 21/16 7/5 3/2 7/4 15/8 2/1]
3. efficiency 0.42
4. not proper
5. no, but 6th is 3:2 in 5 of 8 modes
6. yes; 7th is 8:5, 7:4, 9:5 in 8 of 8 modes

09- Paul Hahn's 3-3-4 trichordal scale in 31-tet
[0 3 6 10 13 18 21 25 28 31]
3. efficiency 0.45
4. proper, R. stability 0.92
5. no, but 6th is 3:2 in 4 of 9 modes
6. yes; 8th is 5:3, 12:7, 7:4 in 9 of 9 modes

09- Nonatonic MOS in orwell
[0 11 19 30 38 49 57 68 76 84]
3. efficiency 0.74
4. strictly proper
5. yes; 3rd is 7:6 in 8 of 9 modes
6. no.

09- Nonatonic MOS in quadrafourths
[0 3 6 9 12 15 18 21 24 29]
3. efficiency 0.74
4. strictly proper
5. yes; 6th is 3:2 in 5 of 9 modes
6. no.

10- Decatonic MOS in MIRACLE
[0 7 14 21 28 35 42 49 56 63 72]
3. efficiency 0.73
4. strictly proper
5. yes; 9th is 7:4 in 8 of 10 modes
6. no.

-Carl

>> Scales that pass all of the first six criteria
>
>Hey, there are a lot of these!

I mainly searched scales that I thought would pass,
which I've been collecting for years.

>Can any of them be altered to have ambiguous intervals?

If you want.

>> 10- Decatonic MOS in MIRACLE
>> [0 7 14 21 28 35 42 49 56 63 72]
>> 3. efficiency 0.73
>> 4. strictly proper
>> 5. yes; 9th is 7:4 in 8 of 10 modes
>> 6. no.
>
>The "5th" could be either 7:5 or 10:7 for (6). That gives 4:7:10
>1/(4:5:7) as triads.

That's a 6th, I believe, but good catch! I don't normally consider
7:10 consonant, but with a 4: on the bottom, it is very consonant.
Just goes to show that the by-hand method I was using isn't very
good.

>It's (5) I'm worried about because you're overlooking 12:7.

Because the rule says that degree can have no other consonances?
Yes, that is a concern. I'm thinking of dropping that rule, though.

>I suppose it could make some sense to draw the cutoff
>between 10:7 and 12:7.

I think 12:7 can be consonant on a good day, and it can be very
consonant in the septimal minor sixth tetrad, though I believe it
will be able to function as a dissonance next to 7:4 in the above
case. And I'm still deciding how important this "characteristic
dissonance" like rule is.

-Carl

the first six criteria

🔗Carl Lumma <carl@lumma.org>

🔗graham@microtonal.co.uk

🔗Carl Lumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@juno.com>