User:Godtone: Difference between revisions

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== Examples of functions ==

* If [[5/4]] is taken as the generator and dominant, then [[25/16]] is a supertonic. Thus [[8/5]] is a serviant and [[32/25]] is a subtonic. if [[32/25]] is equated with [[9/7]] by tempering [[225/224]], then [[9/7]] is a subtonic and [[14/9]] is a supertonic. This forms a [[3L 7s]] scale.

* If [[5/4]] is taken as the generator and dominant, then [[25/16]] is a supertonic. Thus [[8/5]] is a serviant and [[32/25]] is a subtonic. if [[32/25]] is equated with [[9/7]] by tempering [[225/224]], then [[9/7]] is a subtonic and [[14/9]] is a supertonic. This forms a [[3L 7s|sephiroid (3L7s)]] scale.

* if [[13/8]] is taken as the generator and dominant, then [[169/128]] is a supertonic, which seems rather complex, but this can neatly be equated with [[21/16]] by tempering [[169/168]], which is convenient as we want to preserve the rootedness of the generator when stacking it. Thus [[16/13]] is a serviant and [[32/21]] is a subtonic. This also forms a [[3L 7s]] scale, but the generator is now above the antitonic rather than below.

* if [[13/8]] is taken as the generator and dominant, then [[169/128]] is a supertonic, which seems rather complex, but this can neatly be equated with [[21/16]] by tempering [[169/168]], which is convenient as we want to preserve the rootedness of the generator when stacking it. Thus [[16/13]] is a serviant and [[32/21]] is a subtonic. This also forms a [[3L 7s|sephiroid (3L7s)]] scale, but the generator is now above the antitonic rather than below.

* If [[3/2]] is taken as the generator and dominant, then [[9/8]] is a supertonic. Thus [[4/3]] is a serviant and [[16/9]] is a subtonic. This is a uniquely very low complexity instance.

* If [[3/2]] is taken as the generator and dominant, then [[9/8]] is a supertonic. Thus [[4/3]] is a serviant and [[16/9]] is a subtonic. This is a uniquely very low complexity instance. This forms a [[5L 2s|diatonic (5L2s)]] scale.

* If [[7/4]] is taken as the generator and dominant, then [[49/32]] is a supertonic. Thus [[8/7]] is a serviant and [[64/49]] is a subtonic.

* If [[7/4]] is taken as the generator and dominant, then [[49/32]] is a supertonic. Thus [[8/7]] is a serviant and [[64/49]] is a subtonic. This forms a [[5L 1s|machinoid (5L1s)]] scale which extends to a [[5L 6s]] chromatic/extended scale.

* If [[11/8]] is taken as the generator and dominant, then [[121/64]] is a supertonic. Thus [[16/11]] is a serviant and [[128/121]] is a subtonic. This is a peculiar case where the supertonic can reasonably be equated with the lead (and correspondingly the subtonic can reasonably be equated with the contralead). Given this, it doesn't seem necessary to introduce any temperings, as leads and contraleads are a more dissonant type of function.

* If [[11/8]] is taken as the generator and dominant, then [[121/64]] is a supertonic. Thus [[16/11]] is a serviant and [[128/121]] is a subtonic. This is a peculiar case where the supertonic can reasonably be equated with the lead (and correspondingly the subtonic can reasonably be equated with the contralead). Given this, it doesn't seem necessary to introduce any temperings, as leads and contraleads are a more dissonant type of function. This forms an [[2L 5s|antidiatonic]] scale which extends to a [[2L 7s]] and [[2L 9s]] chromatic/extended scale, or [[11L 2s]] for very accurately tuned [[11/8]]'s.

* If [[15/8]] is taken as the generator and dominant, then [[225/128]] is a supertonic. Thus [[16/15]] is a serviant and [[256/225]] is a subtonic. This is a peculiar case where the dominant can reasonably be equated with the lead (and correspondingly the serviant can reasonably be equated with the contralead). Furthermore, if we temper [[225/224]] then the supertonic is [[7/4]] and the subtonic is [[8/7]], meaning the supertonic and subtonic are noticeably more consonant and stable both harmonically and functionally than the dominant and serviant.

* If [[15/8]] is taken as the generator and dominant, then [[225/128]] is a supertonic. Thus [[16/15]] is a serviant and [[256/225]] is a subtonic. This is a peculiar case where the dominant can reasonably be equated with the lead (and correspondingly the serviant can reasonably be equated with the contralead). Furthermore, if we temper [[225/224]] then the supertonic is [[7/4]] and the subtonic is [[8/7]], meaning the supertonic and subtonic are noticeably more consonant and stable both harmonically and functionally than the dominant and serviant. Due to the close proximity of the generator to the tonic, the MOSSes formed by this get quite big before the generator changes from being the small step to being the large step, with [[1L 9s]] being the last MOSS where [[16/15]] is the small step, and the MOSS after that (assuming near-just tuning of [[16/15]]) is [[10L 1s]]. However, in the case of [[10L 1s]], the large and small steps are so close in size that you may want to use [[11edo]] as a simplified tuning. This does come at the cost of tempering [[225/224]] being suboptimal however.

== Intervals ==

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 == Examples of functions ==
-* If [[5/4]] is taken as the generator and dominant, then [[25/16]] is a supertonic. Thus [[8/5]] is a serviant and [[32/25]] is a subtonic. if [[32/25]] is equated with [[9/7]] by tempering [[225/224]], then [[9/7]] is a subtonic and [[14/9]] is a supertonic. This forms a [[3L 7s]] scale.
+* If [[5/4]] is taken as the generator and dominant, then [[25/16]] is a supertonic. Thus [[8/5]] is a serviant and [[32/25]] is a subtonic. if [[32/25]] is equated with [[9/7]] by tempering [[225/224]], then [[9/7]] is a subtonic and [[14/9]] is a supertonic. This forms a [[3L 7s|sephiroid (3L7s)]] scale.
-* if [[13/8]] is taken as the generator and dominant, then [[169/128]] is a supertonic, which seems rather complex, but this can neatly be equated with [[21/16]] by tempering [[169/168]], which is convenient as we want to preserve the rootedness of the generator when stacking it. Thus [[16/13]] is a serviant and [[32/21]] is a subtonic. This also forms a [[3L 7s]] scale, but the generator is now above the antitonic rather than below.
+* if [[13/8]] is taken as the generator and dominant, then [[169/128]] is a supertonic, which seems rather complex, but this can neatly be equated with [[21/16]] by tempering [[169/168]], which is convenient as we want to preserve the rootedness of the generator when stacking it. Thus [[16/13]] is a serviant and [[32/21]] is a subtonic. This also forms a [[3L 7s|sephiroid (3L7s)]] scale, but the generator is now above the antitonic rather than below.
-* If [[3/2]] is taken as the generator and dominant, then [[9/8]] is a supertonic. Thus [[4/3]] is a serviant and [[16/9]] is a subtonic. This is a uniquely very low complexity instance.
+* If [[3/2]] is taken as the generator and dominant, then [[9/8]] is a supertonic. Thus [[4/3]] is a serviant and [[16/9]] is a subtonic. This is a uniquely very low complexity instance. This forms a [[5L 2s|diatonic (5L2s)]] scale.
-* If [[7/4]] is taken as the generator and dominant, then [[49/32]] is a supertonic. Thus [[8/7]] is a serviant and [[64/49]] is a subtonic.
+* If [[7/4]] is taken as the generator and dominant, then [[49/32]] is a supertonic. Thus [[8/7]] is a serviant and [[64/49]] is a subtonic. This forms a [[5L 1s|machinoid (5L1s)]] scale which extends to a [[5L 6s]] chromatic/extended scale.
-* If [[11/8]] is taken as the generator and dominant, then [[121/64]] is a supertonic. Thus [[16/11]] is a serviant and [[128/121]] is a subtonic. This is a peculiar case where the supertonic can reasonably be equated with the lead (and correspondingly the subtonic can reasonably be equated with the contralead). Given this, it doesn't seem necessary to introduce any temperings, as leads and contraleads are a more dissonant type of function.
+* If [[11/8]] is taken as the generator and dominant, then [[121/64]] is a supertonic. Thus [[16/11]] is a serviant and [[128/121]] is a subtonic. This is a peculiar case where the supertonic can reasonably be equated with the lead (and correspondingly the subtonic can reasonably be equated with the contralead). Given this, it doesn't seem necessary to introduce any temperings, as leads and contraleads are a more dissonant type of function. This forms an [[2L 5s|antidiatonic]] scale which extends to a [[2L 7s]] and [[2L 9s]] chromatic/extended scale, or [[11L 2s]] for very accurately tuned [[11/8]]'s.
-* If [[15/8]] is taken as the generator and dominant, then [[225/128]] is a supertonic. Thus [[16/15]] is a serviant and [[256/225]] is a subtonic. This is a peculiar case where the dominant can reasonably be equated with the lead (and correspondingly the serviant can reasonably be equated with the contralead). Furthermore, if we temper [[225/224]] then the supertonic is [[7/4]] and the subtonic is [[8/7]], meaning the supertonic and subtonic are noticeably more consonant and stable both harmonically and functionally than the dominant and serviant.
+* If [[15/8]] is taken as the generator and dominant, then [[225/128]] is a supertonic. Thus [[16/15]] is a serviant and [[256/225]] is a subtonic. This is a peculiar case where the dominant can reasonably be equated with the lead (and correspondingly the serviant can reasonably be equated with the contralead). Furthermore, if we temper [[225/224]] then the supertonic is [[7/4]] and the subtonic is [[8/7]], meaning the supertonic and subtonic are noticeably more consonant and stable both harmonically and functionally than the dominant and serviant. Due to the close proximity of the generator to the tonic, the MOSSes formed by this get quite big before the generator changes from being the small step to being the large step, with [[1L 9s]] being the last MOSS where [[16/15]] is the small step, and the MOSS after that (assuming near-just tuning of [[16/15]]) is [[10L 1s]]. However, in the case of [[10L 1s]], the large and small steps are so close in size that you may want to use [[11edo]] as a simplified tuning. This does come at the cost of tempering [[225/224]] being suboptimal however.
 == Intervals ==