Deep Scales
Scale Deepness is a term coined by Carlton Gamer and Terry Winograd, identifying pitch class sets that have unique values for each interval in the interval vector. The Vector indicates how many common tones will be present in a scale when it is transposed by some distance in semitones. Deepness indicates that there is a unique distance available for any desired quantity of those common tones, and that there is a clear hierarchy of distance from one rotation to the next. Gamer called this "hierarchization". It basically means the scale is fit for building chord progressions. Winograd established the mathematical foundations for deepness, intending that it would be used as an indicator to find useful musical material in tuning systems other than 12-TET.
| Forte Class | Prime | Other transformations | Interval Vector |
|---|---|---|---|
| 6-1 |  | 126, 252, 504, 1008, 2016, 2079, 3087, 3591, 3843, 3969, 4032 | <5, 4, 3, 2, 1, 0> |
| 6-32 |  | 1197, 1323, 1386, 1449, 1701, 2394, 2646, 2709, 2772, 2898, 3402 | <1, 4, 3, 2, 5, 0> |
| 7-1 |  | 254, 508, 1016, 2032, 2111, 3103, 3599, 3847, 3971, 4033, 4064 | <6, 5, 4, 3, 2, 1> |
| 7-35 |  | 1451, 1453, 1709, 1717, 2741, 2773, 2774, 2902, 2906, 3418, 3434 | <2, 5, 4, 3, 6, 1> |
Citations
- Carlton Gamer. "Some Combinational Resources of Equal-Tempered Systems." In: Journal of Music Theory 11.1 (1967), pp. 32–59
- Carlton Gamer. "Deep Scales and Difference Sets In Equal Tempered Systems." Proceedings of the 2nd Annual Conference of the American Society of University Composers (1967), pp. 113–122
- Terry Winograd. "An Analysis of the Properties of Deep Scales in a T-Tone System." May 1966 (unpublished)