Topic: Some scale analysis
4 scales
| File | Description | Notes | Period (ยข) |
|---|---|---|---|
| euler20 | 3^5 5^4 Euler genus tempered by 225/224-planar | 20 | 1200.0 |
| euler24 | 3^6 5^4 Euler genus tempered by 225/224-planar | 24 | 1200.0 |
| quasi-phillips | Quasi-Phillips 19 note scale | 19 | 1200.0 |
| schis24 | 24 tone schismic temperament in its 94-et incarnation | 24 | 1200.0 |
Thread (18 messages)
From: Gene Ward Smith (2002-10-25) Subject: Some scale analysis If one looks at the 5-limit relationships of the Phillips scale, it can be associated with the scale obtained by adding three notes to the 16-note Euler genus 3^4 5^4, where we take the chain of fifths at the bottom of the chain of thirds, and add two fifths at one end and a fourth at the other. Tempering this b 225/224-planar produces a scale related to Phillips I've called "quasi-phillips", which has some advantages. Here it is: ! quasi-phillips.scl Quasi-Phillips 19 note scale 19 ! 83.67670365 152.1394612 199.6830428 268.1458003 384.1521395 399.3660856 467.8288431 500.1584786 583.8351823 652.2979398 699.8415214 768.3042789 851.9809826 899.5245642 967.9873217 1083.993661 1099.207607 1152.456418 2/1 If we set a margin of accuracy of 2.5 cents, then phillips has 4 major tetrads, 3 minor tetrads, 7 major triads and 7 minor triads. Quasi-phillips has 4 major tetrads, 5 minor tetrads, 10 major triads and 9 minor triads. Raising the error bar to 5 gives phillips two additional major terrads, an additional minor tetrad, and two aditional major triads. Scales we might compare to are the 20 and 24 note Euler genuses for 3^5 5^4 and 3^6 5^4, and the 24 note schismic "2MOS". ! euler20.scl 3^5 5^4 Euler genus tempered by 225/224-planar 20 ! 83.67670365 152.1394612 199.6830428 268.1458003 351.8225040 384.1521395 399.3660856 467.8288431 583.8351823 652.2979398 699.8415214 768.3042789 783.5182250 851.9809826 899.5245642 967.9873217 1083.993661 1152.456418 1167.670365 2/1 This has 6 major and 6 minor tetrads and 12 major and 12 minor tetrads within 2.5 cents. ! euler24.scl 3^6 5^4 Euler genus tempered by 225/224-planar 24 ! 83.67670365 152.1394612 199.6830428 268.1458003 283.3597464 351.8225040 384.1521395 399.3660856 467.8288431 583.8351823 652.2979398 667.5118859 699.8415214 768.3042789 783.5182250 851.9809826 899.5245642 967.9873217 1051.664025 1083.993661 1099.207607 1152.456418 1167.670365 2/1 This has 8 major and 8 minor tetrads and 15 major and 15 minor tetrads within 2.5 cents. ! schis24.scl ! ! [0, 2, 7, 9, 14, 16, 23, 25, 30, 32, 39, 41, ! 46, 48, 53, 55, 62, 64, 69, 71, 78, 80, 85, 87] 24 tone schismic temperament in its 94-et incarnation 24 ! 25.53191489 89.36170213 114.8936170 178.7234043 204.2553191 293.6170213 319.1489362 382.9787234 408.5106383 497.8723404 523.4042553 587.2340426 612.7659574 676.5957447 702.1276596 791.4893617 817.0212766 880.8510638 906.3829787 995.7446809 1021.276596 1085.106383 1110.638298 2/1 This has 9 major and 9 minor tetrads within 5 cents, and 15 major and 15 minor triads within 5 cents (actually, within 3.6 cents with this tuning.) It also has a long chain of 23 consectutive fifths.
From: Gene Ward Smith (2002-10-25) Subject: Re: Some scale analysis --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: > This has 6 major and 6 minor tetrads and 12 major and 12 minor tetrads within 2.5 cents. 12 major and 12 minor *triads*. > This has 8 major and 8 minor tetrads and 15 major and 15 minor tetrads within 2.5 cents. 15 major and 15 minor triads.
From: [email protected] (2002-10-25) Subject: Re: [tuning] Some scale analysis In a message dated 10/25/02 1:35:38 PM Central Daylight Time, [email protected] writes: > Tempering this b 225/224-planar produces a scale related to Phillips I've > called "quasi-phillips", which has some advantages. > I fail to see any advantages of this scale over my Phillips JT scale. The tuning in my scale is generally closer to the theoretical values . Pauline
From: Gene Ward Smith (2002-10-25) Subject: Re: Some scale analysis --- In tuning@y..., prophecyspirit@a... wrote: > In a message dated 10/25/02 1:35:38 PM Central Daylight Time, > genewardsmith@j... writes: > > > > Tempering this b 225/224-planar produces a scale related to Phillips I've > > called "quasi-phillips", which has some advantages. > > > I fail to see any advantages of this scale over my Phillips JT scale. The > tuning in my scale is generally closer to the theoretical values . It has within the limits of 225/224-planar temperament (within 2.5 cents, say) one less major tetrad, two fewer minor tetrads, one less major triad, and three less minor triads. It is also in a little better tune in general, fitting everything in an error bar of 2.2 cents. Those seem to me to be advantages. If you mean your scale is generally closer to JI, it isn't.
From: wallyesterpaulrus (2002-10-25) Subject: Re: Some scale analysis --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote: > --- In tuning@y..., prophecyspirit@a... wrote: > > In a message dated 10/25/02 1:35:38 PM Central Daylight Time, > > genewardsmith@j... writes: > > > > > > > Tempering this b 225/224-planar produces a scale related to Phillips I've > > > called "quasi-phillips", which has some advantages. > > > > > I fail to see any advantages of this scale over my Phillips JT scale. The > > tuning in my scale is generally closer to the theoretical values . > > It has within the limits of 225/224-planar temperament (within 2.5 >cents, say) one less major tetrad, two fewer minor tetrads, one less >major triad, and three less minor triads. It is also in a little >better tune in general, fitting everything in an error bar of 2.2 >cents. Those seem to me to be advantages. If you mean your scale is >generally closer to JI, it isn't. so pauline's scale isn't in 225/224-planar after all? where does it depart? perhaps pauline is only shooting for a subset of the chords you're finding, and is including some other chords as well, including something involving a ratio of 17, if i recall correctly . . .
From: [email protected] (2002-10-25) Subject: Re: [tuning] Re: Some scale analysis In a message dated 10/25/02 5:01:28 PM Central Daylight Time, [email protected] writes: > If you mean your scale is generally closer to JI, it isn't. > > My Phillips JT scale has intervals approx only 2 cents or less off the theoretical values in relation to each other. When you take into account the 5th are - 2 cents short, the other intervals are right on target with each other with no noticeable beats while playing. And I've played hundreds of pieces with my tuning, and know every chord I provide notes for is needed to accurately play music composed for tempered scales, whether MT, WM or 12-ET. And my scale allows playing in septimal minor. What other scale does? Pauline
From: [email protected] (2002-10-25) Subject: Re: [tuning] Re: Some scale analysis In a message dated 10/25/02 6:08:14 PM Central Daylight Time, [email protected] writes: > , including > something involving a ratio of 17, if i recall correctly . . . > I've used 30/17 worth 983 cnets as a large minor 7th for E and B. As more tone generators are needed for the 7/4 values worth 155 cents for them than I have right now. But I plan to eventually use the regular values for these 7ths. Pauline
From: wallyesterpaulrus (2002-10-25) Subject: Re: Some scale analysis --- In tuning@y..., prophecyspirit@a... wrote: > And my scale allows playing in septimal minor. What other scale >does? gene's tetrads were septimal ones. i reckon there's at least as much "septimal minor" in his variation as in your original scale.
From: Carl Lumma (2002-10-26) Subject: Re: Some scale analysis >>It has within the limits of 225/224-planar temperament (within >>2.5 cents, say) one less major tetrad, two fewer minor tetrads, >>one less major triad, and three less minor triads. It is also in >>a little better tune in general, fitting everything in an error >>bar of 2.2 cents. Those seem to me to be advantages. If you mean >>your scale is generally closer to JI, it isn't. > >so pauline's scale isn't in 225/224-planar after all? where does >it depart? perhaps pauline is only shooting for a subset of the >chords you're finding, and is including some other chords as well, >including something involving a ratio of 17, if i recall >correctly . . . I think Gene switched "it"s on us mid-stream, there. First, he's talking about Pauline's scale, then his. No? Also, did anybody see that Gene wrote 171-tet, while Pauline wrote 17-tet? I'm with you, Pauline, on the issue of comma shifts. -Carl
From: Gene Ward Smith (2002-10-26) Subject: Re: Some scale analysis --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: > so pauline's scale isn't in 225/224-planar after all? where does it > depart? perhaps pauline is only shooting for a subset of the chords > you're finding, and is including some other chords as well, including > something involving a ratio of 17, if i recall correctly . . . Pauline's scale has 7-limit intervals tempered by .3587, .6863, .8259, 1.1291, 1.4878, 1.955, 2.174, 2.314, 2.641, 4.129, and 4.488 cents--which ones are a part of the system?
From: Gene Ward Smith (2002-10-26) Subject: Re: Some scale analysis --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: > --- In tuning@y..., prophecyspirit@a... wrote: > > > And my scale allows playing in septimal minor. What other scale > >does? > > gene's tetrads were septimal ones. i reckon there's at least as > much "septimal minor" in his variation as in your original scale. Five vs three tetrads and nine vs six triads if you confine yourself to errors under 2.5 cents. It would be nice to have a list of which chords are essential to this business in Pauline's understanding.
From: [email protected] (2002-10-26) Subject: Re: [tuning] Re: Some scale analysis In a message dated 10/25/02 10:51:28 PM Central Daylight Time, [email protected] writes: > I'm with you, Pauline, on the issue of comma shifts. > > -Carl > While the difference in cents betwee 386 (384) and 408 (400) looks large in print, the difference in pitch is very small in midrange and below. Pauline
From: [email protected] (2002-10-26) Subject: Re: [tuning] Re: Some scale analysis In a message dated 10/26/02 5:26:30 AM Central Daylight Time, [email protected] writes: > Pauline's scale has 7-limit intervals tempered by 4.129, and 4.488 cents-- None of my inervals are tempered that much. Becaue the tempered 5ths reduce the apparent tempering. For example, 9/7 worth 435 cents is 432 cents in my JT scale. and just as smooth as the other. Some intervals can be tempered a great deal and sill be smooth. Pauiine
From: [email protected] (2002-10-26) Subject: Re: [tuning] Re: Some scale analysis In a message dated 10/26/02 5:33:39 AM Central Daylight Time, [email protected] writes: > Five vs three tetrads and nine vs six triads if you confine yourself to > errors under 2.5 cents. It would be nice to have a list of which chords are > essential to this business in Pauline's understanding. > My C scale has 7c, 7d, 7 e, 7f, 7g, and 7a minor triads. How nany are needed depends on how a composition is written. Pauline
From: Gene Ward Smith (2002-10-26) Subject: Re: Some scale analysis --- In tuning@y..., prophecyspirit@a... wrote: > In a message dated 10/26/02 5:26:30 AM Central Daylight Time, > genewardsmith@j... writes: > > > > Pauline's scale has 7-limit intervals tempered by 4.129, and 4.488 cents-- > > None of my inervals are tempered that much. They are there if you want to use them, at any rate.
From: Gene Ward Smith (2002-10-26) Subject: Re: Some scale analysis --- In tuning@y..., prophecyspirit@a... wrote: > In a message dated 10/26/02 5:33:39 AM Central Daylight Time, > genewardsmith@j... writes: > > > > Five vs three tetrads and nine vs six triads if you confine yourself to > > errors under 2.5 cents. It would be nice to have a list of which chords are > > essential to this business in Pauline's understanding. > > > My C scale has 7c, 7d, 7 e, 7f, 7g, and 7a minor triads. That's six, all right. What are the major triads? Major and minor tetrads?
From: [email protected] (2002-10-27) Subject: Re: [tuning] Re: Some scale analysis In a message dated 10/26/02 7:27:23 PM Central Daylight Time, [email protected] writes: > My C scale has 7c, 7d, 7e, 7f, 7g, and 7a minor triads. > > What are the major triads? Major and minor tetrads? Major = C, D, E, F#, G, A, B minor = d, d#, e, f #, g#, a, a#, b . The F major and a tertian tminor triad are in the F scale. Pauline
From: Gene Ward Smith (2002-10-28) Subject: Re: Some scale analysis --- In tuning@y..., prophecyspirit@a... wrote: > In a message dated 10/26/02 7:27:23 PM Central Daylight Time, > genewardsmith@j... writes: > > > > My C scale has 7c, 7d, 7e, 7f, 7g, and 7a minor triads. > > > > What are the major triads? Major and minor tetrads? > > Major = C, D, E, F#, G, A, B > > minor = d, d#, e, f #, g#, a, a#, b . The F major and a tertian tminor triad > are in the F scale. I'm having trouble sorting this out. Could you give chains of fifths for your C scale for major and minor triads, and *also* for major and minor tetrads--meaning things intended to play a version of 1--5/4--3/2--7/4 (major tetrad) and 1--6/5--3/2--12/7 (minor tetrad.) You've given a chain for the minor triads, and I presume the next chain is for major triads. What about tetrads? To what does minor = d, d#, e, f #, g#, a, a#, b refer?